On the JLO Character and Loop Quantum Gravity. Chung Lun Alan Lai

Transcription

1 On the JLO Character and Loop Quantum Gravity by Chung Lun Alan Lai A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Copyright 211 by Chung Lun Alan Lai

2 Abstract On the JLO Character and Loop Quantum Gravity Chung Lun Alan Lai Doctor of Philosophy Graduate Department of Mathematics University of Toronto 211 In type II noncommutative geometry, the geometry on a C -algebra A is given by an unbounded Breuer Fredholm module (ρ, N, D) over A. Here ρ : A N is a - homomorphism from A to the semi-finite von Neumann algebra N B(H), and D is an unbounded Breuer Fredholm operator affiliated with N that satisfies certain axioms. Each Breuer Fredholm module assigns an index to a given element in the K-theory of A. The Breuer Fredholm index provides a real-valued pairing between the K-homology of A and the K-theory of A. When N = B(H), a construction of Jaffe-Lesniewski-Osterwalder associates to the module (ρ, N, D) a cocycle Ch JLO(D) in the entire cyclic cohomology group HE (A) for D is θ-summable. The JLO character and the K-theory character ch : K (A) HE (A) intertwine the K-theoretical pairing with the pairing between HE (A) and HE (A). If (ρ, N, F ) is a finitely summable bounded Breuer Fredholm module, Benameur- Fack defined a character formula ch n (F ) generalizing the Connes character formula for bounded Fredholm modules. On the other hand, given a finitely-summable unbounded Breuer Fredholm module (ρ, N, D), there is a canonically associated bounded Breuer Fredholm module. The first main result of this thesis extends the JLO theory to Breuer Fredholm modules (possibly N = B(H)) in the graded case, and proves that the JLO character formula Ch JLO(D) and Connes character formula ch n (F ) define the same class in HE (A). This extends a result of Connes-Moscovici[21] for ordinary Fredholm modules. ii

3 An important example of an unbounded Breuer Fredholm module is given by the noncommutative space of G-connections due to Aastrup-Grimstrup-Nest[3, 4]. In their original work, the authors limit their construction to the case that the group G is either U(1) or SU(2). Another main result of the thesis extends AGN s construction to any connected compact Lie group G; and generalizes by considering connections defined on sequences of graphs, using limits of spectral triples. Our construction makes it possible to equip the module (ρ, N, D) with a Z 2 -grading. The last part of this thesis studies the JLO character of the Breuer Fredholm module of AGN. The definition of this Breuer Fredholm module depends on a divergent sequence. A concrete condition on possible perturbations of the sequence ensuring that the resulting JLO class remains invariant is established. The condition implies a certain functoriality of AGN s construction. iii

4 Acknowledgements I thank my mother for giving me endless support and a stress-free environment to pursue my education. Without her, I would not have been able to complete my Ph.D. studies. I thank my adviser Eckhard Meinrenken for supervising me, a very rebellious and trouble making student. I thank Nigel Higson for numerous patient and fruitful discussions. Finally, I thank my colleagues for keeping me entertained for the duration of my Ph.D. program. It is a pure coincidence that our world-lines/sheets come close together in this universe, and it looks as if they will start to diverge at this point. Regardless of whether they will converge ever again, I will treasure this intersection of our neighbourhoods. iv

5 Contents 1 Introduction 1 2 Preliminaries Breuer Fredholm Theory Index Pairing in K-Theory Entire Cyclic Homology The Homological Chern Character JLO Theory The JLO Character Formula Homotopy Invariance of the JLO Class The JLO Character as an Index Formula The Cohomological Chern Character The Connes Character Formula The JLO Character formula for p-summable unbounded Breuer Fredholm modules From unbounded to bounded Breuer Fredholm modules From the JLO Character formula to the Connes Character Formula The Noncommutative Space of Connections of AGN Inductive Limit of Spectral Triples v

