Equilibrium States on Toeplitz Algebras

Transcription

1 Equilibrium States on Toeplitz Algebras Zahra Afsar a thesis submitted for the degree of Doctor of Philosophy at the University of Otago, Dunedin, New Zealand. 10 July 2015

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3 Abstract This thesis describes the equilibrium states the KMS states of dynamical systems arising from local homeomorphisms. It has two main components. First, we consider a local homeomorphism on a compact space and the associated Hilbert bimodule. This Hilbert bimodule has both a Toeplitz algebra and a Cuntz-Pimsner algebra, which is a quotient of the Toeplitz algebra. Both algebras carry natural gauge actions of the circle, and hence one can obtain natural dynamics by lifting these actions to actions of the real numbers. We study KMS states of these dynamics at, above, and below a certain critical value. For inverse temperature larger than the critical value, we find a large simplex of KMS states on the Toeplitz algebra. For the Cuntz-Pimsner algebra the KMS states all have inverse temperatures below the critical value. Our results for the Cuntz-Pimsner algebra overlap with recent work of Thomsen, but our proofs are quite different. At the critical value, we build a KMS state of the Toeplitz algebra which factors through the Cuntz-Pimsner algebra. To understand KMS states below the critical value, we study the backward shift on the infinite path space of an ordinary directed graph. Merging our results for the Cuntz-Pimsner algebra of shifts with the recent work about KMS states of the graph algebras, we show that Thomsen s bounds on of the possible inverse temperature of KMS states are sharp. In the second component, we consider a family of -commuting local homeomorphisms on a compact space, and build a compactly aligned product system of Hilbert bimodules in the sense of Fowler. This product system also has two interesting algebras, the Nica-Toeplitz algebra and the Cuntz-Pimsner algebra. For these algebras the gauge action is an action of iii

4 a higher-dimensional torus, and there are many possible dynamics obtained by composing with different embeddings of the real line in the torus. We use the techniques from the first component of the thesis to study the KMS states for these dynamics. For large inverse temperature, we describe the simplex of the KMS states on the Nica-Toeplitz algebra. To study KMS states for smaller inverse temperature, we consider a preferred dynamics for which there is a single critical inverse temperature, which we can normalise to be 1. We then find a KMS 1 state for the Nica-Toeplitz algebra which factors through the Cuntz-Pimsner algebra. We then illustrate our results by considering different backward shifts on the infinite path space of some higher-rank graphs. iv

5 Acknowledgements The work presented in this thesis is the result of three years of research as a doctoral candidate at the Department of Mathematics and Statistics at the University of Otago. Throughout my research project many people contributed, but I would like to thank some of them in particular. Foremost, I want to thank my supervisors Professor Astrid an Huef and Professor Iain Raeburn for their dedicated guidance during my PhD research. They remind me, both consciously and unconsciously, the beauty of mathematics. I appreciate all their contributions that made my Ph.D. experience productive and enjoyable. I learned so much from our long and frequent meetings and their advice on my research project and my career have been invaluable. Through these years not only I took the advantages of their ideas on my thesis but also it was a unique opportunity to improve my writing via their corrective feedbacks. I am also thankful to my co supervisor Dr Lisa Orloff Clark for the excellent example she has provided as a successful mathematician woman. I owe special thanks to the members of the operator algebras group who have contributed significantly to my time at the university of Otago. The social cohesion within the group is truly unique and has been a source of friendships as well as collaboration. I am especially grateful for the weekly lunch with this group and for inspirational and fun conversations with them. I would like to especially thank all of them for helping to create a great atmosphere: Professor John Clark, Dr Sooran Kang, Richard McNamara, Ilija Tolich, Danie van Wyk and Yosafat Pangalela. They gave me expert advice and share with me both professional and personal tips. v

6 I have been lucky to have been given the chance to experience long-standing friendships with people in the science building. It was a real pleasure to discuss science as well as life in the common room of science III building. These discussions were the perfect mini-breaks away from my Ph.D. I could fill a couple of pages acknowledging every single one of them, but I just want to say a special thanks to Madhuri Kumari, Mounica Balvoĉiūtȯ, Paula Bran, Leon Escobar Dias, Dr Harish Sankaranarayanan, Adrian Pen and his lovely partner Janet Paola Cruz. Thanks for their inspiration to continue my directions and for being there when I needed advice. I want to wish them good luck for the rest of their lives. I would also like to acknowledge the science building administrative assistants Lenette Grant, Leanne Kirk and Marguerite Hunter. Who kept us organized and were always ready to help. I really enjoyed working with them. I gratefully acknowledge the funding sources that made my Ph.D. work possible. It was funded by University of Otago for three years. Also thanks the staff for their warm and involved way of administrative works. I have never regretted the decision that I chose Otago. My time at Otago was especially enjoyable in large part due to the many friends and groups that became a part of my life in Abbey College. We always had fun with heaps of social events. I would like to thank all of them for creating such amazing memories. Special thanks to Gretchen Kivell, John Seaton and Dr Charles Tustin whose awesome hospitality contributed in my enjoyable life in the beautiful New Zealand. Lastly, I would like to thank my family for all their love and encouragement. Of course, my parents Esmaeil Afsar and Fezze Fathi. I owe them special thanks for completing this journey. They taught me hard work, persistence and being independent. There are no words that can express my appreciation for all they have done and been for me. Also, my loving, supportive and encouraging partner, Mehdi. I could not finish this journey without his faithful support during the all the stages of this Ph.D. Our friendship began in Abbey College and evolved into a deep loving relationship during our PhDs. vi

7 Contents 1 Preliminaries Hilbert bimodules Internal tensor products of Hilbert bimodules Product systems of Hilbert bimodules C -algebras associated to product systems of Hilbert bimodules The Fock representation Topological graphs Measures A product system associated to a family of local homeomorphisms Notations Building a product system from local homeomorphisms The gauge action KMS states on the C -algebras of product systems associated to - commuting local homeomorphisms KMS states commuting local homeomorphisms A characterization of KMS states KMS states and subinvariance relation KMS states at large inverse temperatures KMS states at the critical inverse temperature Ground states and KMS states The shifts on the infinite path spaces of 1-coaligned higher rank graphs Basics of Higher-rank graphs C -algebras associated to higher rank graphs coaligned higher rank graphs and the associated C -algebras KMS states on the Toeplitz algebras References 91 A Realising the universal Nica-covariant representation as a doubly commuting representation 97 vii

8 B KMS states on C -algebras associated to a local homeomorphism 101 viii

9 Introduction Given an action α of the real line R by automorphisms of a C -algebra A, the C - dynamical system A, R, α provides an algebraic model for studying a physical system in quantum statistical physics [5]. In this framework, the observables are the selfadjoint elements of the C -algebra A, the states are positive linear functionals on A with norm 1, and the time evolution is given by the action α. Work of Kubo, Martin and Schwinger shows that equilibrium states of the physical system are exactly those states on A which satisfy a certain commutation relation the so called KMS condition. This relation involves a real number β, which is interpreted as the inverse temperature of the physical system. The KMS condition makes sense for abstract dynamical systems and operator algebraists study KMS states of dynamical systems regardless of applications in physics. Many authors have studied KMS states in different contexts. For example: in systems constructed from number theory [4, 32, 33, 34], in systems associated to graph algebras [12, 15, 28, 29], in systems arising from groupoids [31, 39], and in topological systems built from local homeomorphisms [56, 57]. In most of the contexts mentioned above, there are two main C -algebras: a Cuntz- Pimsner type algebra and its Toeplitz extension. There has been profound progress in characterising KMS states of Cuntz-Pimsner algebras in the literature [12, 13, 43, 56], and interesting work of Exel, Laca and Neshveyev [15, 34] shows that Toeplitz algebras are expected to have a much greater supply of KMS states. This thesis focuses on characterising KMS states on Toeplitz algebras associated to local homeomorphisms. It is organised in two main parts. The first part is allocated to dynamical systems arising from a single local homeomorphism and their KMS states. The result of this part is published in [1] and here we provided it as an Appendix chapter see Appendix A. In the second part, we study KMS states of dynamical systems associated to a family of local homeomorphisms in the context of product systems of Hilbert bimodules. This part occupies the main body of this thesis. 1

10 The notion of a product system was initially introduced by Arveson as a continuous product system of Hilbert spaces [2]. Then several authors generalised this to discrete product systems in [11, 20, 22]. We follow Fowler s extension [20] which is about discrete product systems of Hilbert bimodules over semigroups [20]. Roughly speaking, for a semigroup P with identity e, a product system of Hilbert bimodules over P is a semigroup X = p P X p such that each X p is a right Hilbert bimodule and x y xy implements an isomorphism from X p X q onto X pq for all p, q P \ {e}. For such a product system X, Fowler defined Toeplitz representations of X as multiplicative maps whose restriction on each fibre X p is a Toeplitz representation in the sense of [21]. Then he associated the Toeplitz algebra T X as the universal algebra for Toeplitz representations of X. He defined the Cuntz-Pimsner algebra OX as a quotient of T X. When G, P is a quasi-lattice ordered group in the sense of Nica [40], he imposed a covariance condition Nica-covariance on Toeplitz representations, and defined the Nica-Toeplitz algebra N T X 1 as the universal algebra for Nica-covariant Toeplitz representations. He noticed that N T X is only tractable for certain class of product systems called compactly aligned product systems. For such a product system, he showed that 1 N T X = span{ψ p xψ q y : p, q P, x X p, y X q, } where ψ is the universal Nica-covariant representation. Viewing N k as an additive semigroup, there are many interesting examples for the product systems over N k in the literature. For these examples, by universal properties of N T X, and OX, respectively we can get strongly continuous gauge actions of k-torus T k on these algebras. Then we can lift these actions to the actions of the real line via the embedding t e itr = e itr 1, e itr 2,..., e itr k for some r 0, k. Well known examples of product systems over N k are the ones constructed from the higher-rank graph of Kumjian-Pask [30]. It is observed in [22, page 1492] that we can view a k-graph Λ as a product system over N k. Soon after Sims and Raeburn showed that by putting particular combinatorial condition on the underlying higherranks graph we can get a compactly aligned product system over the quasi-lattice ordered group Z k, N k [45]. They imposed a Nica-covariance condition by adding an extra relation to the usual Cuntz-Krieger relations. They called the associated Nica-Toeplitz algebras the Cuntz-Krieger-Toeplitz algebra T C Λ. The Cuntz-Krieger C Λ can be viewed as a quotient of T C Λ. Thus the C -algebras of higher-rank 1 In Fowler s paper the Nica-Toeplitz algebra is denoted by T cov X. 2

11 graphs and their KMS states can be a rich supply of test examples for analysing KMS states of product systems. In particular there has been recently great progress in analysing the KMS structure of these dynamics for example [26, 28]. There are also intriguing examples for Nica-Toeplitz algebras in number theory, for example, the Toeplitz algebra T N N studied by Laca and Raeburn in [35]. It is observed in [7] that T N N and the associated additive and multiplicative quotients are all Nica-Toeplitz algebras. Then the KMS structure of these algebras is analysed by applying the technique developed in [35]. Following the same approach, Hong, Larsen and Szymański characterized the KMS structure of a product system over a general semigroup [24]. But the authors of [24] used the strong condition finite type product system in their hypothesis. This condition requires the existence of a finite orthonormal basis for all fibres in the product system. In [53, 54], Solel used different notation to study the product systems over N k. He used the term c.c. completely contractive covariant representation for Fowler s Toeplitz representation see [54, Defnition 2.3, Definition 3.1] and defined the doubly commuting relation [54, Defnition 3.8]. He showed in [54, Lemma 3.11] that this relation is equivalent to Fowler s Nica-covariance relation and that the universal Nicacovariant representation ψ satisfies his doubly commuting relation. Here we are interested in the dynamical systems arising from local homeomorphisms. We first show that a family of surjective and commuting local homeomorphisms h 1,..., h k on a compact Hausdorff space Z induces a compactly aligned product system X over N k see Chapter 2. Letting h m := h m 1 1 h m k k, each fibre X m in this product system is the graph correspondence associated to the topological graph Z, Z, id, h m. We know very well from our work in [1] what each fibre looks like. So we think about generalizing the results of [1] from one Hilbert bimodule to a product system of Hilbert bimodules. Our approach is inspired by [28] which is again a refinement of original technique introduced in [34]. So we first look for a characterization of KMS states of N T X which makes it easier to recognise the KMS states. To do this, having looked at similar results in the literature for example [28, Proposition 3.1] and [24, Theorem 4.6], we noticed that it is crucial to express elements of the form ψ n y ψ m x in terms of usual spanning elements ψ p sψ q t in the algebra N T X. For a general product system over a semigroup, Fowler provided an approximation [20, Proposition 5.10], but this is not enough because we need an exact formula; in the dynamics associated to a higher-rank graph [28] this formula already exists as one of the Toeplitz-Cuntz-Krieger 3

12 relations; in [24], since each fibre in the product system has an orthonormal basis, it is easier to find such a formula see [24, Lemma 4.7]. To solve this problem, we impose an extra hypothesis of -commutativity on the local homeomorphisms. Two maps f, g : Z Z, -commute if for every z, z Z such that fz = gz, there exists unique z Z such that z = gz and z = fz see [3]. Recently, there have been great interest in studying C -algebras of -commuting maps and associated dynamics [16, 37, 55]. The -commutativity hypothesis allows us to find Parseval frames for each fibre. Given m N k, since the fibre X m is the graph correspondence associated to the local homeomorphisms h m, there is a well-known Parseval frame {τ i } d i=0 for X m which comes from a partition of unity [17, Proposition 8.2]. We observed that for n N k with m n = 0, the composition of elements of this Parseval frame with h n form another Parseval frame for X m. Then we prove ψ n y ψ m x =, 2 ψ m y, τj h m τ i ψn x, τi h n τ j 0 i,j d getting the formula we need. This formula is for fibres X m and X n with m n = 0. However by using proper isomorphisms between fibres we can apply 2 and rewrite ψ n y ψ m x in terms of elements of the for ψ p sψ q t for general m, n N k. Then we use the formula 2 and provide a characterization of KMS states in Proposition In fact the equation 2 is a translation of Solel s doubly commuting relation from his notation to Fowler s notation. The difficulty of this translation is that the doubly commuting relation contains a flip map between fibres. Notice that the existence of such a flip map is a consequence of definition of the product system. Solel used the doubly commuting relation in his approach without any explicit formula for the flip map. We find a nice formula for this flip map in Lemma 3.1.1c and therefore we can translate the doubly commuting relation to get 2 see Appendix A. Let Λ be a k-graph and A i 1 i k be the associated vertex matrices. The vectors that are subinvariant for all A i in the sense of Perron-Frobenius theory [50], play a very important role in analysing KMS states of T C Λ. For dynamics determined by r 0, k, we follow the same idea and define a subinvariance relation using a family of Ruelle operators. When β is large enough, that is β > β c for β c := max{r 1 j β ci } and β ci := lim sup j 1 ln max i z Z h j i z, we describe all solutions of our subinvariance relation in Proposition If in addition r has rationally independent coordinates, we show that there is a bijection between 4 j

13 the simplex of KMS β states on N T X and the probability measures satisfying our subinvariance relation Theorem A rational independency condition on r is crucial when we prove the surjectivity of our isomorphism in Theorem So whenever we need to get a probability measure satisfying the subinvariance relation from a KMS sates we have to impose this hypothesis. To study KMS states for smaller β, in order to have satisfactory results, we pay careful attention in choosing r 0, k. Following recent conventions in graph algebras [26, 28, 59, 60], we consider a preferred dynamics where r := β c1,..., β ck. Notice that in this case β c = 1. We call β c = 1 the critical inverse temperature. At the critical inverse temperature, we show that by taking limits of KMS βj states as the β j decrease to 1, there is a KMS 1 state on N T X, and at least one such a state factors through OX Theorem Finally, we provide an example of -commuting maps. Let Λ be a 1-coaligned k- graph in the sense that for each pair of paths µ, ν with the same source there is a unique pair of paths ξ, η such that ξµ = ην. It is observed in [37, Theorem 2.3] that the shift maps on the infinite path space of Λ -commute. Now writing XΛ for the associated product system, we apply our result in the previous chapters to study the KMS structure of the associated Nica-Toeplitz algebra N T XΛ and the Cuntz- Pimsner algebra OXΛ. We first prove that, as we expect from our results for a 1-graph, the Cuntz-Pimsner algebra OXΛ is isomorphic to the Cuntz-Krieger algebra C Λ. We also prove that the Nica-Toeplitz algebra N T XΛ contains an injective copy of T C Λ Proposition Furthermore, we prove that every KMS state of T C Λ is the restriction of a KMS state of N T XΛ Proposition Thesis outline This thesis is broken up to 4 chapters and 2 appendices: In Chapter 1, we provide an overview of product systems of Hilbert bimodules and the associated dynamical systems. We present the basic definitions and notation and discuss the properties of these dynamical systems in details. In Chapter 2, we show that a family of commuting and surjective local homeomorphisms gives a compactly aligned product system of Hilbert bimodules. Chapter 3 allocated to characterising KMS states and ground states of dynamical systems arising from a family of -commuting and surjective local homeomorphisms. In Chapter 4, we discuss the shifts on the infinite path space of 1-coaligned higer-rank graphs. We show the relationships between the 5

14 KMS states of graph algebras and the KMS states of the C -algebras of the shifts. In appendix A, we reconcile our results with those of Solel s. We show that for the dynamical system considered in Chapter 3, the universal Nica-covariant representation satisfies Solel s doubly commuting relation. Finally, we attach our published paper [1] as Appendix B. This appendix presents our results about the KMS states of dynamical systems associated to a single local homeomorphism. 6

15 Chapter 1 Preliminaries 1.1 Hilbert bimodules The following definitions are taken from chapter 2 of [46]. Given a complex vector space X and a C -algebra A, by a right action of A on X we mean a pairing x, a x a : X A X satisfying the consistency conditions: x + x a = x a + x a; x aa = x a a and λx a = λx a = x λa for all λ C, x, x X and a, a A. Definition Let A be a C -algebra and X be a complex vector space with a right action of A on X. A right A-valued inner product on X is a function, A : X X A which is linear in the second variable and satisfies: a x, y a A = x, y A a, b x, y A = y, x A, c x, y A is a positive element of A, and d x, x A = 0 implies that x = 0. We may write x, y for x, y A if it is clear from the context which C -algebra A is meant. Remark Since, A is linear in second variable, we deduce that x = 0 implies x, x A = 0. It also follows from condition b that, A is conjugate linear in the first variable. 7

16 It follows from [46, Corollary 2.7] that the formula x A := x, x A 1 2 defines a norm on X. If X is complete in this norm we call it a right Hilbert A-module. Suppose X is a right Hilbert A-module. An operator T : X X is adjointable, if there is an operator T : X X such that T x, y = x, T y for all x, y X. A A We denote by LX the set of all adjointable operators on X. It follows from [46, Lemma 2.18] that every adjointable operator T on a right Hilbert A-module X is a linear bounded operator. [46, Proposition 2.21] says that the adjoint T is unique and the set LX is a C -algebra with respect to the operator norm, and with the involution given by T T. Given x, y X, we define Θ x,y : X X by Θ x,y z = x y, z A. Then Θ x,y is adjointable and Θ x,y = Θ y,x see [46, page 18]. The set KX := span{θ x,y : x, y X}. is a C -algebra and we call it the algebra of compact operators on X. Definition Let A be a C -algebra. A right Hilbert A A bimodule X or a correspondence over A is a right Hilbert A-module X together with a homomorphism ϕ : A LX. We view ϕ as implementing a left action of A on X and we usually write a x for ϕax. We say X is essential if X = span{ϕax : a A, x X}. Remark Since ϕa LX for all a A, it follows that a x, y A = x, a y A. Now let x, y X and a, a A. The statements b and c of Definition imply that a x a, y A = x a, a y A = a y, x a A = a y, x A a = y, a x A a = y, a x a A = a x a, y A. Thus a x a = a x a and the actions of A on X are compatible. Example Let A be a C -algebra. The multiplication in A gives a right action of A on itself. The formula a, a A = a a defines a right A-valued inner product on A. To see this, first note that it is linear in the second variable. 2nd, conditions a c of Definition are immediate. Third, to check d, let a, a A = a a = 0. It follows that aa = a 2 = 0. This implies a = 0. Thus a, a A = a a is a right A-valued inner product on A. Since a A = a, A is complete in the norm A and therefore 8

17 is a Hilbert A-module. Next define ϕ : A LA by ϕaa = aa. To see that ϕ is adjointable, observe that ϕaa, a A = aa a = a a a = a, a a A = a, ϕa a A. Thus ϕa is adjointable and ϕa = ϕa. Clearly ϕ is a homomorphism. Thus A is a right Hilbert A A bimodule which we call the standard bimodule and denote by AA A. Example Let A be a unital C -algebra with identity I A and suppose X is a right Hilbert A A bimodule. If ϕi A x = x for all x X, then X is essential. Definition Let A be a C -algebra and X be a right Hilbert A A bimodule. A representation ψ, π of X in a C -algebra B consists of a linear map ψ : X B and a homomorphism π : A B such that ψa x b = πaψxπb and π x, y A = ψx ψy. for every x, y X and a, b A. Remark A representation ψ, π induces a homomorphism ψ, π 1 : KX T X such that ψ, π 1 Θ x,y = ψxψy see page 202 of [42]. Definition Suppose X is a right Hilbert A A bimodule. Following [17, 23], we refer to a sequence {x i } d i=0 in X such that d 1.1 x i x i, x A = x for all x X. i=0 as a finite Parseval frame for X. The formula 1.1 is known as the reconstruction formula. 1.2 Internal tensor products of Hilbert bimodules In this section, we show how we can define the internal tensor product X A Y for right Hilbert A A bimodules X, Y. We also show that X A Y has a right Hilbert A A bimodule structure. We write X Y for the algebraic tensor product of X and Y. We use X A Y for the quotient of X Y by the subspace 1.2 N := span{x a y x a y : x X, y Y, a A.} To avoid possible confusion, we temporary write x y for the elements of X Y and x A y for the elements X A Y. Then by definition each x A y has the form x y +N. 9

18 Lemma Let A be a C -algebra and let X, Y be two right Hilbert A A bimodules. Then there is a well defined right action x A y, a x A y a : X A Y A X A Y such that x A y a = x A y a for all x A y X A Y, a A. Proof. Fix a A. The map x, y x A y a is a bilinear map from X Y into X A Y. Then the universal property of X Y gives us a linear map L a : X Y X A Y satisfying L a x y = x y a. Since L a vanishes on N, it induces a linear map L a : X A Y X A Y such that L a x A y = x A y a. Now x A y, a L a x A y is a well defined map from X A Y A into X A Y. Write x A y a := L a x A y. To see that this map is a right action, let x A y, x A y X A Y and a, a A. Since L a is linear, it follows that x y + x y a = L a x y + x y = L a x y + L a x y = x y a + x y a. We also have λx y a = L a λx y = λ L a x y = λx y a. A similar calculation shows x y λa = λx y a. Finally, we have x y aa = L aa x y = x y aa = x y a a = L a L a x y = x y a a, as required. The next lemma shows that we can equip the space X A Y with a right A-valued inner product. Proposition [36, Proposition 4.5]. Let A be a C -algebra and let X, Y be two right Hilbert A A bimodules. Suppose that ϕ Y : A LY is the homomorphism which defines the left action of A on Y. Then there is a unique right A-valued inner product on X A Y such that x A y, z A w = 1.3 y, ϕ Y x, z w for x A y, z A w X A Y. 10

19 Let X A Y be the completion of X A Y with respect to the inner product 1.3. It then follows from [46, Lemma 2.16] that 1.3 is a right A-valued inner product on X A Y as well. Thus X A Y is a right Hilbert A-module. The next lemma shows that we can define a left action of A on X A Y. Proposition [58, Proposition I.1]. Let A be a C -algebra and let X, Y be two right Hilbert A A bimodules. Suppose that ϕ Y : A LY is the homomorphism which defines the left action of A on Y. Then for every S LX, there is a unique operator S 1 Y LX A Y such that 1.4 S 1 Y x y = Sx y for x y X A Y. The map S S 1 Y is a homomorphism of LX into LX A Y. In particular the map a ϕ X a 1 Y determines a homomorphism of A into LX A Y. We can view the homomorphism a ϕ X a 1 Y as a left action of A on X A Y. Thus X A Y is a right Hilbert A A bimodule. We call X A Y the balanced tensor product of right Hilbert A A bimodules X, Y. For convenience, in the rest of thesis we keep x y for the elements of X Y and we write x y for the elements of both X A Y and X A Y Product systems of Hilbert bimodules We use conventions of [20] for the basics of product systems of Hilbert bimodules. For convenience, we use the following equivalent formulation from [52, page 6]. Definition Suppose P is a multiplicative semigroup with identity e, and let A be a C -algebra. For each p P let X p be a right Hilbert A A bimodule and suppose that ϕ p : A LX p is the homomorphism which defines the left action of A on X p. A product system over P of right Hilbert A A bimodules or a product system over P with fibres X p is the disjoint union X := p P X p such that: P1 The identity fibre X e equals the standard bimodule A A A. P2 X is a semigroup and for each p, q P \{e} the map x, y xy : X p X q X pq, extends to an isomorphism σ p,q : X p A X q X pq. P3 The multiplications X e X p X p and X p X e X p satisfy ax = ϕ p az, xa = x a for a X e and x X p. 11

20 If each fibre X p is essential, then we call X a product system over P of essential right Hilbert A A bimodules. Let p, q P \ {e} and S LX p. Then the isomorphism σ p,q : X p A X q X pq together with Proposition give us a homomorphism ι pq p by ι pq p S = σ p,q S 1 Xq σ 1 p,q. : LX p LX pq defined Definition Suppose P is a subsemigroup of a group G such that P P 1 = {e}. Then p q p 1 q P defines a partial order on G. Following [40], we say G, P is a quasi-lattice ordered group if for any two elements p, q G which have a common upper bound in P there is a least upper bound p q P. We write p q = when p, q G have no common upper bound. Example Z k, N k is a quasi-lattice ordered group. Observe that for all m, n N k, there is a least upper bound m n with ith coordinate m n i := max{m i, n i }. Definition Let G, P be a quasi-lattice ordered group. A product system over P of right Hilbert A A bimodules is compactly aligned, if for all p, q P with p q <, S KX p and T KX q, we have ιp p q Sι p q q T KX p q. Proposition [20, Proposition 5.8 ]. Let G, P be a quasi-lattice ordered group and suppose that X is a compactly aligned product system over P of right Hilbert A A bimodules. Suppose that the left action of A on each fibre X p is by compact operators. Then X is compactly aligned. 1.3 C -algebras associated to product systems of Hilbert bimodules Definition Let P be a multiplicative semigroup with identity e, and let X be a product system over P of right Hilbert A A bimodules. Let B be a C -algebra, and let ψ be a function from X to B. Write ψ p for the restriction of ψ to X p. We call ψ a Toeplitz representation of X if: T1 For each p P \{e}, ψ p : X p B is linear, and ψ e : A B is a homomorphism, T2 ψ p x ψ p y = ψ e x, y for p P, and x, y X p, T3 ψ pq xy = ψ p xψ q y for p, q P, x X p, and y X q. 12

21 Remark Conditions T1 and T2 imply that ψ p, ψ e is a Toeplitz representation for the fibre X p which is a right Hilbert A A bimodule. Then Remark gives us a homomorphism ψ p : KX p B such that ψ p Θ x,y = ψ p xψ p y. Fowler showed in [20, Proposition 2.8] that there exists a C -algebra T X and a Toeplitz representation ω of X in T X such that: U1 For any other Toeplitz representation T of X in a C -algebra B, there exists a unique homomorphism T : T X B such that T ω = T, and U2 T X is generated by {ωx : x X}. It then follows that the pair T X, ω is unique up to canonical isomorphism. We say the pair T X, ω is universal for the Toeplitz representations. The C -algebra T X, is called the Toeplitz algebra of X and the representation ω is known as the universal Toeplitz representation of X. We keep ω for the universal Toeplitz representation of X. Definition Let P be a semigroup with identity e, and let X be a product system over P of right Hilbert A A bimodules. A Toeplitz representation ψ of X is Cuntz-Pimsner-covariant if 1.5 ψ e a = ψ p ϕ p a for all p P, a ϕ 1 p KX p. The Cuntz-Pimsner algebra OX is the quotient of T X by the ideal { } 1.6 ωa ω p ϕ p a : p P, a ϕ 1 p KX p. Let q O : T X OX be the quotient map. It is observed in [20, Proposition 2.9] that q O ω is a Cuntz-Pimsner-covariant representation of X in OX. Moreover the pair OX, q O ω is universal for the Cuntz-Pimsner-covariant representations of X. Let G, P be a quasi-lattice ordered group and suppose that X is a product system of essential right Hilbert A A bimodules over P. Suppose that ψ is Toeplitz representation of X on a Hilbert space H. It follows from [20, Proposition 4.1] that there is a unique action α ψ : P End ψ e A such that 1.7 α ψ p T ψ p x = ψ p xt for all T ψ e A, x X p, and 1.8 α ψ p 1 p r = 0 for r ψ p X p H, where 1 p is the identity operator on X p. 13

22 Lemma Let G, P be a quasi-lattice ordered group and X be a product system of essential right Hilbert A A bimodules over P. Suppose that ψ is a Toeplitz representation on a Hilbert space H. Let p P and suppose that {x i } d i=0 is a Parseval frame for the fibre X p. Let αp ψ 1 p be as in 1.8. Then α ψ p 1 p = d ψ p x i ψ p x i. i=0 Proof. By uniqueness in [20, Proposition 4.1], it suffices to prove that 1.9 d ψ p x i ψ p x i ψ p x = ψ p x for all x X p, and i= d ψ p x i ψ p x i r = 0 for all r ψ p X p H = 0. i=0 To see 1.9, let x X p. We compute by applying the reconstruction formula for x: d d d ψ p x i ψ p x i ψx = ψ p x i ψ p x i ψ x j x j, x i=0 i=0 = 0 i,j d = 0 i,j d = 0 i,j d j=0 ψ p x i ψ p x i ψx j ψ 0 x j, x ψ p x i ψ 0 x i, x j ψ 0 x j, x ψ p x i x i, x j ψ 0 x j, x. using T3 using T2 Rearranging this and two applications of the reconstruction formula give d ψ p x i ψ p x i ψx = i=0 This is precisely 1.9. d d ψ p x i x i, x j ψ 0 x j, x = j=0 = i=0 d ψ p x j x j, x = ψ p x. j=0 To check 1.10, fix r ψ p X p H. Notice that for all r H we have d ψ p x j ψ 0 x j, x j=0 d ψ p x i ψ p x i r r = i=0 d r ψ p x i ψ p x i r = 0. i=0 It then follows d i=0 ψ px i ψ p x i r = 0 and we have proven

23 Definition Let G, P be a quasi-lattice ordered group and suppose that X is a product system of essential right Hilbert A A bimodules over P. Suppose that ψ is Toeplitz representation of X on a Hilbert space H. We say ψ is Nica-covariant if for every p, q P, we have α αp ψ 1 p αq ψ p ψ 1 p q if p q < 1 q = 0 otherwise. Fowler showed in [20, Proposition 5.6] that the Nica-covariance condition can be expressed in terms of compact operators. Then for the class of compactly-aligned product systems, he extended the Nica-covariance condition for the representations over C -algebras. Definition Let G, P be a quasi-lattice ordered group and suppose that X is a compactly aligned product system over P of right Hilbert A A bimodules. Toeplitz representation ψ of X is Nica-covariant if for every p, q P, S KX p, and T KX q, we have ψ p q ι ψ p Sψ q p p q Sι p q q T if p q < T = 0 otherwise. It follows from [20, Theorem 6.3] that there exists a C -algebra N T X and a Nica-covariant representation ψ of X in N T X such that N T X, ψ is universal for the Nica-covariant representations of X. Moreover, we have A 1.11 N T X = span{ψ p xψ q y : p, q P, x X p, y X q }. The C -algebra N T X, is called the Nica-Toeplitz algebra of X. Throughout we will keep ψ for the universal Nica-covariant representation of X. The next lemma shows that N T X is a quotient of T X. Lemma Let G, P be a quasi-lattice ordered group, and let X be a compactly aligned product system over P of right Hilbert A A bimodules. Suppose J is the ideal in T X such that 1.12 J := {ker θ : θ is Nica-covariant representation of X}, and let q N T : T X T X/J be the quotient map. Then T X/J, q N T ω is universal for Nica-covariant representation, and is canonically isomorphic to N T X, ψ. 15

24 Proof. Since q N T is a homomorphism and ω satisfies T1 T3, it follows that q N T ω satisfies T1 T3 as well. To see that q N T ω is Nica-covariant, let p, q P, S KX p, and T KX q. Notice that q N T ω p = q N T ω p. Then 1.13 q N T ω p Sq N T ω q T = q N T ω p Sω q T. If p q <, since ω p Sω q T ω p q ι p q p q N T ω p Sω q T = q N T Now 1.13 implies that Sιq p q T J, it follows that ω p q ι p q p Sι p q q T. q N T ω p Sq N T ω q T = q N T ω p q ιp p q Sι p q q T. Similarly, for p q =, we have q N T ω p Sω q T = 0. Putting this in 1.13 gives q N T ω p Sq N T ω q T = 0. Thus q N T ω is a Nica-covariant representation. Since {ωx : x X} generates T X, we have that {qωx : x X} generates T X/J. To see U1, suppose that T is another Nica-covariant representation of X in a C -algebra B. Notice that T is in particular a Toeplitz representation of X. Then the universal property of pair T X, ω gives a unique homomorphism T : T X B such that T ω = T. Notice that T vanishes on J because by definition J ker T. Thus there is a homomorphism T : T X/J B such that T q N T ω = T. The Cuntz-Pimsner algebra OX is by definition a quotient of T X. Since we are interested in studying the C -algebra N T X, it would be very helpful to explain OX as a quotient of N T X. The next lemma shows that, under some assumptions, we can express OX as a quotient of N T X. Lemma Let G, P be a quasi-lattice ordered group, and let X be a compactly aligned product system over P of right Hilbert A A bimodules. Suppose that every Cuntz-Pimsner-covariant representation of X is a Nica-covariant representation. Then OX is the quotient of N T X by the ideal generated by 1.14 Proof. Let 1.15 { } ψ e a ψ p ϕ p a : p P, a ϕ 1 p KX p. I := { } ker π : π is a Cuntz-Pimsner-covariant representation of X. 16

25 Following the same argument of Lemma 1.3.7, we can view OX as the quotient of T X by the ideal I. Let J and q N T be the ideal and the quotient map as in Lemma It then follows q N T b = b + J for all b T X and q N T ω = ψ. Since every Cuntz-Pimsner-covariant representation of X is also a Nica-covariant representation, it follows that J I. An application of the third isomorphism theorem in algebra gives us a quotient map q : N T X OX such that ker q = I/J. We now have 1.16 I/J = { i + J : i I } = { q N T i : i I }. An argument in set theory shows that I is the same as the ideal 1.6. Now using elements 1.6 in 1.16 and applying q N T ω = ψ, we have I/J = ψ e a ψ p ϕ p a : p P, a ϕ 1 p KX p. Thus we can consider OX as the quotient of N T X, by the ideal ψ e a ψ p ϕ p a : p P, a ϕ 1 p KX p. Proposition [20, Proposition 5.4]. Let G, P be a quasi-lattice ordered group such that every p, q P have a common upper bound. Let X be a compactly aligned product system over P of right Hilbert A A bimodules. Suppose that each fibre X p is essential and the left action of A on X p is by compact operators. Then every Toeplitz representation of X which is Cuntz-Pimsner-covariant is also Nica-covariant. Remark Let G, P be a quasi-lattice ordered group and X be a compactly aligned product system over P of right Hilbert A A bimodules. In [52, Proposition 3.12], Sims and Yeend defined their Cuntz-Pimsner algebra N OX as a quotient of N T X. In general N OX and OX are different. But we can deduce from [52, Remark 3.14, Proposition 5.1] that if a each pair p, q in P, has an upper bound and automatically a least upper bound, b for each p P the homomorphism ϕ p : A LX p is injective, and c the Cuntz-Pimsner-covariance 1.5 implies the Nica-covariance, then the two C -algebras N OX and OX coincide. In our set-up these conditions are satisfied see Remark But we found it easier to work with OX and the quotient map mentioned in Lemma

26 1.4 The Fock representation We take the definition of Fock representation from [20, page 340]. Let P be a semigroup with identity e and suppose X is a product system over P of right Hilbert A A bimodules. Define r : X P by rx := p for x X p. Let p P X p be the subset of p P X p consisting of all elements x p such that p P x p, x p A converges in norm. Write x p for elements of p P X p. It follows from [20, page 340] that p P X p is a right Hilbert A A bimodule with the right action given by x p a := x p a, the inner product by x p, y p := p P x p, y p, and the left action by the map ϕ p : A LF X defined by ϕ p x p = ϕ p x p for x p F X. We write F X := p P X p and call it the Fock module. Fowler shows in [20, page 340] that for x X there is an adjointable operator T x such that T x x p = xx p for x p F X. The adjoint T x is zero on any summand X p for which p / rxp. When p rxp, there is an isomorphism σ rx,p rx : X rx A X p rx X p, and the adjoint T x is determined by the formula 1.17 T x σ rx,p rx y z = x, y z. He also shows that T is a Toeplitz representation of X and calls it the Fock representation. Remark Let X be a compactly aligned product system over N k of right Hilbert A A bimodules and suppose the left action of A on each fibre is by compact operators. Then the homomorphism T : N T X LF X induced from the Fock representation is faithful see [24, Remark 4.8]. 1.5 Topological graphs A topological graph E = E 0, E 1, r, s consists of two locally compact Hausdorff spaces, a continuous map r : E 1 E 0 and a local homeomorphism s : E 1 E 0. The map r is called the range map and s is called the source map. Given such a graph, let 18

27 A := C 0 E 0. It is observed in [44, Chapter 9] that there is a right action of A on C c E 1 and there is a well-defined right A-valued inner product on C c E 1 such that x az = xzasz, and x, y A z = xwyw. sw=z It follows that the completion XE is a right Hilbert A-module. The formula a xz := arzxz, defines an action of A by adjointable operators on XE see [44, Chapter 9]. Then XE becomes a right Hilbert A A bimodule. We call XE the graph correspondence associated to the topological graph E. In topological graphs of interest to us, the spaces E 0 and E 1 are always compact. Then C c E 1 = CE 1. Since s is a local homeomorphism on the compact space E 1, D := max z E 0 s 1 z <. Now we have x 2 A = sup xwxw x 2 supd. z E 0 sw=z On the other hand, since E 0 is compact, x sup = xz 0 for some z 0 E 0. Then x sup = xz 0 2 xwxw sup xwxw = x 2 A. sw=sz 0 z E 0 sw=z Thus the norm A on XE is equivalent as a vector-space norm to the supremum norm on CE 1. Thus there is no completion required here and it makes sense to write XE = CE 1. Example Let Z be a locally compact Hausdorff space and id : Z Z be the identity map on Z. Let E be the topological graph Z, Z, id, id. Then XE = CZ = A. The actions of A on XE are by pointwise multiplication which are the same as the actions in A A A. Notice that x, y z = xwyw = xzyz. idw=z This is precisely the inner product in the standard bimodule A A A. Thus XE = A A A. 1.6 Measures All the measures we consider here are positive in the sense that they take values in [0,. We write MZ + for the set of finite regular Borel measures on Z. For us, a probability measure is a Borel measure with total mass 1. 19

28

29 Chapter 2 A product system associated to a family of local homeomorphisms In this chapter we show that a family of surjective and commuting local homeomorphisms h 1,..., h k on a compact Hausdorff space Z induces a compactly aligned product system of Hilbert bimodules over N k. We also prove that the C -algebras of product systems of Hilbert bimodules over N k carry gauge actions of T k Notations We consider N k as a monoid under addition with identity 0. We write N k + for the nonzero elements of N k. We use e 1,..., e k for the standard generators and write n i for i-th coordinate of n. We denote for the partial order in N k defined by m n if and only if m i n i for all 1 i k. We write m n for the coordinate-wise maximum of m and n in the sense that m n i := max{m i, n i }. Similarly we denote by m n the coordinate-wise minimum of m and n. Let h 1,..., h k be surjective and commuting local homeomorphisms on a compact Hausdorff space Z. Then for m N k we write h m := h m 1 1 h m k k and h m := h m 1 1 h m k 1. k 2.1 Building a product system from local homeomorphisms In [1, Lemma 5.2] we proved that for a local homeomorphism f and the associated graph correspondence XE, there is an isomorphism from XE 2 onto the graph 21