6 5.2 Graphs and Spectral Triples on Graphs The Space of Connections on a Graph The Algebra of Holonomies The Quantum Weil Algebra The Dirac Operator and the Hilbert Space A Compactification of the Space of Connections Systems of Graphs Embedded Graphs The Limit of Spectral Triples on Graphs Choice of Generators for G(Γ) Morphisms between Spectral Triples on Graphs The System of Spectral Triples Semi-finite Spectral Triples Limit of Semi-finite Spectral Triples The Z 2 -grading The JLO Class of AGN s Space of Connections Weight Independence of the JLO Class Weak θ-summability Bibliography 94 vi

7 Chapter 1 Introduction In non-commutative geometry, the guiding principle is that the topology of a space is encoded in properties of its algebra of continuous functions. A theorem of Gelfand-Naimark [29] states that any commutative unital C -algebra is of the form C(X) for some compact Hausdorff space X. Therefore, the category of C -algebras (or even more generally Banach -algebras) is seen as an extension of the category of compact Hausdorff topological spaces, and a general C -algebra is sometimes referred to as a non-commutative topological space. The geometric features of a C -algebra A are incorporated by the concept of an an unbounded Fredholm module (ρ, B(H), D) over A, where ρ is a continuous representation of A on the Hilbert space H, and D is an unbounded Fredholm operator on H that satisfies certain axioms. As the prototypical example, consider the algebra A of continuous functions on a closed Riemannian manifold, acting in the natural way on the space of L 2 spinor sections H, and the associated Dirac operator D. Geometric features of the manifold such as metric, dimension, measures, and differential forms, etc. can be retrieved algebraically in terms of A, B(H), and D [26]. Connes gives a set of five axioms characterizing the Fredholm modules arising in this way [2]. Taking A to be non-commutative thus leads to a notion of non-commutative manifolds. This theory is summarized in Connes s famous book [19]; further details and newer developments are 1

8 Chapter 1. Introduction 2 described in [26] and [29]. Each Fredholm module assigns an integer, the Fredholm index, to a given element in the K-theory of A. The Fredholm index provides a Z-valued pairing between the K-homology of A and the K-theory of A. In the commutative setting, the index can be viewed as the index of the Dirac operator D, twisted by a vector bundle. Suppose that the unbounded Fredholm module (ρ, B(H), D) is finitely summable, a condition that models finite dimensionality according to Connes s axioms. Jaffe- Lesniewski-Osterwalder [28] defined a cocycle Ch JLO in the entire cyclic cohomology HE (A), now known as the JLO character. Together with the K-theory character ch : K (A) HE (A), they intertwine the K-theoretical pairing given by the Fredholm index with the cohomological pairing between HE (A) and HE (A) [24, 25]. The result generalizes to weakly θ-summable Fredholm modules, where θ-summability can be thought of a suitable notion of infinite-dimensionality. Consequently, the JLO character provides a formula for the Fredholm index in terms of entire cyclic (co)homology for infinite dimensional non-commutative manifolds, which was the original motivation of JLO s work [28]. Furthermore, the formula reduces to the index formula of Atiyah-Singer in the commutative setting [7, 23]. The operator D of a Fredholm module plays the role of a Dirac operator, and is typically unbounded. However, there is a canonical way of passing from an unbounded Fredholm module to a bounded one (ρ, B(H), F ), essentially by taking bounded functions of D, and the latter are often easier to work with in practice. When the bounded Fredholm module (ρ, B(H), F ) is finitely summable, there is a character formula ch n due to Connes [18], which again is a cocycle in HE (A), and ch n intertwines the K-theoretical pairing in the same way as the JLO character [18, 21]. When (ρ, B(H), F ) is the associated bounded module of a finitely summable unbounded Fredholm module (ρ, B(H), D), Connes-Moscovici proved that in fact the cocycle ch n (F ) of (ρ, B(H), F ) defines the same cohomology class as the cocycle Ch JLO(D) of (ρ, B(H), D)

9 Chapter 1. Introduction 3 in HE (A) [21]. Type II non-commutative geometry replaces the algebra B(H) with a (possibly) type II von Neumann algebra N B(H), using Breuer s Fredholm theory relative to the von Neumann algebra N [9, 1]. A type II non-commutative geometry on the algebra A is given by an unbounded Breuer Fredholm module (ρ, N, D) over A, where D is a Breuer Fredholm operator affiliated with N that satisfies certain axioms. Examples of unbounded Breuer Fredholm modules arise from foliations or geometry with degeneracies. A number of examples can be found in [6]. Parallel to the type I setting, the Breuer Fredholm theory provides an index pairing between Breuer Fredholm modules and K-theory given by the Breuer Fredholm index [9, 1, 15]. As a characteristic of the type II von Neumann algebra N, the Breuer Fredholm index now takes values in R as opposed to Z as in the type I case. In this thesis, we will develop the even JLO character for type II non-commutative geometry (the odd JLO character had been introduced in the work of Carey-Phillips [11]). In analogy with the type I case, I show how to pass from unbounded Breuer Fredholm modules to bounded ones. Extending the argument for the type I case, I show that this correspondence takes the cohomology class given by the (both even and odd) JLO character formula to that of the Connes character formula in HE (A), as defined in the type II case by Benameur-Fack [6]. An important example of an unbounded Breuer Fredholm module is given by the non-commutative space of connections due to Aastrup-Grimstrup-Nest (AGN) [3, 4]. In attempts to combine non-commutative geometry and quantum gravity, AGN construct an unbounded Breuer Fredholm module (ρ, N, D) over the algebra of holonomy loops B, modeling the moduli space of G-connections. AGN show that the interaction between the algebra of holonomy loops B and the Dirac type operator D quantizes the Poisson structure of General Relativity [3, 4]. They argue that (ρ, N, D) incorporates quantum gravity in this model.

10 Chapter 1. Introduction 4 One key ingredient in the construction of AGN s non-commutative connection space is a certain left-invariant Dirac operator over the group G, and it is necessary to have control over the kernel of this Dirac operator. AGN determine the kernel for G = U(1), SU(2) their result is generalized to arbitrary compact connected Lie groups here. The AGN construction approximates the manifold by a nested sequence of lattices (e.g. coming form a triangulation and successive refinements). At each stage one defines a space of lattice connections, and an associated algebra of holonomy loops and a Breuer Fredholm module. AGN s construction is suitably generalized here so that it is possible to equip the Breuer Fredholm (ρ, N, D) with a Z 2 -grading in the case that G is even dimensional. The operator D in AGN s Breuer Fredholm module (ρ, N, D) is an infinite sum of the Dirac operator on G with appropriate weights assigned to it. Consequently, (ρ, N, D) depends on the weight assignment. I will give a concrete condition on possible perturbations of the weight assignments so that the resulting cohomology class [Ch JLO(D)] HE (A) remains invariant. In particular, the allowable perturbation shows that AGN s construction is functorial at the cohomology level. Namely, if one re-runs AGN s construction on a sub-manifold, the pull-back Breuer Fredholm module defines the same cohomology class as the Breuer Fredholm module constructed from the submanifold. The thesis is organized in the following way. Chapter 2 is a preliminary chapter that reviews Breuer s Fredholm theory and the index pairing. Chapter 3 develops a JLO theory for unbounded Breuer Fredholm modules. In Chapter 4, it will be shown that the JLO character formula coincides with Connes s character formula at the level of cohomology. Chapter 5 constructs an important example of type II non-commutative geometry, which is the semi-finite spectral triple on the space of connections due to Aastrup-Grimstrup-Nest. Chapter 6 studies the JLO character formula of the semi-finite spectral triple of AGN, and its class in entire cyclic cohomology.