30 correspondence associated to f f. The next lemma generalizes this to graph correspondences of two different local homeomorphisms. There is also a similar result in the dynamics arising from graph algebras see [6, Proposition 2.2]. Lemma Let f, g be surjective local homeomorphisms on a compact Hausdorff space Z. Let A := CZ and suppose XE 1, XE 2 and XF are the graph correspondences related to topological graphs E 1 = Z, Z, id, f, E 2 = Z, Z, id, g, and F = Z, Z, id, g f. Then there is an isomorphism σ f,g from XE 1 A XE 2 onto XF such that 2.1 σ f,g x yz = xzyfz for all z Z. Proof. Define the map σ : CZ CZ CZ by 2.2 σx, yz = xzyfz for all x, y CZ. We first show that σ is bilinear and onto. Take c, c C and x, x, y, y CZ. Then σ cx + c x, y z = cx + c x zyfz = cxzyfz + c x zyfz = cσx, yz + c σx, yz. Similarly we have σ x, cy + c y = cσx, y + c σx, y. So σ is bilinear. Taking y = 1 in 2.2 implies that σ is surjective. Now the universal property of the algebraic tensor product gives a unique surjective linear map σ : CZ CZ CZ satisfying σx yz = xzyfz for all x y CZ CZ. Since σ vanishes on the element of the form 1.2, we can extend it to a surjective linear map σ f,g : CZ A CZ CZ such that σ f,g x yz = xzyfz for all x y CZ A CZ. Next we show that σ f,g preserves the actions and the inner products. Let x y CZ A CZ, a CZ and z Z. To check that σ f,g preserves the right action, we have σ f,g x y a z = xzy afz = xzyfzag fz = σ f,g x yzag fz = σ f,g x y a z. Similarly for the left action, we have σ f,g a x y z = a xzyfz 22

31 = azxzyfz = azσ f,g x yz = a σ f,g x y z. To see that σ f,g preserves the inner products, take x y, x y CZ A CZ. Then remembering that the range functions are identity and the source functions are f, g, we have σ f,g x y, σ f,g x y z = 2.3 g fw=z = g fw=z = gv=z σ f,g x ywσ f,g x y w xwyfwx wy fw [ fw=v = x, x vyvy v gv=z = gv=z xwx w ] yfwy fw yv x, x y v = y, x, x y z = x y, x y z. Next a quick calculation shows that σ f,g is an isometry. Take a typical element v = d i=0 x i y i CZ A CZ. We have σ f,g v 2 = σ f,g v, σ f,g v d = σ f,g x i y i, i=0 = 0 i,j d = 0 i,j d d = x i y i, i=0 = v 2. d j=0 σ f,g x j y j σ f,g x i y i, σ f,g x j y j x i y i, x j y j d x j y j j=0 by 2.3 Thus σ f,g is an isometry on CZ A CZ, and then it extends to an isomorphism σ f,g of XE 1 A XE 2 onto XF which satisfies

32 Corollary Let h 1,..., h k be surjective and commuting local homeomorphisms on a compact Hausdorff space Z. For each m N k, let X m be the graph correspondence associated to the topological graph Z, Z, id, h m. Suppose X := m N X k m and A := CZ. Let σ m,n : X m A X n X m+n be the isomorphism obtained by applying Lemma with the local homeomorphisms h m, h n. Then X is a compactly aligned product system over N k of essential right Hilbert A A bimodules with the multiplication given by 2.4 xy := σ m,n x y for x X m, y Y n, that is xyz = xzyh m z for all z Z. Furthermore, the left action of A on each fibre X m is by compact operators. Proof. To see that X is a semigroup, let m, n, p N k and take x X m, x X n and x X p. Then by applying the definition of multiplication, we have A similar computation shows xx x z = σ m+n,p σm,n x x x z = σ m,n x x zx h m+n z = xzx h m zx h m+n z. xx x z = σ m,n+p x σn,p x x z = xzσ n,p x x h m z = xzx h m zx h m+n z. Thus xx x = xx x and X is a semigroup. Next we check conditions P1 P3 of the Definition P1 follows from Example which says that X e = A A A. P2 is immediate by definition of X. To check P3, let a A and x X m. Then axz = σ 0,m a xz = azxz = a xz, similarly xaz = σ m,0 x az = xzah m z = x az. To see that the fibre X m is essential, notice that A = CZ is unital with the identity I CZ : Z C defined by I CZ z = 1 for all z Z. Since the left action is by pointwise multiplication, ϕ m I CZ x = x for all x X m. Thus X m is essential. 24

33 To prove that the left action of A on the fibre X m is by compact operators, let {U j } d j=0 be an open cover of Z such that h m Uj is injective. Choose a partition of unity {ρ j } subordinate to {U j } and define ξ j := ρ j. We claim that for each a A, 2.5 ϕ m a = d Θ a ξj,ξ j. j=0 Take x X m and z Z, we compute the right-hand side of 2.5 d j=0 Θ a ξj,ξ j x z = Since h m is injective on each supp ξ j, d j=0 = = d j=0 a ξ j ξ j, x z d a ξ j z ξ j, x h m z j=0 d azξ j z j=0 Θ a ξj,ξ j x z = h m w=h m z d azξ j zξ j zxz j=0 = azxz = azxz, d ξ j z 2 which is equal to the left-hand side of 2.5, as we required. j=0 ξ j wxw. Finally, it follows from [20, Proposition 5.8]that X is a compactly aligned product system. Remark Let h 1,..., h k be surjective and commuting local homeomorphisms on a compact Hausdorff space Z and let X be the associated product system as in Corollary We aim to show that the two Cuntz-Pimsner algebra N OX and OX coincide. We check the conditions a b of Remark Condition a is clear because each pair in N k has an upper bound. To prove b, notice that for each m N k the homomorphism ϕ m : A LX m is injective. To see this, let ϕ m a = ϕ m a for a, a A. Let I CZ be the identity in CZ. Then ϕ m ai CZ = ϕ m a I CZ. It follows that az = a z for all z Z and therefore a = a. To check c, notice that X is a compactly aligned product system of essential Hilbert A A bimodule and the left action is by compact operators. Then 25

34 Proposition implies that every Cuntz-Pimsner covariant representation is a Nicacovariant representation. Thus we have checked all conditions Remark , and hence the two Cuntz-Pimsner algebra N OX and OX coincide. 2.2 The gauge action By a strongly continuous action of a locally compact group G on a C -algebra A, we mean a homomorphism g α g : G AutA such that g α g a is continuous for each fixed a A. It is well known that the Nica-Toeplitz algebra of a product system over N k of right Hilbert A A bimodules carries an action of the k-torus T k. But we could not find an explicit reference for this. The next lemma shows this fact. Lemma Let A be a C -algebra and X be a compactly aligned product system over N k of right Hilbert A A bimodules. Then there is a strongly continuous action γ : T k AutN T X, called the gauge action, such that γ z ψ n x = z n ψ n x for all n N k, z T k, x X n. Proof. Fix z T k and define θ : X N T X by θ n x = z n ψ n x for n N k, x X n. We claim that θ is a Toeplitz representation of X. To see this, we check the conditions T1 T3 of Definition That θ is a Toeplitz representation follows because ψ is. Each θ n is linear and θ e is a homomorphism. We have θ n x θ m y = z n ψ n x z n ψ n y = ψ 0 x, y = θ 0 x, y. and θ n xθ m y = z n+m ψ n xψ m y = z n+m ψ n+m xy = θ n+m xy. Thus conditions T1 T3 of Definition are satisfied. To see that it is Nica-covariant, we consider θ n : KX n N T X. For x, x X n, we have θ n Θ x,x = θ n xθ n x = z n ψ n xz n ψ n x = ψ n Θ x,x. Thus θ n S = ψ n S for all S KX. 26

35 Now let S KX n, T KX m. Since ψ is Nica-covariant, we have θ n Sθ m T = ψ n Sψ m T = ψ m n ιn m n Sι m n m T = θ n m ιn m n Sι m n m T. Now it follows from the universal property of N T X that there is a homomorphism γ z : N T X N T X such that θ = γ z ψ. This gives an explicit formula for γ z on the generators of N T X: 2.6 γ z ψ n = θ n = z n ψ n. Notice that γ z γ z ψ n x = γ z γ z ψ n x = ψ n x. But the universal property of N T X implies that the identity map on N T X is the only homomorphism with this property. It then follows γ z 1 = γ z and hence γ z AutN T X. Next let I T k be the identity element in T k. Then γ ψ IT k nx = I T k n ψ n x = ψ n x. Then γ IT k is the identity map on N T X. Finally, for z, w Tk, we have γ z γ w ψ n = zw n ψ n = γ zw ψ n. Thus γ is a homomorphism of T k into the AutN T X. To see that γ is strongly continuous, we must prove that z γ z b is continuous for all b N T X. Fix ɛ > 0 and b. There is a linear combination c of generators in N T X such that b c < ɛ 3. Equation 2.6 implies that, z γ zc is continuous. Then there exists some δ > 0 such that z w < δ γ w c γ z c < ɛ. Now for 3 z w < δ we have as we require. γ w b γ z b γ w b c + γ w c γ z c + γ z b c < ɛ, Remark Let q : N T X OX be the quotient map as in Lemma Since the gauge action on N T X fixes the kernel of q, it then induces a natural gauge action γ of T k on the quotient OX. 27

36

37 Chapter 3 KMS states on the C -algebras of product systems associated to -commuting local homeomorphisms In this chapter we consider a family of -commuting local homeomorphisms and the associated product system as in Corollary We study KMS states and ground states on the C -algebras of this product system. Our object here is to generalize the results in [1] to our product system. When we have only one local homeomorphism, the results here except those associated to ground states reduce to those in [1] KMS states A C -algebraic dynamical system is a triple A, R, α consisting of a C -algebra A, the real line R and an action α : R AutA. Given such a C -algebraic dynamical system, we say an element a of A is analytic if t α t a is the restriction of an entire function z α z a on C. It follows from [41, Sec. 8.12] that the analytic elements form a dense subalgebra of A. Definition Let A, R, α be a C -algebraic dynamical system and φ be a state of A. We say φ is a KMS state with inverse temperature β 0, or a KMS β state of A, α if it satisfies the following KMS condition: 3.1 φab = φbα iβ a for all analytic elements a, b. 29

38 It suffices to check the KMS condition on a set of analytic elements which span a dense subspace of A see [41, Proposition ]. We now look at the product system X associated to the local homeomorphisms h 1,..., h k as in Corollary We have shown in Lemma that the Nica-Toeplitz algebra N T X carries a gauge action of T k. We can lift this action to an action of R on N T X as follows: Fix r 0, k and embed R in T k via the map t e itr = e itr 1, e itr 2,..., e itr k. Then define α : R AutN T X by α t = γ e itr. Considering the system N T X, α, notice that for each ψ m xψ n y N T X, the function t α t ψm xψ n y = e itr m n ψ m xψ n y on R extends to an entire function on all of C. Thus each ψ m xψ n y is an analytic element of N T X. The elements ψ m xψ n y span a dense subalgebra of N T X as in Thus it suffices for us to check the KMS condition on these spanning elements. Remark We could get the action α directly without passing through T k by applying [24, Proposition 3.1] with the homomorphism N : Z K 0, defined by Nn = n r = k i n ir i commuting local homeomorphisms The notion of -commuting maps was first introduced in [3] and then expanded by Exel and Renault in [16, 10]. The next definition is taken from [16, 10]. Definition Let f, g be commuting maps on a set Z. We say f, g -commute, if for every x, y Z satisfying fx = gy, there exists a unique z Z such that x = gz and y = fz. The following digram illustrates this property beautifully. z f g y x g f fx = gy We also say that a family of maps -commute if any two of them -commute. 30

39 Lemma Let f, g and h be -commuting maps on a space Z. Then a For i, j N, f i and g j -commute, b f and g h -commute. Proof. For part a, see the proof of [16, Proposition 10.2]. To prove b, we apply the method used in [16, Proposition 10.2]. Suppose u, v Z satisfying fu = g hv. We have to show that there exists a unique z Z such that 3.2 u = g hz and v = fz. Since f, g -commute, it follows from fu = g hv that there exists a unique w Z such that 3.3 u = gw and hv = fw. Similarly, since h, f -commute, the equation hv = fw gives a unique z Z satisfying 3.4 v = fz and w = hz. Now combining 3.3 and 3.4, we deduce that z satisfies 3.2. To see the uniqueness, suppose z Z satisfies 3.2. Let w := hz. It follows from 3.2 that u = gw and hv = hfz = fw. The uniqueness property in 3.3 implies that w = w. Considering this with the fact that z satisfies 3.2, we have w = w = hz and v = fz. Now the uniqueness in 3.4 implies that z = z. Remark There is another proof for part b of Lemma in [55, Lemma 1.3]. Corollary Let h 1,..., h k be -commuting local homeomorphisms on a space Z. Fix m, n N k such that m n = 0. Then h m and h n -commute. Proof. Remember that h m = h m 1 1 h m k k and h n = h n 1 1 h n k k. Since m n = 0, the local homeomorphisms appearing in h m = h m 1 1 h m k k do not appear in h n = h n 1 1 h n k. Now applying Lemma finitely many times gives the proof. k Remark The condition m n = 0 in Corollary is crucial. When the local homeomorphisms h 1,..., h k -commute, it does not imply that they -commute with themselves. Thus we can not deduce from Lemma that h m and h n -commute for all m, n N k. 31

40 3.1 A characterization of KMS states In this section we provide a characterization of KMS β states on N T X, α in Proposition The characterization formula 3.19 says that KMS states vanish on most of the spanning elements of N T X. Thus Proposition enables us to recognise KMS states easier. To prove this proposition, we first show that the -commutativity condition on h 1,..., h k allows us to find interesting Parseval frames for each fibre in X. Then we use these Parseval frames to find a formula which expresses elements of the form ψ n y ψ m x as linear combinations of the elements ψ m sψ n t for suitable s X m, t X n Proposition 3.1.2b. This formula plays an important role in proving that the KMS condition holds. We also provide two simple lemmas which are again helpful when we discuss KMS condition. Lemma Let f, g be -commuting local homeomorphisms on a compact Hausdorff space Z. Suppose XE 1, XE 2 are the graph correspondences related to topological graphs E 1 = Z, Z, id, f and E 2 = Z, Z, id, g. Let {ρ i } d i=0 be a partition of unity such that f supp ρi, g supp ρi are injective and suppose that τ i := ρ i. Then a {τ i } d i=0,{τ i g} d i=0 are Parseval frames for XE 1, b {τ i } d i=0,{τ i f} d i=0 are Parseval frames for XE 2, and c there exists an isomorphism t f,g : XE 1 A XE 2 XE 2 A XE 1 such that 3.5 t f,g τ i g τ j = τ j f τ i for 0 i, j d. We will call this isomorphism the flip map. Proof. Parts a and b are quite similar. We only prove a. It follows from [17, Proposition 8.2] that {τ i } d i=0 is a Parseval frame for XE 1. To see that {τ i g} d i=0 is a Parseval frame for XE 1, we take x XE 1 and check the reconstruction formula: d τ i g τ i g, x = x. i=0 Take z Z. Using the definition of the left action and the inner product, we have 3.6 d τ i g τ i g, x z = i=0 = d τ i gz τ i g, x fz i=0 d [ τ i gz i=0 fw=fz 32 τ i gwxw. ]

41 Suppose fw = fz. Notice that the i-summand vanishes unless gz, gw supp τ i. So suppose that gz, gw supp τ i. Then fw = fz g fz = g fw f gz = f gw gw = gz f is one-to-one on supp τ i. Now we consider the digram f g fz gz g f gfz = fgz Notice that both w, z fit in the box. Then the -commutativity of f, g implies that w = z. Thus the interior sum in the last line of 3.6 will collapse to τ i gzxz and hence the reconstruction formula follows from d τ i g τ i g, x z = i=0 d τ i gzτ i gzxz i=0 = xz = xz. d τ i gz 2 Next we look at part c. Applying Lemma implies that there are isomorphisms σ f,g : XE 1 A XE 2 XF and σ g,f : XE 2 A XE 1 XF. Now set t f,g := σ 1 g,f σ f,g. It is clear that t f,g is an isomorphism from XE 1 A XE 2 onto XE 2 A XE 1. To check 3.5, note that i=0 3.7 σ f,g τ i g τ j = σ g,f τ j f τ i. Thus t f,g τ i g τ j = σ 1 g,f σ f,gτ i g τ j = τ j f τ i, as required. The next Proposition is an analogue of [20, Proposition 5.10] and [24, Lemma 4.7]. In fact Proposition is more general because the formula of [20, Proposition 5.10] is an approximation and [24, Lemma 4.7] holds only for product systems where each fibre is required to have an orthonormal basis. 33

42 Proposition Let h 1,..., h k be -commuting and surjective local homeomorphisms on a compact Hausdorff space Z and let X be the associated product system as in Corollary Take m, n N k such that m n = 0. Let {ρ i } d i=0 be a partition of unity such that h m supp ρi, h n supp ρi are injective and suppose that τ i := ρ i. a Let σ m,n : X m A X n X m+n and σ n,m : X n A X m X m+n be the isomorphisms induced by the multiplication in X. Then for all x y X m A X n, we have σ m,n x y = x, σ n,m τj h m τ i τi h n 3.8 τ j, y. 0 i,j d b Then for all x X m, y X n, we have ψ n y ψ m x = ψ m y, τj h m τ i ψn x, τi h n. 3.9 τ j 0 i,j d Proof. For part a, it suffices to prove 3.8 for x y X m A X n. Notice that X m, X n are graph correspondences associated to the topological graphs Z, Z, id, h m and Z, Z, id, h n. Since m n = 0, h m and h n are -commuting. It then follows from Lemma that {τ i h n } d i=0 and {τ j } d j=0 form Parseval frames for X m, X n respectively. Also notice that the formula for multiplication in X implies that 3.10 σ m,n τ i h n τ j = σ n,m τ j h m τ i. We use this to prove 3.8. So we must write x y in terms of the elements {τ i h n τ j } i,j. To do this we start by writing the reconstruction formulas for the Parseval frames {τ i h n } d i=0 and {τ j } d j=0. d x y = τ i h n τ i h n, x d τ j τ j, y. i=0 Since the tensors are balanced, we have x y = τ i h n τ i h n, x 3.11 τ j τ j, y We then claim that 0 i,j d τi h n, x τ j τ j, y = τ j j=0 x, τi h n τ j, y. To see the claim, we evaluate both sides of 3.12 on z Z. For the left-hand side we have τi h n, x τ j τ j, y z = τ i h n, x zτ j z τ j, y h n z 34

43 = τ i h n, x zτ j z h n w=h n z τ j wyw = τ i h n, x zτ j zτ j zyz h n is injective on supp τ j. Similarly, we compute the right-hand side of 3.12: x, τ j τi h n x, τ j, y z = τ j z τi h n τ j, y h n z = τ j z x, τi h n wτ j wyw So we have proven the claim. h n w=h n z = τ j z τ i h n, x zτ j zyz. Now putting 3.12 in 3.11 gives x y = x, τ i h n τ j τi h n τ j, y, 0 i,j d which express x y in terms of the elements {τ i h n τ j } i,j. Next we compute σ m,n x y using Notice that σ m,n is an isomorphism of correspondences. Then σ m,n x y = Now applying 3.10 gives as required. σ m,n x y = 0 i,j d 0 i,j d x, σ m,n τi h n τ j τi h n τ j, y. x, σ n,m τj h m τ i τi h n τ j, y, For part b, we use the Fock representation T of X. Remark implies that the induced homomorphism T : N T X LF X is an injection. Then by the universal property of ψ, it suffices for us to prove that 3.13 T n y T m x = 0 i,j d T m y, τj h m τ i Tn x, τi h n τ j. To do this, we evaluate both sides of 3.13 on an arbitrary s X p where p N k. An application of the formula 1.17 for the adjoint shows that the right-hand side of 3.13 vanishes unless p n. For the left hand-side, the definition of the Fock representation says that T n y T m x s = T n y σ m,p x s. Now equation 1.17 implies that 35