11 Chapter 2 Preliminaries Breuer s Fredholm theory [9, 1] forms the basis for type II noncommutative geometry; we will review it in Section 2.1. Section 2.2 defines the index pairing between a Fredholm operator in the Breuer sense with K-theory. Then entire cyclic (co)homology and the group homomorphism ch from K-theory to entire cyclic homology is introduced in Sections 2.3 and 2.4 respectively. 2.1 Breuer Fredholm Theory The presentation in this section follows closely [22] and [6] to which we refer for proofs and further details. A von Neumann algebra with underlying Hilbert space H is a unital -subalgebra of the algebra of bounded operators B(H) on H that is closed under the weak operator topology. A positive linear functional on a von Neumann algebra is said to be normal if it preserves sup of any increasing nets of positive operators in the von Neumann algebra; faithful if it is positive-definite on positive operators; semi-finite if the -subalgebra generated by positive elements with finite value under the functional is σ-weakly dense in the von Neumann algebra [8]. A von Neumann algebra is called semi-finite if it admits 5

12 Chapter 2. Preliminaries 6 a faithful, semi-finite normal trace. A von Neumann algebra is type I if it is semi-finite and every (non-zero) projection contains a minimal sub-projection with the same central support; type II if it is semi-finite but not type I [8]. Let N be a semi-finite von Neumann algebra with underlying Hilbert space H and a faithful semi-finite normal trace τ. Definition A densely defined closed operator T on H with polar decomposition T = U T [35] is said to be affiliated with N if U N and also the spectral projection 1 [,λ] ( T ) of T lie in N for all λ, where 1 [,λ] is the characteristic function supported on the closed interval [, λ] R. For a positive self-adjoint operator T = λde λ affiliated with N with E λ = 1 [,λ] ( T ), its semi-finite trace is by τ(t ) = λdτ(e λ ). From now on, when an operator T is said to be affiliated with N, it is implicitly demanded that T is densely defined and closed. Definition For an operator T affiliated with N and x >, the generalized singular number µ x (T ) with respect to (N, τ) is defined to be µ x (T ) := inf E { T E : τ(1 E) x}, where the infimum is taken over projections E N. Definition Let T be an operator affiliated with N, < p <, and x >. Then T is said to be ˆ p-summable if T p := τ( T p ) 1/p <, ˆ τ-compact if lim µ x(t ) =, x

13 Chapter 2. Preliminaries 7 ˆ τ-measurable if for each ε > there exists a projection E N such that Ran(E) Dom(T ) and τ(1 E) < ε. Remark Anything in N is τ-measurable. If a self-adjoint operator T is affiliated with N and its resolvent is τ-compact, then T is τ-measurable [6]. Proposition ([22]). Let T, S, R be τ-measurable operators. 1. The map: x (, ) µ x (T ) is non-increasing and continuous from the right. Moreover, lim µ x(t ) = T [, ]. x 2. µ x (T ) = µ x ( T ) = µ x (T ) and µ x (zt ) = z µ x (T ) for x > and z C. 3. µ x (T ) µ x (S), x >, if T S. 4. µ x (f( T )) = f(µ x ( T )), x > for any continuous increasing function f on [, ) with f(). 5. µ x (ST R) S R µ x (T ), x >. Proposition ([22]). Let T be a positive τ-measurable operator. Then τ(t ) = µ x (T )dx. Proposition ([22]). Let T, S, and R be operators in N. Then for < p <, ST R p S R T p. Denote by L p N the space of all p-summable operators in N. For < p <, the space L p N forms a norm closed two-sided ideal in N with norm given by p +. Denote by K N the space of all τ-compact operators in N. The space K N forms a norm closed two-sided ideal in N.

14 Chapter 2. Preliminaries 8 Theorem ([22]). Let T, S be τ-measurable operators. Then 1. T S r T p S q for p, q, r > and p 1 + q 1 = r T + S p T p + S p for p 1. For N = B(H) with τ the operator trace, then p-summability and τ-compactness are the usual notion of p-summability and compactness, and the ideals L p N and K N are the usual ideal of Schatten p-class and the ideal compact operators. 2.2 Index Pairing in K-Theory To define the index pairing, let us adapt the more general account appearing in [11, 13, 15, 16]. Definition An odd Breuer Fredholm module over a unital Banach -algebra A is a triple (ρ, N, F ) for which N is a (separable) semi-finite von Neumann algebra with faithful semi-finite normal trace τ, ρ : A N a continuous -representation, and F N an operator such that F 2 = 1 and [F, ρ(a)] K N for all a A. If (ρ, N, F ) is equipped with a Z 2 grading χ N such that all ρ(a) are even and F is odd, then we call (ρ, N, F ) an even Breuer Fredholm module. If N = B(H) and τ is the standard operator trace, the prefix Breuer is dropped. As Fredholm modules are representatives of K-homology classes in Kasparov s sense [27], they are also referred to as K-cycles. Technically speaking, Breuer Fredholm modules do not define K-homology classes in the usual sense; however one can still consider its classes given by the equivalence relations in K-homology. In other words, up to degenerate modules, two such modules are equivalent if their Fredholm operators are connected by a norm continuous homotopy of Fredholm operators (in N ) (see for example [27] for a precise definition). We think of Breuer Fredholm modules as representatives of elements in some semi-finite or type

15 Chapter 2. Preliminaries 9 II K-homology as [13, 15] did. Whenever we write [(ρ, N, F )] K (A), it is implicitly meant that the K-homology is in the semi-finite sense. Definition Given two projections e, f N, a (possibly unbounded) operator T affiliated with N is called (e, f)-fredholm if there is an operator S N, such that e esft e K en e and f ft esf K fn f, where K en e denotes the set of τ-compact operators in en e, likewise for K fn f. The operator S is called an (e, f)-parametrix for T. Example ˆ Let (ρ, N, F ) be a Breuer Fredholm module. If u N is a unitary, then u is ( F +1, F )-Fredholm with ( F +1 2, F +1 2 )-parametrix u 1. ˆ Suppose that (ρ, N, F ) comes equipped with a Z 2 grading χ and that the projection p N is even with respect to χ. Then F is (p +, p )-Fredholm with (p +, p )- parametrix F again. The following Proposition can be found in [6]. We have adapted it to the (e, f)- parametrix case. Proposition Let T be a (e, f)-fredholm operator, and P ker T eh and P ker(t ) fh be the projections onto the kernels of T eh and T fh respectively. Then P ker T eh and P ker(t ) fh have finite trace with respect to τ. Proof. Let S be a (e, f)-parametrix of T as in Definition We have (e esft e)p ker T eh = P ker T eh and P ker(t ) fh(f ft esf) = P ker(t ) fh. By the ideal property of K en e, P ker T eh is a τ-compact projection. As projections only have eigenvalue {, 1}, τ-compactness forces the singular values of projections to have support in a bounded region, whence τ of any τ-compact projection must be finite, and τ(p ker T eh ) <. Likewise, τ(p ker(t ) fh) <.