44 the left hand side of 3.13 is zero unless m + p n. Since m n = 0, m + p n is equivalent to p n. Thus both sides of 3.13 are zero unless p n. So we assume p n from now. It suffices to check 3.13 for s = σ n,p n s s where s s X n A X p n. To do this we first compute the right-hand side of 3.13 by using the adjoint formula 1.17 and the definition of the Fock representation: i,j d T m y,τj h m τ i Tn x, τi h n τ j σn,p n s s = 0 i,j d = 0 i,j d T m y, τj h m τ i x, τi h n τ j, s s y, σ m,p n τj h m x, τ i τi h n τ j, s s. Next we evaluate the left-hand side of 3.13 at σ n,p n s s. For convenience, let := T n y T m x σ n,p n s s. We start by applying the definition of the Fock representation. Then = T n y σ m,p x σ n,p n s s. The associativity of multiplication in X implies that = T n y σ m+n,p n σ m,n x s s. In order to apply the adjoint formula 1.17, we must write σ m,n x s in terms of the elements of X n X m. To do this, we apply part a for x s X m A X n. Then = x, T n y σ m+n,p n σ n,m τ j h m τ i τi h n τ j, s s. 0 i,j d Since the tensors are balanced, we have = 0 i,j d T n y σ m+n,p n σ n,m τ j h m τ i x, τi h n τ j, s s. Another application of associativity of the multiplication in X gives = x, T n y σ n,m+p n τ j h m σ m,p n τ i τi h n τ j, s s. 0 i,j d Now, we can apply the adjoint formula 1.17 = y, τj h m x, σ m,p n τ i τi h n τ j, s s. 0 i,j d 36

45 Since σ m,p n is an isomorphism of correspondences, = y, σ m,p n τj h m x, τ i τi h n τ j, s s. 0 i,j d This equals Thus 3.13 holds for all n N k and s X n. Then it holds for all elements of F X. Now the injectivity of T gives 3.9. Remark In [54], Solel studied the product systems over N k via different notations see Appendix A. He used the notion doubly commuting representation [54, 3.12] as an alternative for Nica-covariance representation. Then he proved in [54, Theorem 3.15] that the universal Nica-covariant representation ψ satisfies his doubly commuting relation. The doubly commuting relation involves a flip map between fibres of the product system. Since we have an explicit formula for the flip as in 3.5, we can translate his results to our notation. In Appendix A, we reconcile our result with [54, Theorem 3.15]. We show that ψ satisfies [54, Lemma 3.9i] by using our formula 3.9 and the flip map 3.5. Lemma Let h 1,..., h k be -commuting and surjective local homeomorphisms on a compact Hausdorff space Z and let X be the associated product system as in Corollary Suppose m, n, p, q N k x X m, y X n, s X p, and t X q. Then there exist {ξ i,j } 0 i,j d X m+p n p and {η i,j } 0 i,j d X n+q n p such that 3.15 ψ m xψ n y ψ p sψ q t = ψ m+p n p ξ i,j ψ n+q n p η i,j. 0 i,j d Proof. Let N := n n p and P := p n p. It suffices for us to prove 3.15 for y = σ n p,n y y and s = σ n p,p s s, where y y X n p A X N, s s X n p A X P. Routine calculation shows that ψ n y ψ p s = ψ N y ψ n p y ψ n p s ψ P s = ψ N y ψ 0 y, s ψ P s 3.16 = ψ N y ψ P y, s s. Let {U i } d i=0 be an open cover of Z such that h N Ui and h P Ui are injective. Choose a partition of unity {ρ i } d i=0 subordinate to {U i } d i=0 and define τ i := ρ i. Since N P = 0, applying Proposition to {τ i } d i=0 implies that 3.17 ψ N y ψ P y, s s = 0 i,j d ψ P y, τ j h P τ i ψn y, s s, τ i h N τ j. 37

46 Combining equations 3.16 and 3.17, we have ψ m xψ n y ψ p sψ q t [ ] = ψ m x ψ P y, τ j h P τ i ψn y, s s, τ i h N τ j ψ q t = 0 i,j d 0 i,j d ψ m+p σ m,p x y, τ j h P τ i ψ q+n σ N,q t. y, s s, τ i h n τ j Now labelling ξ i,j := σ m,p x y, τ j h P τ i and ηi,j := σ N,q t y, s s, τ i h τ n j completes the proof of Lemma Suppose m, n, p, q N k satisfying m + p = n + q and n p = 0. Then m m q = n and q m q = p. Proof. We first prove m m q = n. Fix 1 i k. Since n p = 0, either n i = 0 or p i = 0. If n i = 0, then m + p = n + q implies that m i q i and hence m i m q i = m i m i = 0 = n i. If p i = 0, then m i q i and m i m q i = m i q i = n i p i = n i. Thus m i m q i = n i for all i, as required. To prove q m q = p, it suffices to apply the construction of the previous paragraph to the equality q + n = p + m. Now we are ready to prove a generalization of [1, Proposition 3.1] to our product system. There is also a similar Proposition for the higher-rank graph algebras see [27, Proposition 3.1]. Proposition Let h 1,..., h k be -commuting and surjective local homeomorphisms on a compact Hausdorff space Z and let X be the associated product system as in Corollary Suppose r 0, k and α : R AutN T X is given in terms of the gauge action by α t = γ e itr. Let β > 0 and φ be a state on N T X. a If φ satisfies 3.18 φ ψ m xψ n y = δ m,n e βr m φ ψ 0 y, x for x X m, y X n, then φ is a KMS β state of N T X, α. b If φ is a KMS β state of N T X, α and r 0, k has rationally independent coordinates, then φ satisfies

47 Proof of a. Suppose state φ satisfies To show that φ is a KMS β state, it suffices to check the KMS condition 3.19 φbc = e βr m n φcb for elements b = ψ m xψ n y and c = ψ p sψ q t from N T X. Let M := m m q, N := n n p, P := p n p and Q := q m q. It is also enough to prove 3.19 for elements of the form x = σ m q,m x x, y = σ n p,n y y, s = σ n p,p s s and t = σ m q,q t t where x x X m q A X M, y y X n p A X N, s s X n p A X P, and t t X m q A X Q. During the proof, we will need the following equations occasionally 3.20 ψ n y ψ p s = ψ N y ψ P y, s s, and 3.21 ψ q t ψ m x = ψ Q t ψ M t, x x ; they are obtained by a calculation similar to the one done to establish To prove 3.19, first note that Lemma together with the equation 3.18 imply that both of φbc and φcb vanish unless m + p = n + q. So we assume this from now. Next we claim that it suffices for us to check 3.19 for n p = 0. To see this, suppose we have proven the case n p = 0 and consider m, n, p, q such that m+p = n+q. Then 3.20 implies that φbc = φ ψ m xψ N y ψ P y, s s ψ q t. Since N P = 0, we are back into the other case. Thus φbc = e βr m N φ ψ P y, s s ψ q t ψ m xψ N y. Applying a similar calculation twice by using 3.21 and 3.20 gives: φcb = φ ψ p sψ Q t ψ M t, x x ψ n y = e βr p Q φ ψ M t, x x ψ n y ψ p sψ Q t since Q M = 0 = e βr p Q φ ψ M t, x x ψ N y ψ P y, s s ψ Q t = e βr p Q+M N φ ψ P y, s s ψ Q t ψ M t, x x ψ N y. Since m + p = n + q, we have e βr m N = e βr m n e βr p Q+M N. Now 3.21 and our calculations imply that φbc = e βr m n φcb. So it is enough to prove 3.19 when n p = 0. Now we assume that m+p = n+q and n p = 0. Let {U i } d i=0 be an open cover of Z such that h n Ui and h p Ui are injective. Choose a partition of unity {ρ i } d i=0 subordinate 39

48 to {U i } d i=0 and define τ i := ρ i. To compute φbc, we start by using 3.9 to rewrite ψ n y ψ p s to get φbc = φ ψ m xψ n y ψ p sψ q t [ ] = φ ψ m x ψ p y, τj h p τ i ψn s, τi h n τ j ψ q t = 0 i,j d 0 i,j d φ ψ m+p σ m,p x y, τ j h p τ i ψ q+n σ q,n t s, τ i h n τ j. By our assumption 3.18, we get φbc = e βr m+p φ ψ 0 0 i,j d σ q,n t s, τ i h n τ j, σ m,p x y, τ j h p τ i. To calculate φcb, notice that φcb = φ ψ p sψ Q t ψ M t, x x ψ n y by Since m + p = n + q and n p = 0, Lemma implies that Q = p and M = n. Then φcb = φ ψ p sψ p t ψ n t, x x ψ n y. Now we use the formula 3.9 and the identity ψξψη = ψη ψξ to rewrite ψ p t ψ n t, x x. [ φcb = φ ψ p s ψ n t, τ i h n τ j ψp t, x x, τ j h p ] τ i ψ n y = φ 0 i,j d 0 i,j d ψ p+n σ p,n s t, τ i h n τ j Our assumption 3.18 implies that φcb = e βr n+p φ ψ 0 ψ n+p σ n,p y t, x x, τ j h p τ i 0 i,j d σ n,p y t, x x, τ j h p τ i, σ p,n s t, τ i h n τ j. Since m + p = m n + n + p, it follows that e βr m+p = e βr m n+n+p. Now to check KMS condition 3.19, it suffices to prove that := σ q,n t s, τ i h n τ j, σ m,p x y, τ j h p τ i 0 i,j d and := 0 i,j d σ n,p y t, x x, τ j h p τ i, σ p,n s t, τ i h n τ j 40

49 are equal. To do this we compute z and z for z Z. Since the calculation for z is easier, we compute it first. We start by applying the multiplication formula 2.4 in X: z = i,j d h n+p w=z = 0 i,j d h n+p w=z = h n+p w=z ywsw σ n,p y t, x x, τ j h p τ i wσ p,n s t, τ i h n τ j w yw t, x x, τ j h p h n w τ i hn w sw t, τ i h n h p w τ j h p w d t, τ i h n h p w τ i h n w i=0 d τ j h p w τ j h p, t, x x h n w. j=0 Since n p = 0, h n and h p are -commuting. Now remembering that X n, X p are graph correspondences associated to the topological graphs Z, Z, id, h n and Z, Z, id, h p Lemma implies that {τ j h p } d j=0 and {τ i h n } d i=0 are Parseval frames for X n, X p respectively. We rearrange 3.22 by using the definition of the actions to apply the reconstruction formulas for these Parseval frames: z = d ywsw τi h n τ i h n, t w h n+p w=z = h n+p w=z i=0 d τj h p τ j h p, t, x x w j=0 ywswt w t,x x w. Next we compute z. Using the formula 2.4 for multiplication in X, we have z = σ q,n t s, τ i h n τ j wσ m,p x y, τ j h p τ i w 0 i,j d = 0 i,j d h m+p w=z tw s, τ i h n h q wτ j h q wxw y, τ j h p h m wτ i h m w h m+p w=z = twxw h m+p w=z d s, τ i h n h q wτ i h m w i=0 d τ j h q w y, τ j h p h m w. An application of Lemma implies that q = m q + p and m = m q + n. Then d z = twxw τi h n, s h m q+p w τ i h m q+n w h n+p+m q w=z i=0 41 j=0

50 d τ j h m q+p w y, τ j h p h m q+n w. j=0 We again rearrange this equation to apply the reconstruction formulas for the Parseval frames {τ i h n } d i=0 and {τ j h p } d j=0. z = twxw [ τi h n τ i h n, s ] h m q w h n+p+m q w=z = h n+p+m q w=z 0 i d 0 j d [τ j h p τ j h p, y ] h m q w twxws h m q w y h m q w. Now writing t = σ m q,q t t, x = σ m q,m x x and splitting, we have z = t wt h m q w x wx h m q w s h m q w y h m q w h n+p+m q w=z = sux uyut u h n+p u=z = h n+p u=z t wx w h m q w=u sux uyut u t, x u Thus z = z and hence φ satisfies Proof of b. Suppose φ is a KMS β state on N T X and r has rationally independent coordinates. To show that φ satisfies 3.18, let x X m and y X n. By two application of the KMS condition, we have φ ψ m xψ n y = φ ψ n y α iβ ψ m x = e βr m φ ψ n y ψ m x = e βr m n φ ψ m xψ n y. Now since r has rationally independent coordinates and β > 0, both sides will vanish for m n. For m = n the KMS condition and T2 of Definition1.3.1 imply that φ ψ m xψ m y = e βr m φ ψ m y ψ m x = e βr m φ ψ 0 y, x, and φ satisfies

51 3.2 KMS states and subinvariance relation In this section we introduce a subinvariance relation involving a family of Ruelle operators. We characterize the solutions of this subinvariance relation in Proposition We also show that every KMS β state for β 0, gives a measure which satisfies our subinvariance relation Proposition Lemma Let h 1,..., h k be commuting and surjective local homeomorphisms on a compact Hausdorff space Z. For i {1,..., k}, define Q i : CZ CZ by Q i az = h i w=z aw for a CZ. a The functions Q i : CZ CZ are commuting linear bounded operators. b For n N k, set Q n := Q n k k Qn 1 1. Then 3.23 Q n az = h n w=z aw for a CZ. c For each 1 i k, there is a unique adjoint operator Q i : CZ CZ such that Q i = Q i and Q i f = f Q i for f CZ. Proof. To prove a, take 1 i k and a CZ. It is clear that Q i is linear. Notice that Q i a = sup z Z Q i az = sup z Z = max h i w=z z Z h 1 i aw max z a. z Z h 1 i z sup az Since h i is a local homeomorphism on the compact space Z, max z Z h 1 i z <. It then follows that Q i is bounded and Q i max z Z h 1 i z. For the commutativity, take 1 i, j k. We have Qi Q j a z = Q i Qj a z = Qj a w z Z 3.24 = au = h i w=z h i w=z h j u=w h i h j u=z au. Since h i, h j are commuting, 3.24 implies Q i Q j = Q j Q i. 43

52 For part b, notice that {Q i } are commuting and surjective local homeomorphisms. Then 3.23 follows from Finally part c follows from [19, page 160 Exercise 22] or from [47, Theorem 4.10]. Definition Let h 1,..., h k be commuting and surjective local homeomorphisms on a compact Hausdorff space Z. Let Q 1,..., Q k be as in Lemma For 1 i k, we define R e i : CZ CZ by R e i := Q i. Then R e 1,..., R e k are commuting, linear bounded operators. We write R 0 := id CZ and for n N k +, we use R n := R n ke k R n 1e 1. The operators R e 1,..., R e k are sometimes called Ruelle operators for example see [48, 2.3],[49, 3.1],[14, 2.1]. Remark A finite regular Borel measure ν on Z can be viewed as an element of CZ by νa := az dνz for a CZ. We can then calculate a formula for R n ν. Lemma 3.2.1c implies that R n ν = Q n ν = νq n. It then follows a d R n ν 3.25 = aw dνz for a CZ. h n w=z Remark The operation R in 3.25 is an analogue for the operation R studied in [1]. But here we define it as an operator on the whole of CZ, while in [1] it is only defined on measures which are positive elements of CZ. Definition Let h 1,..., h k be commuting and surjective local homeomorphisms on a compact Hausdorff space Z and suppose ν is a finite regular Borel measure on Z. We say ν satisfies the subinvariance relation if for every subset K of {1,..., k}, we have 3.26 a d 1 e βr i R e i ν 0 for all positive a CZ. i K Given J K, we write e J := j J e j and we interpret R e ν = ν. The following identity is helpful when we work with the subinvariance relation e βr i R e i ν = 1 J e βr e J R e J ν. i K J K 44

53 Remark The subinvariance relation 3.26 is a generalization of the subinvariance relation [1, 4.2] where we have only one local homeomorphism. It also is a variant of the subinvariance relation appearing in the analysis of KMS states of the Toeplitz-Cuntz-Krieger algebras of higer-rank graphs [28, Proposition 4.1 a]. The next Proposition characterizes the solutions of the subinvariance relation It is a generalization of [1, Proposition 4.2] and [28, Theoerem 6.1a]. Proposition Let h 1,..., h k be surjective and commuting local homeomorphisms on a compact Hausdorff space Z. For each 1 i k, let 3.28 β ci := lim sup j 1 ln max j z Z h j i z. Let r 0, k, and suppose β 0, satisfies βr i > β ci. a The series n N k e βr n h n z converges uniformly for z Z to a continuous function f β z 1. b Suppose ε is a finite regular Borel measure on Z. Then the series n N k e βr n R n ε converges in norm in the dual space CZ with sum µ, say. Then µ satisfies the subinvariance relation 3.26 and we have ε = k i=1 1 e βr i R e i µ. Then µ is a probability measure if and only if f β dε = 1. c Suppose µ is a probability measure which satisfies the subinvariance relation Then ε = k i=1 1 e βr i R e i µ is a finite regular Borel measure satisfying n N e βr n R n ε = µ, and we have f k β dε = 1. Before starting the proof, notice that we regard a sum indexed by N k as an integral over N k with respect to the counting measure. All series here have positive summands. Then by Tonelli s theorem, we can consider a sum over N k as iterated sums over N. Moreover, if the iterated sums over N are convergent in one order, then the sum over N k converges as well see for example [19, Theorem 7.27] We will need the following algebraic identities occasionally: m N k i=1 k f i m i = k f i m i. i=1 m i N Also notice that if f i g j = g j f i for all 1 i, j k, then 3.30 k f i i=1 j=1 k g j = k f l g l. l=1 45

54 Proof of Proposition For part a, we first claim that for each 1 i k, there exist 0 < δ i R and M i N such that 3.31 l N, l M i e lβr i max h l i z < e lδ i for all z Z. z To see the claim, since βr i > β ci, applying the calculation of the first paragraph in the proof of [1, Proposition 4.2] with the local homeomorphism h i gives δ i and M i satisfying Now we take M := M 1,..., M k and calculate the N-th partial sum for N M. M n N e βr n h n z = M n N M n N = M n N i=1 e βr n h n 1 1 h n k 1z k e βr n k k i=1 e βr i n i max z Using the identity 3.29 and the equation 3.31, we have max h n i i z z h n i i z e βr n h n z M n N k i=1 M i n i N i k i=1 M i n i N i e δini. e βr i n i max z h n i i z Now let N in N k. This means each N i for 1 i k. Since each sum n i =M i e δ in i is convergent, it follows that n=m e βr n h n z converges uniformly for z Z. Notice that h n = h n k k h n 1 1 is a local homeomorphism on Z for all n N k because each h i 1 i k is. Then [8, Lemma 2.2] implies that z h n z is locally constant and hence is continuous. Thus f β z := n N k e βn h n z is the uniform limit of a sequence of continuous functions, and is therefore continuous. The term corresponding to n = 0 is 1, so f β 1. For part b, take M and δ i 1 i k as in part a. We want to show that n M e βr n R n ε converges in norm in the dual space CZ. To do this, we calculate the N-th partial sum using formula 3.25 for the definition of R n. Let g CZ, we have M n N e βr n g dr n ε = M n N e βr n 46 h n w=z gw dεz

55 3.33 M n N ε CZ g e βr n h n z ε CZ g k i=1 M i n i N i e δini by Now when N, all the series n i =M i e δ in i are convergent and hence the series n=0 e βr n R n ε converges in the norm of CZ to a measure µ, say. Since ε is a measure on Z, it is a positive functional on CZ. The formula 3.25 for definition of R n, says that µ is positive functional on CZ and therefore is a Borel measure on Z by the Riesz-representation theorem. To prove that µ satisfies the subinvariance relation 3.26, let K {1,..., k}. We first simplify the N-th partial sum j K 1 e βr j R e j 0 n N e βr n R n for N N k. We have, j K1 e βr j R e j 0 n N k = 1 e βr j R e j j K = 1 e βr j R e j j K e βr n R n = N i i=1 n i =0 N i i K n i =0 j K e βr in i R n ie i e βr in i R n ie i Relabelling the indices in products, we have 1 e βr j R e j e βr n R n 3.34 j K = N i i {1,...,k}\K n i =0 0 n N e βr in i R n ie i j K N j n j =0 1 e βr j R e j 0 n N i=1 by identity 3.29 N i i {1,...,k}\K n i =0 N j k e βr in i R n ie i e βr in i R n ie i by e βr jn j R n je j e βr jn j +1 R n j+1e j. Now we can compute j K 1 e βr j R e j n 0 e βr n R n ε by applying 3.34 to ε and letting N. Notice that for each j K, we have n j =0 n j =0 e βr jn j R n je j ε e βr jn j +1 R n j+1e j ε = ε. n j =0 It then follows that 1 e βr j R e j e βr n R n ε = j K n 0 i {1,...,k}\K n i =0 e βr in i R n ie i ε. The argument in the last paragraph of the proof of [1, Proposition 4.2b], shows that applying each n i =0 e βr in i R n ie i to a finite regular Borel measure gives a finite regular 47