16 Chapter 2. Preliminaries 1 Definition The (e, f)-index Ind τ (ft e) of an (e, f)-fredholm operator T is defined to be Ind τ (ft e) := τ(ep ker T ) τ(p ker(t )f), where P ker T and P ker(t ) are the projections onto the kernel of T and T respectively. Let there be given an even Breuer Fredholm module (ρ, N, F ) over A, and a projection p A. It follows from Example that F is a (ρ(p) +, ρ(p) )-Fredholm operator. Thus it has a well-defined (ρ(p) +, ρ(p) )-index, given by Ind τ (ρ(p) F ρ(p) + ). For a given odd Breuer Fredholm module (ρ, N, F ), and a unitary u A, then ρ(u) is a (Q, Q)-Fredholm operator, where Q = F +1. Thus it has a well-defined (Q, Q)-index, 2 given by Ind τ (Qρ(u)Q). Since the function Ind τ is locally constant [16], the (ρ(p) +, ρ(p) )-index descends to a pairing between the K-homology class [(ρ, N, F )] K (A) and the K-theory class [p] K (A). Likewise, the (Q, Q)-index descends to a pairing between the classes [(ρ, N, F )] K 1 (A) and [u] K 1 (A). Let us extend the pairing to a pairing between K-homology and K-theory of A with the following definition. To simplify our notation, whenever an element a A is mentioned, we think of it as an operator ρ(a) N represented on H, and will stop writing ρ. Similarly, when we have a M N (A), we think of it as an operator in M N (N ) represented on H N = H C N with the obvious representation extended from ρ. Definition ([12, 13, 15]). 1. Let (ρ, N, F ) be an even Breuer Fredholm module over A, representing the K- homology class [(ρ, N, F )] K (A), and p M N (A) be a projection, representing the K-theory class [p] K (A). Let us define the even index pairing to be: [(ρ, N, F )], [p] := Ind τ (p (F 1 N )p + ), where p (F 1 N )p + is an operator from p + H N to p H N.

17 Chapter 2. Preliminaries Let (ρ, N, F ) be an odd Breuer Fredholm module over A, representing the K-homology class [(ρ, N, F )] K 1 (A), and u M N (A) be a unitary, representing the K-theory class [u] K 1 (A). Let us define the odd index pairing to be: [(ρ, N, F )], [u] := Ind τ (QuQ), where Q = F 1 N +1 2 is a projection in M N (N ), and QuQ is an operator from QH N to QH N. In commutative geometry, that is when A = C(M), a source of Fredholm modules consists of taking the function sign(x) of self-adjoint degree 1 elliptic operators on M, +1 if x where sign(x) =. For those Fredholm modules arising in this fashion, 1 if x < they can be defined directly as follows. Definition An odd unbounded Breuer Fredholm module over a unital Banach -algebra A is a triple (ρ, N, D) for which N is a (separable) semi-finite von Neumann algebra in B(H) with a faithful semi-finite normal trace τ, ρ : A N a continuous -representation, and D is an unbounded self-adjoint operator on H such that 1. D is affiliated with N, 2. For all a A, the commutator [D, ρ(a)] extends to an operator in N and there is a constant C such that [D, ρ(a)] C a. 3. (1 + D 2 ) 1/2 K N. If (ρ, N, D) is equipped with a Z 2 grading χ N such that all ρ(a) are even and D is odd, then we call (ρ, N, D) an even unbounded Breuer Fredholm module. If N = B(H) and τ is the standard operator trace, the prefix Breuer is dropped. To avoid confusion, the Breuer Fredholm module from Definition will sometimes be refered to as a bounded Breuer Fredholm module. As for its bounded counterpart, an unbounded Fredholm module is sometimes called an unbounded K-cycle.

18 Chapter 2. Preliminaries 12 The term (semi-finite) spectral triple seems to be popular among physicists. It is a convenient term for the package consisting of the algebra A and an unbounded (Breuer )Fredholm module. In this thesis, our algebra A is always fixed and we view the JLO character formula and the Connes character formula as maps from K-homology classes to some cohomology classes that respect group addition. Hence, the term unbounded Breuer Fredholm module is more convenient and suitable in our setting. An example of an unbounded Breuer Fredholm module is given by the semi-finite spectral triple over a space of G-connections due to Aastrup, Grimstrup, and Nest. A detailed construction will be given in Chapter 5. Similar to the Breuer Fredholm module case, we think of an element a A as an operator ρ(a) N represented on H, and will stop writing ρ. In Section 4.3, it will be explained in detail how one associates a bounded Breuer Fredholm module to an unbounded one. Definition For a given even unbounded Breuer Fredholm module (ρ, N, D) over A, define its pairing with the even K-theory K (A) of A given by the index: ( [(ρ, N, D)], [p] := Ind τ p (D 1 N )p +) for a projection p M N (A) representing the class [p] K (A), where p (D 1 N )p + : p + H N p H N. 2. For a given odd unbounded Breuer Fredholm module (ρ, N, D) over A, define its pairing with the odd K-theory K 1 (A) of A given by the spectral flow: [(ρ, N, D)], [u] := sf ( D 1 N, u(d 1 N )u 1) for a unitary u M N (A) representing the class [u] K 1 (A), where sf (D 1 N, u(d 1 N )u 1 ) is the spectral flow from (D 1 N ) (1 + (D 1 N ) 2 ) 1 2 to (u(d 1 N )u 1 ) (1 + (u(d 1 N )u 1 ) 2 ) 1 2 defined in [12].

19 Chapter 2. Preliminaries Entire Cyclic Homology Entire cyclic homology is not as well known as its cohomology counterpart. We adopt the bicomplex construction from [24] and use the entire growth control given in [34]. Under this definition, the homology theory is precisely (pre-)dual to the cohomology counterpart [25] in the sense that their pairing produces a finite value. If B is a topological unital algebra over C, define C n (B) := B ˆ (B/C) ˆ n, where ˆ denotes the projective tensor product. Denote the element a a n of C n (B) by (a,..., a n ) n ; when the context is clear the subscript n will be omitted. The operators b : C n (B) C n 1 (B) and B : C n (B) C n+1 (B) are given in terms of simple tensors by the formulas b(a,..., a n ) n := B(a,..., a n ) n := n 1 ( 1) j (a,..., a j a j+1,..., a n ) n 1 + ( 1) n (a n a, a 1,..., a n 1 ) n 1, j= n ( 1) nj (1, a j,..., a n, a,..., a j 1 ) n+1. j= Simple calculations show that b 2 =, B 2 =, and Bb+bB =. Therefore (b+b) 2 =

20 Chapter 2. Preliminaries 14 and we obtain the following bicomplex:.... B B B B b b C 3 (B) b C 2 (B) b C 1 (B) C (B) B B B b b C 2 (B) C 1 (B) b C (B) B B b b C 1 (B) C (B) (b+b) B b 8 C (B). The space C (B) := n= C n(b) has a natural Z 2 grading given by C + (B) = k= C 2k(B) and C (B) = k= C 2k+1(B). We obtain a chain complex (C (B), b + B) with the odd boundary map b + B. However, the homology of this chain complex is trivial [34]. In order to make it nontrivial, one needs to control the growth of a chain as n varies. The following definition is taken from [24, 34]. Definition Define the space of entire chains C ω (B) := { ) } λ A C (B) : sup ( A n n π n Γ( n) < for some λ > 2 where π is the projective tensor norm. (C ω (B), b + B) forms a subcomplex of (C (B), b + B). The homology defined by (C ω (B), b + B) is the entire cyclic homology

21 Chapter 2. Preliminaries 15 of B, denoted by HE (B) = HE + (B) HE (B). HE (B) is equipped with the obvious group structure inherited from the addition on C n (B). Set C n (B) := Hom(C n (B), C) and let (b + B) : C (B) C (B) denote the transpose of the odd boundary map (b + B) : C (B) C (B) where C (B) := n= Cn (B). Then one obtains a similar diagram as above with the arrows reversed. The space C (B) has a natural Z 2 grading given by C + (B) = k= C2k (B) and C (B) = k= C2k+1 (B). (C (B), b + B) forms a cochain complex with the odd boundary map b + B, which gives trivial cohomology [34]. Definition Define the space of entire cochains { } C ω(b) := ϕ C (B) : Γ( n 2 ) ϕ n z n is an entire function in z n= where ϕ n := sup { ϕ n (a,..., a n ) : a j 1 j}. (C ω(b), b + B) forms a subcomplex of (C (B), b + B). The cohomology defined by (C ω(b), b + B) is the entire cyclic cohomology of B, denoted by HE (B) = HE + (B) HE (B). HE (B) is equipped with the obvious group structure inherited from the addition on Hom(C n (B), C). It is known that the de Rham homology (over C) on a closed manifold M is a direct summand of the entire cyclic cohomology of the algebra C (M). They are expected to be equal; however this has not been proven except for the case when M is one-dimensional [23]. 2.4 The Homological Chern Character The homological Chern character considered in this section takes values in entire cyclic homology as opposed to cyclic homology as in Connes s original construction [18]. Let Tr : C n (M N (A)) C n (A) denote the map defined by Tr (m, m 1,..., m n ) := ((m ) i i 1, (m 1 ) i1 i 2,..., (m n ) in i ), i,...,i n N