56 Borel measure. It then follows that j K 1 e βr j R e j n 0 e βr n R n ε is a finite regular Borel measure. Thus a d 1 e βr i R e i i K 0 n N e βr n R n ε 0 for all positive a CZ. Thus µ satisfies the subinvariance relation To prove that ε = k i=1 1 e βr i R e i µ, it suffices to apply the argument of the previous two paragraphs with K = {1,..., k}. To see the relation between µ and f β, we compute using 3.25: µz = n N k e βr n R n εz = n N k e βr n = n N k e βr n 1 dr n ε h n z dεz. An application of Tonelli s theorem implies that µz = e βr n h n z dεz = n N k f β dε. Since Z is compact and f β is continuous on Z, µz = f β dε <. Also µ is a probability measure if and only if f β dε = 1. We now look at c. First note that the measure ε is obtained by finitely many times applications of the bounded operators R e i 1 i k on the measure µ. Since µ is a finite measure, ε is a finite measure as well. The subinvariance relation 3.26 says that ε is a positive measure. An application of the Riesz-representation theorem implies that ε is a Borel measure on Z. Since ε is finite, it is regular as well see [19, Theorem 7.8]. To check 3.35 e βr n R n ε = µ, n N k we calculate the N-th partial sum using the identity 3.29: k e βr n R n ε = e βr n R n 1 e βr i R e i µ 0 n N 0 n N = 0 n N i=1 i=1 k e βr in i R n ie i 1 e βr i R e i µ 48

57 3.36 k = N i i=1 n i =0 e βr i n i R n ie i e βr in i +1 R n i+1e i µ. Let i {1,..., k} and N i. Applying each sum in the last line of 3.36 to µ, we have e βr in i R e in i µ n i =0 e βr in i +1 R e in i +1 µ = µ. n i =0 Taking the product over 1 i k, completes the proof of The next Proposition shows that every KMS states on N T X, α gives a probability measure on Z satisfying the subinvariance relation This proposition is an extension of our result in [1, Proposition 4.1] for a single local homeomorphism. There is also a similar result for the Toeplitz-Cuntz-Krieger algebra of a higher-rank graph in [28, Proposition 4.1a]. Proposition Let h 1,..., h k be commuting and surjective local homeomorphisms on a compact Hausdorff space Z and let X be the associated product system over N k, as in Corollary Let r 0, k and suppose that α : R AutN T X is given in terms of the gauge action by α t = γ e itr. Suppose φ is a KMS β state of N T X, α, and µ is the probability measure on Z such that φψ 0 a = a dµ for all a CZ. Let K be a subset of {1,..., k} and write e J := j J e j for all J K. Then 3.37 a dµ + J K 1 J e βr e J a d R e J µ 0 for all positive a CZ. To prove the Proposition 3.2.8, we need the following simple lemma. Lemma Let h 1,..., h k be -commuting and surjective local homeomorphisms on a compact Hausdorff space Z and let X be the associated product system as in Corollary Let T be the Fock representation of X. Take n N k, and let {ρ l } d l=0 be a partition of unity such that h n supp ρl is injective for each l. Set τ l := ρ l. Then the restriction of d l=0 T nτ l T n τ l to each m-summand X m of the Fock module is the identity map if m n, and is otherwise 0. Proof. Let m N k. If m n, then the adjoint formula for the Fock representation 1.17 implies that d l=0 T nτ l T n τ l vanishes on X m. Now let m n. It suffices to prove d T n τ l T n τ l x = x, l=0 49

58 for x = σ n,,m n x x where x x X n A X m n. To see this, we compute by using the definition of the Fock representation and the adjoint formula 1.17: d l=0 T n τ l T n τ l σ n,m n x x = = d T n τ l τ l, x x l=0 d σ n,n m τl τ l, x x. l=0 Lemma 3.1.1a implies that {τ l } d l=0 is a Parseval frame for the fibre X m. Applying the reconstruction formula for {τ l } d l=0, we have d T n τ l T n τ l d σ n,m n x x = σ m,m n τ l τ l, x x l=0 which is precisely x as required. l=0 = σ n,m n x x Proof of Proposition Let a be a positive element of CZ. If K =, since a is positive, a dµ 0. So we assume K. We apply the method of the proof of [1, Proposition 4.1]. So we first write each integral in 3.37 in terms of elements of N T X and then use the Fock representation to show that the sum of these integrals is positive. The first integral in 3.37 by assumption is 3.38 a dµ = φ ψ 0 a. Now consider J-summand. To write the integral a d R e J µ in terms of elements of N T X, let {Ul J}d l=0 be an open cover of Z such that he J U J l is injective and choose a partition of unity {ρ J l }d l=0 subordinate to {U l J}d l=0. Define τ l J := ρ l. Remember that the fibre X ej in X is the graph correspondence Z, Z, id, h e J. Then applying the calculation in the first two paragraphs of [1, Proposition 4.1], to X ej shows that 3.39 a d R e J µ d = e βr e J φ ψ ej a τl J ψ ej τl J l=0 d = e βr e J φ ψ 0 aψ ej τl J ψ ej τl J. l=0 50

59 Putting 3.38 and 3.39 in the left-hand side of 3.37, we have a dµ+ 1 J e βr e J a d R e J µ 3.40 J K = φψ 0 a + J K = φ ψ 0 a + J K d 1 J φ ψ 0 aψ ej τl J ψ ej τl J 1 J d l=0 l=0 ψ 0 aψ ej τ J l ψ ej τ J l, which express the integrals in 3.37 in terms of elements of N T X. Next we show that the right-hand side of 3.40 is positive. Since φ is a state, it suffices to show that 3.41 ψ 0 a + J K 1 J ψ 0 a d l=0 ψ ej τ J l ψ ej τ J l 0. To do this, we use the Fock representation T of X. We aim to prove 3.42 T 0 a + J K 1 J T 0 a d l=0 T ej τ J l T ej τ J l x n 0 for all x n X n, n N k. Fix n N k and x n X n. Let I := {i i K, n i 0}. Applying Lemma with {τ J l }d l=0 implies that the J-summands with n e J vanishes. Since n e J is equivalent to J I, the outer sum in 3.42 reduces to T 0 a + J I Now we compute using Lemma T 0 a + J I 1 J T 0 a 1 J T 0 a d l=0 d l=0 = T 0 ax n + = J I T ej τ J l T ej τ J l x n. T ej τ J l T ej τ J l x n J I 1 J T 0 ax n. 1 J T 0 ax n This vanishes because the number of subsets with odd cardinality equals with the number of subsets with even cardinality. Thus T 0 a + J K 1 J d l=0 T ej a τ J l T ej τ J l x n 0. 51

60 We now deduce that T 0 a + J K 1 J d l=0 T e J a τ J l T e J τ J l is a positive operator on F X. Since the induced homomorphism T : N T X LF X is an injection see Remark 1.4.1, it follows that as required. ψ 0 a + J K 1 J ψ 0 a d l=0 ψ ej τ J l ψ ej τ J l 0, 3.3 KMS states at large inverse temperatures In this section we prove our main theorem which characterizes the KMS β states of N T X, α for large β. We found a one-to-one correspondence between the KMS states on N T X, α and the probability measures on Z satisfying the subinvariance relation This theorem is a generalization of our result in [1, Theorem 6.1] for dynamical system associated to a single local homeomorphism. There is also a similar characterization in [27, Theorem 6.1] for the dynamics arising from higher-rank graphs. As a corollary, we also obtain some results for the dynamical system OX, α. Theorem Let h 1,..., h k be -commuting and surjective local homeomorphisms on a compact Hausdorff space Z. Let X be the associated product system over N k, as in Corollary For 1 i k let β ci satisfies βr i > β ci α : R AutN T X by α t = γ e itr. be as in 3.28, and suppose that r 0, k for all i. Let f β be the function in Proposition 3.2.7a and define a Suppose that ε is a finite regular Borel measure on Z such that f β dε = 1, and take µ := n N k e βr n R n ε. Then there is a KMS β state φ ε on N T X, α such that 3.43 φ ε ψm xψ p y 0 if m p = e βr m y, x dµ if m = p. b If in addition r has rationally independent coordinates, then the map ε φ ε is an affine isomorphism of Σ β := { ε MZ + : } f β dε = 1 onto the simplex of KMS β states of N T X, α. Given a state φ, let µ be the probability measure such that φψ 0 a = a dµ for a CZ. Then the inverse of ε φ ε takes φ to ε := k i=1 1 e βr i R e i µ. 52

61 Proof of a. Let ε be a finite regular Borel measure on Z. We follow the structure of the proof of [1, Theorem 5.1]. Thus we aim to construct the KMS state φ ε by using a representation θ of X on H θ := n N L 2 Z, R n ε. Notice that here each R n is a k bounded operator on CZ while in [1, Theorem 5.1] the operation R was defined only on measures positive functionals. We write ξ = ξ n for the elements of the direct sum. For m N k and x X m, we claim that there is a well-defined operator θ m x on H θ such that 0 if n m 3.44 θ m xξ n z = xzξ n m h m z if n m. Let ξ = ξ n n N L 2 Z, R n ε. Then k θ m xξ 2 = θ m xξ n 2 n N k = xz 2 ξ n m h m z 2 dr n εz n m = xz 2 ξ n h m z 2 dr n+m εz n N k x 2 ξ n h m w 2 dr n εz n N k = x 2 n N k h m w=z h m w=z ξ n z 2 dr n εz x 2 c m ξ n z 2 dr n εz where c m = max z h m z n N k 3.45 = c m x 2 ξ 2. Thus θ m x BH θ. Next we apply a similar calculation to compute the adjoint θx. Take η H θ, then θm xξ θm η = xξ η n n n N k = n N k = n m = n N k θ m xξ n zη n z dr n εz xzξ n m h m zη n z dr n εz xzξ n h m zη n+m z dr n+m εz 53

62 = n N k = n N k xwξ n h m wη n+m w dr n εz h m w=z ξ n z h m w=z xwη n+m w dr n εz. Thus θ m x satisfies 3.46 θm x η z = xwη n n+m w for η H θ. h m w=z Next we claim that θ is a Toeplitz representation of X. We check conditions T1 T3 of Definition For T1, since each θ m : X m BH θ is clearly linear, we need only check that θ 0 : A BH θ is a homomorphism on A = CZ here. Since the multiplication in A is pointwise multiplication, for a, a A, we have θ0 aa ξ z = n aza zξ n z = az θ 0 a ξ z = θ n 0 aθ 0 a ξ z. n Thus θ 0 : A BH θ is a homomorphism. To check T2, fix m and x 1, x 2 X m. Then θ0 x 1, x 2 ξ z = x n 1, x 2 zξ n z = x 1 wx 2 wξ n z h m w=z = h m w=z Since θ m x 2 ξ n+m w = x 2 wξ n h m w, θ0 x 1, x 2 ξ n z = Now formula 3.46 implies that h m w=z x 1 wx 2 wξ n h m w. x 1 wθ m x 2 ξ n+m w. θ0 x 1, x 2 ξ n z = θ m x 1 θ m x 2 ξ n z. Thus θ 0 x 1, x 2 = θ m x 1 θ m x 2, giving T2. For T3, let x X m and y X p. If n m + p, then θ m+p xyξ z = 0. Also n we have θm xθ p yξ n z = xz θ p yξ n m h m z, which vanishes for n m p. So we assume n m + p. Using the definition of multiplication in X, we have θm+p xyξ n z = θ m+p σm,p x y ξ n z 54

63 This complete our proof of T3. = σ m,p x y zξ n m+p h m+p z = xzy h m z ξ n m+p h m+p z = xz θ p yξ n m z h m z = θ m xθ p yξ n z. Next we show that θ is Nica-covariant. Let 1 m be the identity operator on the fibre X m and α θ : N k End θ 0 A be the action as in [20, Proposition 4.1]. Since each fibre X m is essential and θ is a representation on the Hilbert space H θ, we must show that α m p1 θ αm1 θ m αp1 θ m p if m p < 3.47 p = 0 otherwise. for all m, p N k. To do this, fix m, p N k. Clearly m p <. So we check α θ m1 m α θ p1 p = α θ m p1 m p. Choose a partition of unity {ρ j : 1 j d} for Z such that h m p is injective on each supp ρ j and take τ j := ρ j X m. Notice that {τ j } d j=0 can be viewed as a Parseval frame for the fibres X m, X p and X m p. Now to check 3.47, Lemma implies that it suffices to prove that 3.48 d d d θ m τ i θ m τ i θ p τ j θ p τ j = θ m p τ l θ m p τ l. i=1 j=1 l=1 To see this, let ξ H θ and z Z. We evaluate both sides of 3.48 at ξ: For the right-hand side of 3.48, notice that the definition of θ m p implies that d l=1 θ m pτ l θ m p τ l ξ vanishes unless n m p. So we assume n m p and n compute using the definition of θ m p and the adjoint formula 3.46: d l=1 θ m p τ l θ m p τ l ξ z = n = = d l=1 d l=1 d l=1 Since h m p is injective on each supp τ l, we have d l=1 θ m p τ l θ m p τ l ξ z = n d l=1 θ m p τ l θ m p τ l ξ nz τ l z θ m p τ l ξ n m p h m p z τ l z h m p w=h m p z τ l zτ l zξ n z = ξ n z 55 τ l wξ n w. d τ l z 2 = ξ n z. l=1

64 Thus 3.49 d l=1 θ m p τ l θ m p τ l ξ = n ξ n if n m p 0 otherwise. For the left-hand side of 3.48, notice that {τ i } d i=0 is a Parseval frame for X m and h m is injective on each supp τ i. Then applying the same calculation of the previous paragraph using formula for θ m τ i and θ m τ i, we have d d d θ m τ i θ m τ i θ p τ j θ p τ j ξ j=1 = θ pτ j θ p τ j ξ if n m n n 0 otherwise. i=1 j=1 Now suppose n m. Again since {τ j } d j=0 is a Parseval frame for X p and h p is injective on each supp τ j. A similar computation for d j=1 θ pτ j θ p τ j ξ, implies that n d d ξ θ m τ i θ m τ i θ p τ j θ p τ j ξ n if n m p 3.50 = n 0 otherwise. i=1 j=1 Comparing 3.49 and 3.50 gives 3.48, and hence θ is a Nica-covariant representation. Now the universal property of N T X [20, Theorem 6.3], gives us a homomorphism θ : N T X BH θ such that θ ψ = θ. For each q N k, we choose a finite partition {Z q,i : 1 i I q } of Z by Borel sets such that h q is one-to-one on each Z q,i. 1 χ q,i = χ Zq,i, and define ξ q,i n N L 2 Z, R n ε by k 0 if n q ξn q,i = χ q,i if n = q. We now define φ ε : N T X C by We take Z 0,1 = Z and write I 0 = 1. Let 3.51 φ ε b = q N k I q i=1 e βr q θ bξ q,i ξ q,i for b N T X, To see φ ε is well-defined, we need to show that the series converges. Notice that elements of C -algebras can be written as a linear combination of positive elements, 1 To see that there is such a partition, notice that since h q is a local homeomorphism on Z, there is an open cover {U l } d l=0 of Z such that each hq Ul is injective. Now set V 0 := U 0 and for each l let V l := U l \ l 1 j=0 V j. Clearly {V l } d l=0 is a Borel partition of Z. Since this partition is dependent on q, we relabel it as {Z q,i } Iq i=1. 56

65 and a positive element b satisfies b b 1. Thus it suffices for us to show that the series defining φ ε 1 is convergent. By definition φ ɛ 1 is I q e βr q χ Zq,i χzq,i = I q q N k i=1 q N k i=1 = q N k Since {Z q,i } i is a partition of Z, we have I q i=1 e βr q e βr q R q εz q,i. χ Zq,i zχ Zq,i z dr q εz q N k I q i=1 e βr q χ Zq,i χ Zq,i = q N k e βr q R q εz. By Proposition 3.2.7b, the sum q N k e βr q R q ε converges to a measure µ. Since fβ dε = 1, µ is a probability measure. Then q N k I q i=1 e βr q χ Zq,i χzq,i = µz = 1. Thus φ ɛ 1 = 1, and the formula 3.51 gives us a well-defined state on T XE. To see that φ ε satisfies 3.43, take x X m, y X p and b = ψ m xψ p y. Since ξ q,i is zero in all except the qth summand of n N L 2 Z, R n ε, k θ bξ q,i = θ ψ m xψ p y ξ q,i = θ m xθ p y ξ q,i is zero in all but the q p + mth summand. Thus θ bξ q,i ξ q,i = 0 for all q, i whenever p m, and φ ε satisfies 3.43 when p m. Then we assume p = m. If q m, then θ m xθ m y ξ q,i = 0. Now suppose q m. Since h q is injective on Z q,i, it follows that h m is injective on each Z q,i. Then θm xθ m y ξ q,i ξ q,i = xz = h m w=h m z xzyzχ q,i z dr q εz. Since the Z q,i partition Z, summing over i, we have I q i=1 θ ψ m xψ m y ξ q,i ξ q,i = 57 ywχ q,i w χ q,i z dr q εz xzyz dr q εz.

66 Now using the formula 3.25 for R m, we have φ ε ψm xψ m y = 3.52 e βr q xzyz dr q εz q m = e βr q xwyw dr q m εz q m h m w=z = e βr m+q y, x z dr q εz q N k = e βr m y, x d e βq R q ε. q N k Recall that q N k e βr q R q ε = µ. Then φ ε ψm xψ m y = e βr m y, x dµ. Thus φ ε satisfies To see that φ ε is a KMS β state, we apply Proposition with m = p = 0 and x = y = a A to get φ ε ψ0 a a = φ ε ψ0 a ψ 0 a = a, a A dµ = a a dµ. This implies that φ ε ψ 0 a = a dµ for all positive a A and so for all elements of A. It then follows that φ ε ψ0 y, x = y, x dµ. Now φ ε ψm xψ n y = δ m,n e βr m φ ε ψ0 y, x, and the Proposition 3.1.6a says that φ ε is a KMS β state. Proof of Theorem b. Now assume that r has rationally independent coordinates. We first claim that Σ β is a compact subset of CZ in the weak norm. Then to prove that ε φ ε is an isomorphism, it suffices to show that it is injective, surjective, and continuous. For the claim, we show that Σ β is a closed subset of the compact unit ball of CZ. Let ε Σ β. Recall that f β = f β 1. Thus ε CZ = sup f 1, f CZ f dε sup f 1, f CZ f dε f β dε = 1. Then Σ β is a subset of the unit ball in CZ. To check that it is closed, take a sequence {ε j } j=1 Σ β and ε CZ such that ε j ε in weak topology. Since εf = lim j ε j f 0 for all positive f CZ, 58

67 the Riesz-representation theorem implies that ε MZ +. Also note that f β dε = lim j Then ε Σ β and that Σ β is closed, as required. f β dε j = 1. For the surjectivity of ε φ ε, let φ be a KMS β state, and let µ be the probability measure such that φ ψ 0 a = a dµ for a CZ. Since r has rationally independent coordinates, Proposition implies that φ satisfies φ ψ m xψ n y = δ m,n e βr m φ ψ 0 y, x = e βr m 3.53 y, x dµ. On the other hand, since µ satisfies subinvariance relation 3.26 by Proposition 3.2.8, Proposition 3.2.7c implies that ε := k i=1 1 e βr i R e i µ belongs to Σβ and satisfies n N e βr n R n ε = µ. Now applying part a to ε gives a KMS k β state φ ε such that 3.54 φ ε ψm xψ n y 0 if m n = e βr m y, x dµ if m = n. Comparing equations 3.54 and 3.53, we have φ = φ ε. This shows that ε φ ε is surjective. To show the injectivity of ε φ ε, let φ ε1 = φ ε2 be two KMS β states. Suppose µ 1, µ 2 are probability measures such that φ ε1 ψ 0 a = a dµ 1 and φ ε2 ψ 0 a = a dµ 2 for all a A. Then µ 1 = µ 2. Now the construction of the previous paragraph shows that ε 1 = k 1 e βr i R e i µ 1 = k 1 e βr i R e i µ 2 = ε 2. i=1 Thus ε φ ε is one-to-one. Finally, to check the continuity of ε φ ε, suppose ε j ε in Σ β. Let µ := n N e βr n R n ε and µ k j := n N e βr n R n ε k j. Remember from the calculation 3.33 that e βr n R n ε ε CZ. CZ n N k i=1 It then follows µ j µ in weak topology. Now the formula 3.43 for φ ε shows that φ εj φ ε in weak topology. The next Corollary is a generalization of [1, Corollary 5.3] to the dynamical system OX, α. 59

68 Corollary Let h 1,..., h k be -commuting and surjective local homeomorphisms on a compact Hausdorff space Z. Let X be the associated product system over N k, as in Corollary For each 1 i k, take β ci as in 3.28 and suppose r 0, k has rationally independent coordinates. Define α : R Aut OX in terms of the gauge action γ by α t = γ e itr. If there is a KMS β state of OX, α, then there exists an 1 i k such that βr i β ci. Proof. Suppose φ is a KMS β state of OX, α. Aiming for a contradiction suppose that βr i > β ci for all 1 i k. Let q : N T X OX be the quotient map as in Lemma Then φ q is a KMS β state for the system N T X, α considered in Theorem Since r has rationally independent coordinates part b of Theorem gives a measure ε on Z such that f β dε = 1 and φ q = φ ε. Since f β 1, fβ dε = 1 implies that εz > 0. We temporarily set K := {1,..., k} and take an open cover {U l : 1 l d} of Z such that h e J Ul is injective for all J K and 1 l d. By applying [46, Lemma 4.32], we can find open cover {V l : 1 l d} for Z such that V l U l. Since εz > 0, there exists at least one l satisfying εv l > 0. Then we can find a function f CZ such that fz 0 for some z V l see [47, Lemma 2.12], for example. Next for each J K, take f J := f X ej We aim to set up a contradiction by showing that b := ψ 0 f 2 + J K and view f 2 as an element of A = CZ. 1 J ψ J f J ψ J f J belongs to ker q while φ ε = φ q does not vanish on it. Since the left action of f 2 on each fibre X ej for b shows that is implemented by the finite-rank operator Θ fj,f J, a routine calculation b = ψ 0 f 2 + = J K = J K J K 1 J +1 ψ 0 f J ψ e J Θ fj,f J J K 1 J ψ 0 f 2 ψ J ϕ ej f 2. Thus b belong to ker q because each summand does. 1 J ψ e J ϕ ej f J 2 Next we compute φ ε b using the measure µ in part b of Theorem 3.3.1: φ ε b = f 2 z dµz + fj 1 J e βr e J, f J z dµz J K 60

69 = f 2 z dµz + 1 J e βr e J J K h e J w=z f 2 w dµz. Using definition of R at equation 3.25 and the notation R e µ = µ, we have φ ε b = f 2 z dµz + 1 J e βr e J f 2 z dr e J µz = J K J K 1 J e βr e J f 2 z dr e J µz. This is precisely f 2 z dεz. It follows that φ ε b > 0, and we have a contradiction. Thus there should be at least one 1 i k satisfying βr i β ci. 61

70 3.4 KMS states at the critical inverse temperature In Theorem 3.3.1, we first chose an r N k and then characterised KMS states of the dynamical system N T X, α for β satisfying 3.55 β > r 1 i β ci for all 1 i k. Thus the range of possible inverse temperature is dependent on the choice of r N k. When r is a multiple of β c1,..., β ck, following the recent conventions for the higher rank graph algebras see [26, 28, 59, 60], we call the common value β c := r 1 i β ci the critical inverse temperature. In particular, we are interested in r := β c1,..., β ck which gives the critical inverse temperature β c = 1. In this case, we refer to the associated dynamics α : t γ e itr as the preferred dynamics. In the next theorem, we consider the preferred dynamics α and discuss the KMS states at the critical inverse temperature. Theorem is a generalization of [1, Theorem 6.1] and the proof follows a similar method. Theorem Let h 1,..., h k be -commuting and surjective local homeomorphisms on a compact Hausdorff space Z. Let X be the associated product system over N k as in Corollary For each 1 i k, let β ci be as in 3.28 and set r := β c1,..., β ck. Define α : R Aut N T X and α : R Aut OX in terms of the gauge actions by α t = γ e itr and α t = γ e itr. Then there is a KMS 1 state on N T X, α, and at least one such state factors through a KMS 1 state of OX, α. To prove this, we need the next lemma from [1]. Lemma [1], Lemma 6.2. Suppose A, R, α is a dynamical system, and J is an ideal in A generated by a set P of positive elements which are fixed by α. If φ is a KMS β state of A, α and φp = 0 for all p P, then φ factors through a state of A/J. Proof of Theorem Choose a decreasing sequence {β j } j N such that β j 1 and a probability measure ν on Z. Then K j := f βj dν belongs to [1,, and ε j := K 1 j ν satisfies f βj dε j = 1. Thus for each j, part a of Theorem gives a KMS βj state φ εj on N T X, α. Since {φ εj } j is a sequence in the compact unit ball of CZ, by passing to a subsequence and relabelling, we may assume that φ εj φ. Now [5, Proposition ] implies that φ is a KMS 1 state. φ for some state To show that at least one such state factors through OX, α, we apply the construction of the previous paragraph to a particular sequence of measures ε j. Take 62

71 one of the local homeomorphisms, for example h 1. Since for all d N, z h d 1 z is continuous see [8, Lemma 2.2], applying Proposition 2.3 of [18] gives u Z such that h d 1 u e dβc 1 for all d N. Now let δ u be the unit point mass at u, and take ε j = f βj u 1 δ u. The argument of the first paragraph gives a sequence of KMS βj to a KMS 1 state φ of N T X, α in the weak* topology. states on N T X, α which converges We aim to show that φ factors through OX, α. By Lemma 3.4.2, it suffices for us to prove that the generators of the kernel of the quotient map q : N T X OX are all positive, are fixed by α, and belong to ker φ. Remember from Lemma that ker q = ψ 0 a ψ n ϕ n a : n N k, a ϕ 1 n KX n. Fix n N k and a ϕ 1 n KX n. Let {Ul n}ln l=0 be an open cover of Z such that h n U n l is injective. Choose a partition of unity {ρ n l } subordinate to {U l n } and define τ n l := ρ n l. The argument of the last paragraph in the proof of Corollary shows that ϕ n a = L n l=0 Θ a τ n l,τ n l. Then L n ψ 0 a ψ n ϕ n a = ψ 0 a ψ n l=0 Θ a τ n l,τ n l L n = ψ 0 a ψ n a τl n ψ n τl n = ψ 0 a 1 l=0 L n l=0 ψ n τ n l ψ n τ n l. Thus the generators of ker q are of the form of 1 L n l=0 ψ nτ n l ψ nτ n l. Clearly they are fixed by α. Next we show that theses generators are positive. Writing T for the Fock representation, Lemma says that L n l=0 T nτl nt nτl n x is either zero or x for all x X. Therefore, L n 1 l=0 T n τ n l T n τ n l is positive in LF X. Since the induced homomorphism T : N T X LF X is injective, it follows that each generator 1 L n l=0 ψ nτ n l ψ nτ n l is positive in N T X. 2 As we mentioned in the proof of [1, Theorem 6.1], the results of [18] are mainly about metric spaces. But it seems that the argument for Proposition 2.3 in [18] does not need this hypothesis. 63

72 Now it remains to prove that 3.57 L n φ ψ n τl n ψ n τl n = 1. l=0 Let µ j be the measure m N k e β jr m R m ε j of Theorem 3.3.1a. We compute using the formula 3.43 for φ εj : L n Ln φ ψ n τl n ψ n τl n = lim φ εj ψ n τl n ψ n τl n j l=0 = lim e β jr n j Since h n is injective on supp τl n, we have L n l=0 L n τ n l, τl n z = l=0 h n w=z Ln l=0 τ n l wτ n l w = l=0 τ n l, τl n dµj z. L n h n w=z l=0 τ n l w 2 = h n w=z 1 = h n z. Thus e β jr n Ln l=0 τ n h l, τl n dµj z = e β jr n n z dµj z = m N k e β jr n+m h n z dr m ε j z. Using formula 3.25 for R m, we have e β jr n Ln l=0 τ n l, τl n dµj z = e βjr n+m m N k h m w=z h n w dεj z. Remember ε j = f βj u 1 δ u. Then e β jr n Ln l=0 τ n l, τl n dµj z = e βjr m+n h m+n u fβj u 1 m N k = m n e β jr m h m u fβj u 1. Since f βj u = m N k e β jr m h m u, e β jr n Ln l=0 τ n l, τl n dµj z = fβj u m<n e β jr m h m u f βj u. 64

73 Now to prove 3.57, it suffices to show that f βj u as j. Fix j. Since the inverse image h m u h m 1 u, we have f βj u m N k e β jr m h m 1 1 u. Recall that r = β c1,..., β ck. Since m N e βjr m = k k i=1 m i N e β jr i m i, we have [ f βj u m 1 N It follows from the equation 3.56 that e β jβ c1 m 1 h m 1 1 u ] k e β jβ ci m i. i=2 m i N [ ] k f βj u e β jβ c1 +β c1 m 1 e β jβ ci m i. m 1 N i=2 m i N Since β j > 1, all series in the right-hand side are convergent geometric series. Computing these series, we have f βj u 1 k 1 1 e β jβ c1 +β c1 1 e β jβ ci i=2 for all j. Now if j, the right hand side goes to infinity. Thus f βj u, as required. 3.5 Ground states and KMS states In this section we describe the ground states and KMS states of N T X, α. We first provide a characterization for the ground states in Lemma Then in Proposition we prove that there is a bijection between the simplex of the probability measures on Z and the ground states of N T X, α. We also show that every ground state on N T X, α is a KMS state. The following definition and remarks have been taken from [35, page 19]. Definition Let A, R, α be a dynamical system. Following [10], we say a state φ is a KMS state if it is the weak limit of a sequence of KMS βi states as β i. A state φ is said to be a ground state, if the entire functions z φaα z b are bounded on the upper half-plan for all analytic elements a, b. Remark Here we distinguish between ground states and the KMS states. But in older literature for example in [5, 41], there was not such a distinction. Considering our set-up, it follows from [10, Theorem ] that every KMS state is a ground state. But a ground state need not be a KMS state see for example [10, page 447] or [35, Theoerem 7.1]. 65

74 Remark Given a dynamical system A, R, α, [41, Proposition ] implies that it suffices to check the ground state condition on a set of analytic elements which span a dense subspace of A. Note that the definition of ground states in [41] is slightly different: A state φ is said to be ground state if all the functions z φaα z b are bounded by a b. But it is shown in the proof of 2 5 in [5, Proposition ] that an entire function which is bounded on the upper half-plane is bounded by the sup norm of its restriction to the real axis. Fortunately, in the dynamical system N T X, α, the sup norm of the restriction of the functions z φaα z b to the real line is bounded by a b. To see this, let a := ψ m xψ n y, b := ψ p sψ q t and φ be a state of N T X. Notice that for each t R φaαt b φ bαt a a b. Since φ is bounded linear functional, we can extend this to all of N T X. Thus any ground state in our set-up is a ground state of [41]. Now [41, Proposition ] implies that it is enough to check the ground state condition on a set of analytic elements which span a dense subspace of N T X. The following lemma is a generalisation of [28, Proposition 3.1c] and [27, Proposition 2.1b] in the dynamical systems of graph algebras. Lemma Let h 1,..., h k be -commuting and surjective local homeomorphisms on a compact Hausdorff space Z and let X be the associated product system as in Corollary Suppose r 0, k and α : R AutN T X is given in terms of the gauge action by α t = γ e itr. Suppose β > 0 and let φ be a state on N T X. Then φ is a ground state of N T X, α if and only if 3.58 φ ψ m xψ n y = 0 whenever r m > 0 or r n > 0. Proof. First notice that for every state φ, a + ib C and m, n, p, q N k, the definition of α implies that φ ψ m xψ n y α a+ib ψp sψ q t = e ia+ibr p q φ ψ m xψ n y ψ p sψ q t 3.59 Now suppose φ is a ground state. Then = e br p q φ ψ m xψ n y ψ p sψ q t. φ ψ m xα a+ib ψn y = e br n φ ψ m xψ n y, 66

75 is bounded on the upper half plane b > 0. Thus φ ψ m xψ n y = 0 whenever r n > 0. Since φ ψ n yψ m x = φ ψ m xψ n y, a symmetric calculation shows that φ ψ n yψ m x = 0 whenever r m > 0. Next suppose that φ satisfies It follows from Lemma that there exist {ξ i,j } 0 i,j d X m+p n p and {η i,j } 0 i,j d X q+n n p such that ψ m xψ n y ψ p sψ q t = Putting this in 3.59, we have 0 i,j d φ ψ m xψ n y α a+ib ψp sψ q t = e br p q φ ψ m+p n p ξ i,j ψ q+n n p η i,j. 0 i,j d ψ m+p n p ξ i,j ψ q+n n p η i,j. The assumption 3.58 implies that the right-hand side is zero consequently is bounded unless r m + p n p = 0 = r q + n n p. So suppose r m + p n p = 0 = r q + n n p. Since r 0, k, it follows that m + p n p = 0 = q + n n p. Then φ ψ m xψ n y α a+ib ψp sψ q t = e br p q 0 i,j d φ ψ 0 ηi,j, ξ i,j. Notice that q and n n p are both positive. Then q + n n p = 0 implies that q = 0. Now we have φ ψ m xψ n y α a+ib ψp sψ q t = e br p 0 i,j d φ ψ 0 ηi,j, ξ i,j. Thus φ is bounded on the upper half plane b > 0, and hence it is a ground state. The next Proposition is an extension of [28, Proposition 8.1] and [27, Proposition 5.1] from dynamical systems of graph algebras to the dynamical system N T X, α. Proposition Let h 1,..., h k be -commuting and surjective local homeomorphisms on a compact Hausdorff space Z and let X be the associated product system as in Corollary Suppose r 0, k and α : R AutN T X is given in terms of the gauge action by α t = γ e itr. For each probability measure ε on Z there is a unique KMS state φ ε such that 67

76 3.60 φ ε ψm xψ n y y, x dε if m = n = 0 = 0 otherwise. The map ε φ ε is an affine isomorphism of the simplex of probability measures on Z onto the ground states of N T X, α, and that every ground state of N T X, α is a KMS state. Proof. Suppose ε is a probability measure on Z. For each 1 i k, let β ci be as in Choose a sequence {β j } j N such that β j and each β j > max i r 1 i β ci. For each β j, let f βj z be the function in Proposition 3.28 a and set K j := f βj dε. Then K j belongs to [1,, and ε j := K 1 j ε satisfies f βj dε j = 1. Now part a of Theorem gives a KMS βj state φ εj on N T X, α. Since {φ εj } j is a sequence in the compact unit ball of CZ, by passing to a subsequence and relabelling, we may assume that φ εj φ ε. Then φ ε is by definition a KMS state. We now show that φ ε satisfies For each φ εj, we have φ εj ψm xψ n y = δ m,n e β jr n y, x dε j for all m, n N k. Thus φ εj ψm xψ n y = 0 for m n and hence φ ε ψm xψ n y = 0 if m n. So we suppose that m = n. If n 0, then r 0, k implies that e βjr n 0, and again φ ε ψm xψ n y = lim j φ εj ψm xψ n y = 0. So we assume that m = n = 0. Fix z Z and let f βj z = p N e βjr p h p z as in Proposition 3.2.7a. We k first show that f βj z 1 as j. For each p N k let 1 if p = 0 gp = 0 if p 0. Clearly e βjr p h p z gp as j. Since for each j, e βjr p h p z is dominated by e β0r p h p z, the dominated convergence theorem implies that f βj z = e βjr p h p z gp = 1, p N k p N k as j. Also notice that each f βj is dominated by f β0 and ε is a probability measure. Then another application of dominated convergence theorem implies that K j = f βj dε 1 dε = 1 68

77 as j. We now compute using the formula 3.43 for φ εj : φ ε ψm xψ n y = lim φ εj ψm xψ n y j = lim y, x dε j. j Since ε j = K 1 j ε, φ ε ψm xψ n y = lim j K j 1 y, x z dεz = y, x z dεz. Thus φ satisfies Since φ ε ψm xψ n y vanishes for all m 0 or n 0, it also does for r m 0 or r n 0. Then Lemma says that φ ε is a ground state. Next let φ be a ground state and suppose that ε is the probability measure satisfying φψ 0 a = a dε for all a A. Then the formulas 3.58 and 3.60 for φ and φ ε imply that φ = φ ε. Thus ε φ ε maps the simplex of the probability measures of Z onto the ground states, and it is clearly affine and injective. Since each φ ε is by construction a KMS state, it follows that every ground state is a KMS state. 69

78

79 Chapter 4 The shifts on the infinite path spaces of 1-coaligned higher rank graphs In this chapter we consider a special type of k-graph called 1-coaligned k-graph. The shift maps on the infinite path space of this kind of graph -commute. So by Corollary we have a product system over N k. We study the relationships between the C -algebras associated to this product system and the C -algebras of the k-graph. Then we use our results in Chapter 3 and some others from the literature to compare the KMS states of these C -algebras. 4.1 Basics of Higher-rank graphs Most of the following definitions have been taken from [44, Chapter 10] and [30]. A countable category C consists of two countable sets C 0 and C, two functions r C, s C : C C 0, a partially defined product f, g fg from {f, g C C : s C f = r C g} to C, and an injective map id : C 0 C, which satisfies a r C fg = r C f and s C fg = s C g, b fgh = fgh when s C f = r C g and s C g = r C h, c r C idv = v = s C idv for all v C 0, and 71

80 d idvf = f and g idv = g when r C f = v and s C g = v. The elements of C 0 are called the objects of the category, the elements of C are called the morphisms of the category, r C is the range map, s C is the source map, the operation f, g fg is called composition, and idv is called the identity morphism on the object v. When it is clear from the context we may write r, s for r C, s C. Example Let k N. We can view N k as morphisms of a countable category with a single object {v}. For each m, n N k, we can define rn := v, sn := v, mn := m + n, and idv := 0. Suppose that C and D are two countable categories. A functor F : C D is a pair of maps F 0 : C 0 D 0 and F : C D such that a F 0 r C f = r C F f and F 0 s C f = s C F f for all f C, b F fg = F gf f for all f, g C C, and c idf 0 v = idf 0 v for all v C 0. Definition Let k N \ {0}. A k-graph Λ, d consists of a countable category Λ and a functor d from Λ to N k view N k as the category of Example satisfying the factorization property: For all λ Λ and m, n N k such that d λ = m + n, there exist unique elements µ Λ and ν Λ such that λ = µν. Since the category N k has only one object, the map d 0 of the functor d is trivial. So we write d for both d and d 0 and call it the degree map. We usually use Λ for Λ, d. Let Λ be a k-graph. For any n N k, we define Λ n := {λ Λ : dλ = n} and we say Λ is a finite k-graph if Λ n is finite for all n N k. We say Λ has no sinks if for every v Λ 0 and every n N k, there is a λ Λ such that sλ = v and dλ = n. Similarly, Λ has no sources if for every v Λ 0 and every n N k, there is a λ Λ such that rλ = v and dλ = n. For µ, ν Λ, we write Λ min µ, ν for the set of ξ, η Λ Λ such that µξ = νη and dµξ = dµ dν. Given v, w Λ 0, vλ n w denotes the {λ Λ n : rλ = v and sλ = w}. For 1 i k, let A i be the matrix in M Λ 0N with entries A i v, w = vλ e i w. We call the A i 1 i k the vertex matrices. Notice that A i A j v, w = vλ e i+e j w. Then the factorisation property in Λ implies that A i A j = A j A i, and therefore we can define A n := k i=1 An i i for all n N k. 72

81 Example Let Ω k be the category with objects Ω 0 k = Nk, morphisms Ω k : {m, n N k N k : m n}, range and source maps rm, n = m, sm, n = n, identity morphisms idm = m, m, and the composition m, nn, p = m, p. If we equip the category Ω k with the degree map dm, n = n m, then Ω k, d becomes a k-graph. Let Λ 1, d 1 and Λ 2, d 2 be two k-graphs. A k-graph morphism is a functor F form the category Λ 1 to the category Λ 2 preserving the degree maps, in the sense that d 2 F = d 1. For a k-graph Λ, we refer to the infinite path space of Λ as Λ := {z : Ω k Λ : z is a k-graph morphism}. For p N k, we define the shift map σ p : Λ Λ by σ p zm, n = zm + p, n + p for all z Λ and m, n Ω k. Clearly σ p σ q = σ p+q = σ q σ p. Notice that for every z Λ and p N k we have 4.1 z = z0, pσ p z. For each λ Λ, let Zλ := {z Λ : z0, dλ = λ}. Endow Λ with the topology generated by the collection {Zλ : λ Λ}. For finite Λ, [30, Lemma 2.6] shows that Λ is compact in this topology. For each p N k, [30, Remark 2.5] implies that the shift map σ p is a local homeomorphism on Λ C -algebras associated to higher rank graphs Definition Let Λ be a finite k-graph. Following [27, 45], we say a collection of partial isometries {S λ : λ Λ} in a C -algebra B forms a Toeplitz-Cuntz-Krieger Λ-family if TCK1 {S v : v Λ 0 } is a collection of mutually orthogonal projections, TCK2 S λ S µ = S λµ whenever sλ = rµ, TCK3 Sλ S λ = S sλ for all λ, TCK4 for all v Λ 0 and n N k, we have S v S λ Sλ, λ vλ n 73

82 TCK5 for all µ, ν Λ, we have SµS ν = S ξ Sη. ξ,η Λ min µ,ν They form a Cuntz-Krieger Λ-family if they also satisfy CK S v = λ vλ S n λ Sλ for all v Λ0 and n N k. We interpret any empty sums as 0. Remark Conditions TCK1 TCK3 and CK implies TCK5 see [30, Lemma 3.1]. Then to see that a family of partial isometries is a Toeplitz-Cuntz- Krieger Λ-family, we can either check TCK1 TCK5 or check TCK1 TCK4 together with CK. The next lemma shows that it suffices to check TCK5 for a subset of {S λ : λ Λ}. Lemma Let Λ be a finite k-graph. Suppose that {S λ : λ Λ} is a collection of partial isometries in a C -algebra B which satisfies TCK1 TCK3. Suppose that for all µ, ν Λ with dµ dν = 0 we have SµS ν = ξ,η Λ min µ,ν S ξsη. Then {S λ : λ Λ} satisfies TCK5. Proof. Fix µ, ν Λ. By the factorisation property we can write µ = µ µ and ν = ν ν such that dµ = dν = dµ dν, dµ = dµ dµ dν, and 4.2 dν = dν dµ dν. Now using TCK1 TCK3 and the identity S rλ S λ = S λ, we have SµS ν = Sµ S µ S ν S ν = Sµ δ µ,ν S sµ S ν = δ µ,ν S µ S rµ S ν since sµ = rµ = δ µ,ν S µ S ν. Since dµ dν = 0, applying TCK5 for µ, ν gives 4.3 SµS ν =δ µ,ν S ξ Sη. ξ,η Λ min µ,ν 74

83 Next we aim to show that 4.4 µ = ν and ξ, η Λ min µ, ν ξ, η Λ min µ, ν. Suppose µ = ν and ξ, η Λ min µ, ν. Then µ ξ = ν η implies that µξ = νη. Since dµ dν = 0, it follows from dµ ξ = dµ dν that dµ ξ = dµ + dν and hence dξ = dν. Now 4.2 implies that dµξ = dµ + dν = dµ + dν dµ dν = dµ dν. Thus ξ, η Λ min µ, ν. Next let ξ, η Λ min µ, ν. Since µξ = νη, the factorization property implies that µ = ν. Notice that dµ ξ = dµξ dµ = dµ dν dµ = dµ + dν dµ dν dµ. Now 4.2 implies that dµ ξ = [dµ dµ dν] + [dν dµ ] = dµ + dν. Thus ξ, η Λ min µ, ν. Next we finish off by putting 4.4 in 4.3. We have S µs ν = which is precisely TCK5 for µ, ν. ξ,η Λ min µ,ν S ξ S η, Kumjian and Pask showed in [30] that for a finite k-graph Λ, there is a C -algebra C Λ and a Cuntz-Krieger Λ-family {t λ : λ Λ} on C Λ such that U1 For any other Cuntz-Krieger Λ-family {T λ : λ Λ} in a C -algebra B, there exists a unique homomorphism π T : C Λ B such that π T t λ = T λ. U2 C Λ is generated by {t λ : λ Λ}. We say the pair C Λ, t λ is universal for Cuntz-Krieger Λ-families. The C - algebra C Λ is called the C -algebra of Λ and the family {t λ : λ Λ} is called a universal Cuntz-Krieger Λ-family. The universal property shows that there exists a strongly continuous gauge action γ : T k AutC Λ such that γ z t λ = z dλ t λ in multi-indexed notation, so that 75

84 z n = k i=1 zn i i for z = z 1..., z k T k and n Z k. It also follows from [30, Lemma 3.1] that C Λ = span{t λ t µ : sλ = sµ}. Raeburn and Sims showed in [45, Corollary 7.5] that there exists a C -algebra T C Λ and a Toeplitz-Cuntz-Krieger Λ-family {s λ : λ Λ} on T C Λ such that T C Λ, s λ is universal for Toeplitz-Cuntz-Krieger Λ-families. We call T C Λ the Toeplitz-Cuntz-Krieger algebra and call {s λ : λ Λ} a universal Toeplitz-Cuntz-Krieger Λ-families. γ : T k The universal property shows that there is a strongly continuous gauge action AutT C Λ such that γ z s λ = z dλ s λ using multi-indexed notation. Furthermore, by a standard argument and using T CK5, we can show that see [51, Lemma 3.1.2, Proposition 3.2.1]. T C Λ = span{s λ s µ : λ, sλ = sµ}. Remark We can lift the gauge actions of T C Λ and C Λ to actions of R via the maps t γ e itr and t γ e itr for some r 0, k. Notice that for each s λ s µ T C Λ, the function t γ e itr sλ s µ = e itr dµ dν s λ s µ on R extends to an entire function on all of C. Thus s λ s µ is an analytic element of T C Λ. The elements s λ s µ span a dense subalgebra of T C Λ. So when we study the KMS states of the system T C Λ, γ e itr, it suffices to check KMS condition on these elements. Similarly, we can show that {t λ t µ : sλ = sµ} spans a dense subspace of analytic elements of the system C Λ, γ e itr. The next lemma shows that we can view C Λ as a quotient of T C Λ. Lemma Let Λ be a finite k-graph. Suppose I is the ideal in T C Λ generated by {s v = λ vλ n s λ s λ, v Λ 0, n N k and let q : T C Λ T C Λ/I be the quotient map. Then T C Λ/I, qs λ is universal for Cuntz-Krieger Λ-families, and is canonically isomorphic to C Λ, t λ. Proof. Since q is a homomorphism and {s λ : λ Λ} satisfy TCK1 TCK3, the family {qs λ : λ Λ} satisfies TCK1 TCK3 as well. Clearly {qs λ : λ Λ} satisfies CK. Since {s λ : λ Λ} generates T C Λ, we have that {qs λ : λ Λ} generates T C Λ/I. 76 },

85 To see U2, suppose that {T λ : λ Λ} is another Cuntz-Krieger Λ-family, in a C - algebra B. Notice that {T λ : λ Λ} is in particular a Toeplitz-Cuntz-Krieger Λ-family. Then the universal property of the pair T C Λ, s λ gives a unique homomorphism π T : T C Λ B such that π T s λ = T λ. Notice that {T λ : λ Λ} satisfies CK. Then we can descends π T to a homomorphism of T C Λ/I such that π T qs λ = T λ for all λ Λ coaligned higher rank graphs and the associated C -algebras Definition [37, Definition 2.2]. A k-graph Λ is 1-coaligned if for all 1 i j k and λ, µ Λ e i Λ e j such that ηλ = ζµ. with sλ = sµ there exists a unique pair η, ζ Λ e j Λ e i It is observed in [37, Theorem 2.3] that a k-graph Λ is 1-coaligned if and only the shift maps σ e 1,..., σ e k on the infinite path space Λ -commute. Let Λ be a 1-coaligned k-graph and let XΛ be the product system associated to σ e 1,..., σ e k as in Corollary We write N T XΛ and OXΛ for the Nica-Toeplitz algebra and the Cuntz-Pimsner algebra of XΛ. In this section, we show that the Cuntz-Pimsner algebra OXΛ is isomorphic to the Cuntz-Krieger algebra C Λ and the Nica-Toeplitz algebra N T XΛ contains an isomorphic copy of the Toeplitz Cuntz-Krieger algebra T C Λ. The next lemma is contained in [37, Theorem 2.3]; since [37, Theorem 2.3] has not been published, we provide a brief proof here. Lemma Let Λ be a finite 1-coaligned k-graph. Suppose 0 i j k. Then the shift maps σ e i and σ e j -commute. Proof. Let w, z Λ such that 4.5 σ e i z = σ e j w. It follows from 4.1 that z = z0, e i σ e i z and w = w0, e j σ e j w. Now equation 4.5 implies that z0, e i and w0, e j have the same sources. Since Λ is 1-coaligned there exists a unique pair η, ζ Λ e j Λ e i such that ηz0, e i = ζw0, e j. Let λ be the element of Λ e i+e j identified by ηz0, e i or ζw0, e j, then u := λσ e i z Λ satisfies σ e j u = z and σ e i u = w. 77

86 Since λ is determined uniquely, so is u. Thus σ e i and σ e j -commute. Notation Let Λ be a finite 1-coaligned k-graph with no sinks. Then the shift maps σ e 1,..., σ e k are surjective -commuting maps. As we mentioned before, we write XΛ for the product system associated to σ e 1,..., σ e k. We use ψ for the universal Nica-covariant representation. We write X m Λ for the fibre associated to m N k. We write ϕ m for the left action of A on the fibre X m Λ. Recall that the multiplication formula in XΛ is 1 xyz = xzyσ m z for x X m, y X n, z Λ. In this section we work with four C -algebras: N T XΛ, OXΛ, T C Λ, and C Λ. All of these C -algebras carry a gauge action of T k. To avoid possible clash of notation, we continue to write γ and γ for the gauge actions on N T XΛ and OXΛ, respectively. We write γ and γ for the actions on T C Λ and C Λ, respectively. Lemma Let Λ be a finite 1-coaligned k-graph with no sources. Suppose λ Λ m and µ Λ n such that m n = 0 and sλ = sµ. Then there exists a unique pair η, ξ Λ Λ such that ηλ = ξµ. Proof. We first show that there is such a pair η, ξ Λ Λ. Since Λ has no source, there exists z Λ such that z0, 0 = sλ. Let w := µz and w := λz. Notice that σ n w = z = σ m w. Since m n = 0, Corollary implies that σ m and σ n are -commuting. Then there exists unique w Λ such that w = σ m w and w = σ n w. Now let η := w0, n and ξ := w0, m. Clearly ηλ = ξµ. The uniqueness of pair η, ξ follows from the uniqueness of w. Remark We could have proved the Lemma for a finite 1-coaligned k-graph with sources by the way of induction. But all the k-graphs that we work with have no sources and with this hypothesis the proof of Lemma is easier. Lemma Let Λ be a finite k-graph and suppose m, n N k. Then the collection {χ Zµ } µ Λ m+n is a partition of unity such that σ m supp χzµ and σ n supp χzµ are injective for all µ Λ m+n. 1 In previous chapters we wrote the multiplication in terms of isomorphisms between fibres. For example xyz = σx yz. Unfortunately in this chapter we use letter σ for shifts. Thus here we do not use σ when writing products. 78

87 Proof. Fix m, n N k. Remark 2.5 in [30] says that, the sets {Zµ : dµ = m + n} form a partition of Λ. Then {χ Zµ } µ Λ m+n is a partition of unity. Fix µ Λ m+n. To see that σ m supp χzµ is injective, let σ m w = σ m z for w, z supp χ Zµ. Notice that w0, m = µ0, m = z0, m. On the other hand, 4.1 implies that w = w0, mσ m w and z = z0, mσ m z. Comparing these equations, we deduce that w = z. Thus σ m supp χzµ is injective. A similar argument shows that σ n supp χzµ is injective as well. Proposition Let Λ be a finite 1-coaligned k-graph with no sinks or sources. For each λ Λ, let S λ := ψ dλ χ Zλ. Then a The elements {S λ } λ Λ form a Toeplitz-Cuntz-Krieger Λ-family in N T XΛ. Then the corresponding homomorphism π S : T C Λ N T XΛ is injective and intertwines the respective gauge actions of T k in the sense that π S γ = γ π S. b Let q : N T XΛ OXΛ be the quotient map as in Lemma Then {q S λ } λ Λ is a Cuntz-Krieger Λ-family in OXΛ. The corresponding homomorphism π q S : C Λ OXΛ is an isomorphism and intertwines the respective gauge actions of T k. Proof of a. Let λ Λ. Notice that χ Zλ X dλ Λ. We will need the next formula is our proof: χzλ, χ Zλ z = χ Zλ wχ Zλ w σ dλ w=z { = w : σ dλ w = z and w Zλ} 0 if z / Z sλ = 1 if z Z sλ 4.6 = χ Zsλ z. Next we show that S λ is a partial isometry: S λ SλS ψdλ λ = ψ dλ χzλ ψdλ χzλ χzλ = ψ dλ χzλ ψ0 χzλ, χ Zλ = ψ dλ χzλ χ Zsλ by 4.6. Now the calculation χ Zλ χ Zsλ z = χ Zλ zχ Zsλ σ dλ z = χ Zλ z for z Λ, 79

88 implies S λ Sλ S λ = ψ dλ χ Zλ = S λ. Thus S λ is a partial isometry. Next we aim to check properties TCK1 TCK5. To see TCK1, let v Λ 0. Since ψ 0 is a homomorphism, we have Sv = ψ dv χzv = ψ0 χzv = ψ0 χ Zv = ψ 0 χ Zv = S v. Similarly, S v S v = ψ 0 χzv ψ0 χzv = ψ0 χzv χ Zv = ψ0 χzv = Sv. So S v is a projection. Now let v, w Λ 0. We have S v S w = ψ 0 χzv ψ0 χzw = ψ0 χzv χ Zw = δ w,v ψ 0 χzv = δw,v S v, which implies that the collection {S v : v Λ 0 } are mutually orthogonal projections. To check TCK2, let λ, µ Λ such that sλ = rµ. We have χzλ S λ S µ = ψ dλ χzλ ψdµ χzµ = ψdλµ χzµ. The multiplication formula in XΛ for χ Zλ X dλ Λ and χ Zµ X dµ Λ implies that χzλ χzµ z = χ Zλ zχ Zµ σ dλ z = χ Zλµ z. Then S λ S µ = ψ dλµ χzλµ = Sλµ. To check TCK3, let λ Λ. A routine calculation shows that SλS ψdλ λ = ψ dλ χzλ χzλ = ψ 0 χzλ, χ Zλ = χ Zsλ by 4.6 = S sλ. We will need TCK5 for the proof of TCK4. So we first check TCK5. Lemma says that it suffices to prove TCK5 for µ, ν Λ with dµ dν = 0. For convenience, let m := dν and n := dµ. Let {χ Zξ } ξ Λ m+n be the partition of unity from lemma Applying Proposition to {χ Zξ } ξ Λ m+n gives SµS ν = ψ n χ Zµ ψ m χ Zν = ψ 0 χzµ, χ Zη σ m ψ m χ Zξ ψ n χ Zη ψ 0 χzξ σ n, χ Zν ξ Λ m, η Λ n 80

89 4.7 = ξ Λ m, η Λ n ψ m χzµ, χ Zη σ m χ Zξ ψn χzξ σ n, χ Zν χzη We now consider a summand for fixed ξ and η. We have χzµ, χ Zη σ m χ Zξ z = χ Zµ, χ Zη σ m zχ Zξ z = χ Zξ z χ Zµ wχ Zη σ m w σ n w=z 1 if z Zξ, and µξ = αη for some α Λ m = 0 otherwise χ Zξ if µξ = αη for some α Λ m = 0 otherwise. Similarly χzξ σ n, χ Zν χzη z = χ Zξ σ n, χ Zν zχzη z = χ Zη z χ Zξ σ n w χ Zν w σ m w=z It then follows that the ξ-η summand vanishes unless 1 if z Zη, and νη = βξ for some β Λ n = 0 otherwise χ Zη if νη = βξ for some β Λ n = 0 otherwise. µξ = αη and νη = βξ. This means ξ and η must have the same source. Since Λ is 1-coaligned and dξ dη = 0 note that dξ = m and dη = n, Lemma implies that α = ν and β = µ. Thus the sum in 4.7 collapses to SµS ν = = ξ,η Λ min µ,ν ξ,η Λ min µ,ν which completes our proof of TCK5. ψ m χzξ ψn χzη S ξ S η, To see TCK4, let v Λ 0 and n N k. Suppose that λ, µ vλ n and λ µ. It follows from dλ = dµ that Λ min λ, µ =. Now TCK5 implies that S λ S λs µ S µ = 0. 81

90 Thus S λ Sλ S µsµ. It follows from [44, Corollary A.3] that λ vλ S n λ Sλ is a projection. Thus to check S v λ vλ S n λ Sλ, it suffices to prove S v S λ Sλ = 4.8 S λ Sλ = S λ Sλ S v. λ vλ n λ vλ n λ vλ n For the first equality, we have S v S λ Sλ = S v S λ Sλ λ vλ n λ vλ n = λ vλ n ψ 0 χzv ψdλ χzλ ψdλ χzλ. Since rλ = v, a quick calculation shows that the left action of χ Zv on χ Zλ is χ Zλ. It then follows S v λ vλ n S λ S λ = λ vλ n ψ dλ χzλ ψdλ χzλ = Similarly, the second equation in 4.8 follows from λ vλ n S λ S λ S v = λ vλ n S λ S λs v = λ vλ n S λ S λ. λ vλ n ψ dλ χzλ ψdλ χzλ ψ0 χzv. Since rλ = v, we again have χ Zv χ Zλ = χ Zλ. Then λ vλ n S λ S λ S v = λ vλ n ψ dλ χzλ ψdλ χzλ = λ vλ n S λ S λ. We now have proved TCK4 and therefore the collection {S λ } λ Λ forms a Toeplitz- Cuntz-Krieger Λ-family in N T XΛ. To see that the corresponding homomorphism π S is injective, by [45, Theorem 8.1], it suffices to check S v λ vλ n S λ S λ for all v Λ 0 and n N k +. To do this, we use the Fock representation T of XΛ. Notice that the homomorphism T : N T XΛ LF XΛ satisfies T S v S λ Sλ = T ψ 0 χ Zv ψ dλ χ Zλ ψ dλ χ Zλ λ vλ n λ vλ n = T 0 χ Zv T dλ χ Zλ T dλ χ Zλ. λ vλ n 82

91 Now the adjoint formula 1.17 for the Fock representation says that T dλ χ Zλ vanishes on the 0-summand in Fock module F XΛ. Notice that Λ has no sources, and then the injectivity of T 0 implies that T 0 χ Zv 0. Thus T 0 χ Zv λ vλ n T dλ χ Zλ T dλ χ Zλ. Another application of the injectivity of T gives S v λ vλ n S λ S λ. Finally, since the gauge actions in T C Λ and N T XΛ satisfy γ z s λ = z dλ s λ and γ z ψ m x = z m ψ m x, we have π S γ = γ π S. Proof of b. By Remark 4.1.5, we must check the conditions TCK1 TCK3 and CK. Since the quotient map q is a C -homomorphism, and the family {S λ } λ Λ satisfies TCK1 TCK3, so does {q S λ } λ Λ. To check CK, notice that q ψ is the universal Cuntz-Pimsner-covariant representation of XΛ. For convenience let ρ := q ψ then the restriction ρ on each fibre X n is ρ n = q ψ n. Let µ Λ n, n N k. We first show that the left action of χ Zµ on the fibre X n is by the finite rank operator Θ χzµ,χ Zµ. To see this take x X n Λ and z Λ. We have Θ χzµ,χ Zµ x z = χ Zµ χ Zµ, x z and this vanishes unless z, w Zµ. = χ Zµ z χ Zµ, x σ n z = χ Zµ z χ Zµ wxw, σ n w=σ n z Since µ Λ n, w, z Zµ, the equation σ n w = σ n z has unique solution z and therefore the sum collapses to χ Zµ zxz. Thus 4.9 Θ χzµ,χ Zµ x z = χ Zµ zxz, which equals the left action of χ Zµ on x X n. that Next we check CK. Let v Λ 0 and n N k. Then a routine calculation shows q S λ q S λ = ρ dλ χ Zλ ρ dλ χ Zλ λ vλ n λ vλ n = ρ dλ Θ χzλ,χ Zλ λ vλ n = λ vλ n ρ dλ ϕ dλ χ Zλ by

92 Since ρ is Cuntz-Pimsner-covariant, λ vλ n q S λ q S λ = λ vλ n ρ 0 χ Zλ = ρ 0 = q ψ 0 χ Zv = q S v λ vλ n χ Zλ Thus CK holds and the collection {q S λ } λ Λ forms a Cuntz-Krieger Λ-family in OXΛ. This gives a homomorphism π q S : C Λ OXΛ. Since the gauge actions in C Λ and OXΛ satisfy γ z s λ = z dλ s λ γ z ρ m x = z m ρ m x, we have π q S γ = γ π q S. Thus π q S intertwines the gauge actions. Notice that Λ has no source. Since ρ 0 is injective see for example [52, Lemma 3.15], ρ 0 χ Zv 0 for all v Λ 0. Then the gauge-invariant uniqueness theorem see [30, Theorem 3.4] implies that π q S is injective. To show that π q S is surjective, note that OXΛ is generated by ρxλ. We know from the Stone-Weierstrass theorem that the set {χ Zλ : λ Λ} spans a dense -subalgebra of CΛ. Since the norm of XΛ is equivalent to see argument in the end of the Section 1.5, the elements {χ Zλ : λ Λ} span a dense subspace of XΛ. Thus it is enough for us to show that ρ m χ Zµ lies in the range of π q S for all m, n N k and µ Λ n. We first check this for m = 0 and all µ Λ n. Since ρ is Cuntz-Pimsner-covariant, a routine calculation shows that 4.10 ρ 0 χ Zµ = ρ dµ ϕ dµ χ Zµ = ρ dµ Θ χzµ,χ Zµ = ρ dµ χ Zµ ρ dµ χ Zµ = q S µ q S µ, which belongs to the range of π q S. using 4.9 Now let m 0 and take µ Λ n. Notice that χ Zµ = ν sµλ χ m Zµν. Each ν-summand is the pointwise multiplication of χ and χz σ m. This Z µν0,m µνm,m+n is exactly the right action of χ on χz Xm Λ. It follows Z µνm,m+n µν0,m ρ m χ Zµ = ρ m χ χz Z µν0,m ν sµλ m = χ ρ 0 χ Z µν0,m Z ν sµλ m ρ m 84 µνm,m+n µνm,m+n and

93 = ν sµλ m q S µν0,m ρ 0 which lies in the range of π q S by 4.10, as required. χ, Z µνm,m+n 4.3 KMS states on the Toeplitz algebras In this section we want to see the relationship between KMS states of the C -algebras T C Λ and N T XΛ. The KMS states of T C Λ is described thoroughly in [27, Theorem 6.1]. We apply Theorem to characterise KMS states of N T XΛ. It follows from [1, Proposition 7.3] that for the shift maps σ e i 1 i k on Λ, each β ci in Theorem is exactly ln ρa i used in [27, Theorem 6.1]. Thus the range of possible inverse temperatures studied in Theorem is the same as that of [27, Theorem 6.1]. Now when we view T C Λ as a C -subalgebra of N T XΛ, restricting KMS states of N T XΛ gives KMS states of T C Λ with the same inverse temperature. We expect from our results in [1, Corollary 7.6] to see that for the common inverse temperatures described in Theorem and [27, Theorem 6.1] all KMS states of T C Λ arise as restrictions of KMS states of N T XΛ. We achieve this objective in Proposition We keep our notation in Theorem to emphasise the parallels with [27, Theorem 6.1]. Then we have a clash when we try to use both descriptions at the same time. So we write δ for the measure ε in Theorem 3.3.1, and choose ε for the vectors in [1, Λ0 appearing in [27, Theorem 6.1]. We also choose α for the action of N T XΛ and write α for the action of T C Λ. Otherwise, we use the notation of Theorem Proposition Suppose that Λ is a finite 1-coaligned k-graph with no sources and no sinks. Let A i be the vertex matrices of Λ. Suppose that r 0, k satisfies βr i > ln ρa i for 1 i k. Let α : R AutN T XΛ and α : R AutT C Λ be given in terms of the gauge actions by α t = γ e itr and α t = γ e itr. Let δ be a finite regular Borel measure on Λ such that f β dδ = 1. Define ε = ε v [0, Λ0 by ε v = δzv and take y = y v [0, Λ0 as in [27, Theorem 6.1]. Then y ε = 1, and the restriction of the state φ δ of Theorem to T C Λ, α is the state φ ε of [27, Theorem 6.1]. Proof. We first compute the function f β CΛ. For z Λ, we have f β z = e βr n σ n z n N k 85

94 = e βr n Λ n rz n N k = e βr n Λ n v χ Zv z. n N k v Λ 0 Recall that y v = µ Λv e βr dµ. By applying the Tunelli theorem, we have = f β dδ = e βr n Λ n v δzv = y v ε v = y ε. n N k v Λ 0 v Λ 0 To see that φ δ restricts to φ ε, it suffices to compute both of them on the elements S λ S ν. Equation 3.43 together with [27, 6.1] imply that φ δ S λ S ν = 0 = φ ε S λ S ν for dλ dν. So we assume dλ = dν = p say. It then follows from 3.43 that 4.12 φ δ S λ S ν = φ δ ψ p Zλψ p Zν = e βr p χzν, χ Zλ dµ, where µ = n N k e βr n R n δ. Applying the inner product formula in the fibre X p, we have χzν, χ Zλ z = σ p w=z χ Zν wχ Zλ w = δ λ,ν σ p w=z χ Zλ w. It then follows that χ Zν, χ Zλ = δ λ,ν χ Zsλ. Putting this in 4.12, we have 4.13 φ δ S λ S ν = δ λ,ν e βr p µ Zsλ Next we compute µzv for v Λ 0. Notice that for each n, we have R n δzv = χ Zv dr n δz = χ Zv w dδz. We also have σ n w=z σ n w=z χ Zv w = vλ n rz = A n v, rz = u Λ 0 A n v, uχ Zu z. Thus and R n δzv = A n v, uχ Zu z dδz = A n v, uδzu, u Λ 0 u Λ 0 µzv = n N k e βr n u Λ 0 A n v, uδzv = n N k e βr n u Λ 0 A n v, uε v 86

95 Now we put this into 4.13, and write down = k e βr n A n ε v = 1 e βr i A i 1 ε. n N k k 4.14 φ δ S λ Sν = δ λ,ν e βr p 1 e βr i A i 1 ε i=1 which in the notation of [27, Theorem 6.1] is δ λ,ν e βr p m sλ. Now [27, 6.1] implies that φ δ S λ Sν = φ ε S λ Sν, as required. i=1 sλ, v Corollary Suppose that Λ is a finite 1-coaligned k-graph with no sources and no sinks. Let A i be the vertex matrices of Λ. Suppose that r 0, k satisfies βr i > ln ρa i for 1 i k and let α : R AutN T XΛ and α : R AutT C Λ be given in terms of the gauge actions by α t = γ e itr and α t = γ e itr Suppose that δ 1, δ 2 are regular Borel measures on Λ satisfying f β dδ i = 1. Then φ δ1 T C Λ = φ δ2 T C Λ if and only if δ 1 Zv = δ 2 Zv for all v Λ 0. Proof. Let δ 1, δ 2 be two regular Borel measures on Λ such that f β dδ i = 1. Suppose φ δ1 T C Λ = φ δ2 T C Λ. Proposition implies that for the corresponding ε i [0, Λ0 where ε i v = δ i Zv for all v Λ 0 we have φ ε1 = φ ε2. Now the injectivity of the map ε φ ε from [27, Theorem 6.1c] gives ε 1 = ε 2. But this says precisely that δ 1, δ 2 agree on each Zv. For the other direction, let δ 1 Zv = δ 2 Zv for all v Λ 0. Then the corresponding ε i are equal, and the formula 4.14 implies that φ δ1, φ δ2 agree on T C Λ. Proposition Suppose that Λ is a finite 1-coaligned k-graph with no sources and no sinks. Let A i be the vertex matrices of Λ. Suppose that r 0, k satisfies βr i > ln ρa i for 1 i k and let α : R AutN T XΛ and α : R AutT C Λ be given in terms of the gauge actions by α t = γ e itr and α t = γ e itr. Then every KMS β state of T C Λ, α is the restriction of a KMS β state of N T XΛ, α. Before starting the proof, we first describe a standard way of construction of measures on Λ. We need the notion of inverse limit see for example [9, Section 1, 2]: Let P be a directed partially ordered set. An inverse system of compact spaces {Y p }, {r p,q } p,q P consists of a family {Y p } p P of compact spaces such that for any p, q P, p q there exists a surjection r p,q : Y q Y p such that a r p,p : Y p Y p is the identity map, and 87

96 b r p,q r q,s = r p,s whenever p q s and p, q, s P. The inverse limit limy p, r p,q is the set of all collections {y p : y p Y p, p P } such that for p q, r p,q y q = y p. It follows that for each y p Y p there exists y lim Y p, r p,q with pth coordinate y p. Thus we can define the canonical maps π p : lim Y p, r p,q Y p, by π p y = y p. The next lemma shows how we can construct measures on the inverse limits. Lemma [25, Lemma 5.2]. Let P be a directed partially ordered set with the smallest element 0. For p, q P, let Y p be a compact space and r p,q : Y q Y p be a surjection. Let limy p, r p,q be the inverse limit of the system {Y p }, {r p,q } p,q P and let π p be the canonical map from limy p, r p,q to Y p. Suppose that we have Borel measure δ p on Y p such that δ 0 is finite and 4.15 f r p,q dδ q = f dδ p for p q and f CY p. Then there is a unique finite Borel measure δ on limy p, r p,q such that f π p dδ = f dδ p for f CX p. Remark Given a finite k-graph Λ, let D := 1,..., 1 and M := {ld : l N}. For each m, n M such that m n, define r m,n : Λ n Λ m by r m,n λ = λ0, m. Clearly M is a directed partially ordered set, and each r m,n is a surjection. The argument of [30, Remark 2.2] shows that, by factorisation property, Λ can be viewed as the inverse limit of the system {Λ m }, {r m,n } m,n M. Proof of Proposition Suppose φ is a KMS β state of T C Λ, α. Then [27, Theorem 6.1c] implies that there is a vector ε [0, Λ0 such that y ε = 1 and φ = φ ε. If δ is a measure on Λ such that δzv = ε v for all v Λ 0 and f β dδ = 1, then Proposition implies that φ δ T C Λ = φ ε. So it suffices to show that there is such a measure. To see this, we view Λ as the inverse limit described in Remark 4.3.5, and then we apply Lemma So we must construct a sequence of measures δ m on Λ m satisfying Let D be as in Remark We recursively choose weights {w η : η Λ with dη = ld for some l 1} such that λ vλ D w λ = ε v, 88

97 and 4.16 λ sµλ D w µλ = w µ, for all v Λ 0 and µ Λ ld l 1. Then we set δ 0 := ε and δ m µ = w µ for all µ Λ ld. Next we check 4.15 for these measures. Let m M. Since the characteristic functions of singletons span CΛ m, it is enough to prove 4.15 for f = χ {µ} and µ Λ m. First notice that χ {µ} r m,m+d = λ sµλ D χ {µλ}. Then we have χ {µ} r m,m+d dδ m+d = χ {µλ} dδ m+d λ sµλ D = δ m+d µλ λ sµλ D 4.17 Since for each n M with m n, we have = δ m µ using 4.16 = χ {µ} dδ m. r m,n = r m,m+d r m+d,m+2d r n D,n, applying the calculation 4.17 finitely many times gives χ {µ} r m,m+n dδ m+n = χ {µ} dδ m. This is precisely Now Lemma implies that there is a unique measure δ on Λ such that χ {v} π 0 dδ = χ {v} dδ 0 for v Λ 0. Notice that χ {v} π 0 dδ = δzv and χ {v} dδ 0 = δ 0 v = ε v. It also follows from the calculation 4.11 that f β dδ = y ε = 1. Thus δ has required properties. 89

98

99 References [1] Z. Afsar, A. an Huef, and I. Raeburn, KMS states on C -algebras associated to local homeomorphisms, Internat. J. Math , no. 8, pages. [2] W. Arveson, Continuous analogues of Fock space, Memoirs Amer. Math. Soc [3] V. Arzumanian and J. Renault, Examples of pseudogroups and their C -algebras, Operator algebra and quantum field theory Rome, 1996, International Press, 1997, pp [4] J. B. Bost and A. Connes, Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. New Series , [5] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II, 2nd ed., Springer-Verlag, Berlin, [6] N. Brownlowe, Realising the C -algebra of a higher-rank graph as an Exel crossed product, J. Operator Theory , [7] N. Brownlowe, A. an Huef, M. Laca, and I. Raeburn, Boundary quotients of the Toeplitz algebra of the affine semigroup over the natural numbers, Ergodic Theory Dynam. Systems , [8] N. Brownlowe, I. Raeburn, and S. T. Vittadello, Exel s crossed product for nonunital C -algebras, Math. Proc. Cambridge. Philos. Soc , [9] J. R. Choksi, Inverse limits of measure spaces, Proc. London Math. Soc , [10] A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields, and Motives, Colloq. Publ., vol. 55, Amer. Math. Soc., Providence,

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104

105 Appendix A Realising the universal Nica-covariant representation as a doubly commuting representation In this appendix we use our results from previous chapters and show that the universal Nica-covariant representation ψ satisfies the doubly commuting relation [54, Lemma 3.9 i]. We first need to understand the notations. Let A be a C -algebra and Y be a right Hilbert A A bimodule. Suppose that π is representation of A on BH for a Hilbert space H. Let Y H be the algebraic tensor product of Y and H. It follows from [46, Proposition 2.6] that the formula y r y r = r π y, y r for y r, y r Y H, defines a semi-definite inner product on Y H. Notice that y a r y πar y r = r π y a, y r πar π y, y r = r π y a, y r r πa y, y r = r π y a, y r r π y a, y r = 0. Now let Y π H be the completion 1 of Y H with respect to this semi-definite inner product see [46, Lemma 2.16]. Since the completing process requires modding out element of length 0, in the completion we have y a r y πar. 1 Completion with respect to semi-definite inner products are sometimes called Hausdorff completion for example [54, page 92]. 97

106 Let S LY and U πa. A similar proof to that of [46, Proposition 2.66] shows that there is a well-defined bound operator S U on Y π H such that S Uy r = Sy Ur for y r Y π H. Muhly and Solel showed in [38, Lemma ] that there is a well-defined map π : Y π H H such that πy r = πyr for all y r Y π H. The map π is called a contraction. Let X be a product system of right Hilbert A A bimodules over N k and θ be a Toeplitz representation of X on BH for a Hilbert space H. It follows that for each m N k and fibre X m, there is a contraction θ m : X m θ0 H H such that θ m x r = θ m yr for all x r X m θ0 H. Let I m, I H be the identity maps on X m and H respectively. Suppose that t ei,e j is the flip map between fibres X ei and X ej as in Lemma A representation θ is doubly commuting representation if for every 1 i j k, we have θ e θei j = I ej θ ei t ei,e j I H I ei θ e j. Suppose θ is a doubly commuting representation and let t m,n be the flip map between fibres X m and X n. Write N k + for non zero elements of N k, and suppose m, n N k + satisfying m n = 0. [54, Lemma 3.9i] implies that A.1 I n θ m t m,n I H I m θ n = θn θm. Now we want to show that the universal Nica-covariant representation ψ satisfies A.1. It follows from [54, Remark 3.12] that we can consider ψ as a representation on a C -algebra H. Proposition A.0.6. Let h 1,..., h k be -commuting and surjective local homeomorphisms on a compact Hausdorff space Z and let X be the associated product system as in Corollary Take m, n N k + such that m n = 0. Then A.2 I n ψ m t m,n I H I m ψ n = ψn ψm We first need to calculate the adjoint ψ n : H Xn ψ0 H. The next lemma gives a formula for ψ n in terms of a general Parseval frame of Xn. 98

107 Lemma A.0.7. Let {η j } d j=0 be a Parseval frame for the fibre X n. Then A.3 ψ n r = d η j ψ n η j r for r H. j=0 Proof. Fix r H and let y s X n ψ0 H. We compute: d η j ψ n η j r y s = j=0 d ψ n η j r ψ 0 η j, y s j=0 = r = r d ψ n η j ψ 0 η j, y s j=0 d j=0 = r ψ n ys. ψ n ηj η j, y s This is precisely r ψ n y s. Thus ψ n r = d j=0 η j ψ n η j r. Proof of Proposition A.0.6. Let x r X m ψ0 H. We evaluate both sides of A.2 on x r. To do this we will need to have Parseval frames for fibres X m, X n. Let {ρ i } d i=0 be a partition of unity such that h m supp ρi, h n supp ρi are injective and suppose that τ i := ρ i. Notice that {τ i } d i=0 forms a Parseval frame for both fibres X m, X n. Also since m n = 0, {τ i h n } d i=0 and {τ i h m } d i=0 are Parseval frame for the fibres X m, X n, respectively. We start computing the left-hand side of A.2 by applying the adjoint formula A.3 with Parseval frame {τ j } d j=0 X n. For convenience, set := I n ψ m t m,n I H I m ψ n. We have x r = I n ψ m t m,n I H d x τ j ψ n τ j r. j=0 Writing the reconstruction formula for the Parseval frame {τ i h n } d i=0 X m gives x r = d I n ψ d m t m,n I H τ i h n τ i h n, x τ j ψ n τ j r j=0 = 0 i,j d i=0 I n ψ m t m,n I H τ i h n τ i h n, x τ j ψ n τ j r. 99

108 Applying the reconstruction formula for the Parseval frame {τ l } d l=0 X n, we have x r = 0 i,j d I n ψ m t m,n I H τ i h n Now we continue by using the flip map 3.5 x r = 0 i,j,l d = 0 i,j,l d = 0 i,l d d l=0 τ l τ l, τ i h n, x τ j ψ n τ j r. I n ψ m t m,n I H τ i h n τ l ψ 0 τ l, τ i h n, x τ j ψ n τ j r I n ψ τi m τ l h m τ i ψ n τ j h n, x r τ j, τ l I n ψ d m τ l h m τ i ψ n τ j The reconstruction formula for frame {τ j } d j=0 X n implies that A.4 x r = 0 i,l d j=0 τ j, x, τ i h n τ l r. τ l h m ψ m τ i ψ n x, τi h n τ l r. Next we compute the right-hand side A.2 by applying the adjoint formula A.3 with the Parseval frame {τ l h m } d l=0 X n. ψ n ψm x r = ψ n ψm xr = d τ l h m ψ n τ l h m ψ m xr. l=0 Applying our formula 3.9 implies that d ψ n ψm x r = τ l h m ψ m τl h m, τ j h m τ i ψn x, τi h n τ j r l=0 0 i,j d = τ l h m ψ 0 τl h m, τ j h m r ψ m τ i ψ n x, τi h n τ j 0 i,j,l d = τ l h m τ l h m, τ j h m r. ψ m τ i ψ n x, τi h n τ j 0 i,j,l d Now applying reconstruction formula for the Parseval frame {τ l h m } d l=0 X n gives ψ n ψm x r = r. A.5 τ j h m ψ m τ i ψ n x, τi h n τ j 0 i,j d Comparing A.5 and A.4 completes our proof of A

109 Appendix B KMS states on C -algebras associated to a local homeomorphism In this appendix we provide our result about KMS states on dynamical systems associated to a single local homeomorphism. This work is published in Internat. J. Math. Vol. 25, No pages. 101

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111 International Journal of Mathematics Vol. 25, No pages c World Scientific Publishing Company DOI: /S X KMS states on C -algebras associated to local homeomorphisms Zahra Afsar,AstridanHuef and Iain Raeburn Department of Mathematics and Statistics University of Otago, P. O. Box 56 Dunedin 9054, New Zealand zafsar@maths.otago.ac.nz astrid@maths.otago.ac.nz iraeburn@maths.otago.ac.nz Received 21 February 2014 Accepted 5 June 2014 Published 18 July 2014 For every Hilbert bimodule over a C -algebra, there are natural gauge actions of the circle on the associated Toeplitz algebra and Cuntz Pimsner algebra, and hence natural dynamics obtained by lifting these gauge actions to actions of the real line. We study the KMS states of these dynamics for a family of bimodules associated to local homeomorphisms on compact spaces. For inverse temperatures larger than a certain critical value, we find a large simplex of KMS states on the Toeplitz algebra, and we show that all KMS states on the Cuntz Pimsner algebra have inverse temperature at most this critical value. We illustrate our results by considering the backward shift on the onesided path space of a finite graph, where we can use recent results about KMS states on graph algebras to see what happens below the critical value. Our results about KMS states on the Cuntz Pimsner algebra of the shift show that recent constraints on the range of inverse temperatures obtained by Thomsen are sharp. Keywords: Toeplitz algebra; Cuntz Pimsner algebra; gauge action; KMS state. Mathematics Subject Classification 2010: 46L35 1. Introduction We consider actions α of the real line R by automorphisms of a C -algebra A. When α describes the time evolution in a model of a physical system, the states of the system are given by positive functionals of norm 1. The equilibrium states are the states on A that satisfy a commutation relation called the KMS condition. This condition makes sense for every dynamical system of the form A, R,α, irrespective of its origin, and studying the KMS states of such systems often yields interesting information. This is certainly the case, for example, for the number-theoretic Hecke algebra of Bost and Connes [2] and its generalizations [21, 22], for systems involving gauge actions on graph algebras [8, 11, 18, 15], and for systems associated to local homeomorphisms of the sort arising in topological dynamics [34, 35]