22 Chapter 2. Preliminaries 16 where (m k ) ij denotes the entries of the matrix m k. Definition Let p M N (A) be a projection. Define the even Chern character formula ch + (p) C + (A) of p to be where ch + (p) := ch 2k (p), k= ch (p) := Tr(p), ch 2k (p) := ( 1) k (2k)! 2 k! Tr(2p 1, p,..., p) 2k. 2. Let u M N (A) be a unitary. Define the odd Chern character formula ch (u) C (A) of u to be where ch (u) := ch 2k+1 (u), k= ch 2k+1 (u) := 1 Γ( 1 2 )( 1)k+1 k! Tr(u 1, u,..., u 1, u) 2k+1. For convenience, often Tr (m, m 1,..., m n ) will simply be written as (m,..., m n ). Lemma ([25, 24]). The Chern character formulas ch + (p) and ch (u) define entire cyclic cycles in HE + (A) and HE (A) respectively. That is, ch + (p) C+(A) ω, (b + B) ch + (p) = ; and ch (u) C (A) ω, (b + B) ch (u) =. Furthermore, the homology classes [ch + (p)] and [ch (u)] depend only on the K-theory classes of [p] K (A) and [u] K 1 (A) respectively.

23 Chapter 2. Preliminaries 17 As a result of Lemmas 2.4.1, the assignment ch descends to a character from K (A) to HE (A). It is easy to see that ch respects group addition, and hence it is a group homomorphism. It is called the homological Chern character.

24 Chapter 3 JLO Theory The JLO theory due to Jaffe, Lesniewski and Osterwalder [28] is a formula that assigns a cocycle, hence a class, in entire cyclic cohomology to a weakly θ-summable unbounded Fredholm module. The cohomology class is homotopy invariant, and hence the JLO formula induces a character that maps K-homology classes to entire cyclic cohomology classes. Furthermore, this character together with the homological Chern character ch : K (A) HE (A) introduced in Section 2.4 intertwines the K-theoretic pairing given by the Fredholm index, and hence the JLO character is the Chern character. In this chapter, by adopting the work of Getzler-Szenes [25] we will generalize JLO theory to weakly θ-summable unbounded Breuer Fredholm modules. 3.1 The JLO Character Formula The JLO character formula assigns cocycles in entire cyclic cohomology to unbounded Breuer Fredholm modules satisfying an appropriate summability condition. Let us begin by defining the summability conditions of main interest. Definition An unbounded Breuer Fredholm module (ρ, N, D) over A is: (a) p-summable if τ ( (1 + D 2 ) p/2) < ; 18

25 Chapter 3. JLO Theory 19 (b) θ-summable if τ(e td2 ) < for all t > ; (c) weak θ-summable if τ(e td2 ) < for some < t < 1. Observe that p-summability implies θ-summability, which in turn implies weakly θ- summability. Example Let Γ M M be a Galois cover of a compact p-dimensional manifold M. Let D denote the Γ cover of a generalized Dirac operator on M. Consider the von Neumann algebra N of bounded Γ-invariant operators defined by Atiyah, with its natural trace Tr Γ, and denote by H the Hilbert space N represents on. Then (ρ, N, D) is a p-summable unbounded Breuer Fredholm module over C(M) with ρ given by point-wise multiplication [6]. Example The unbounded Breuer Fredholm module given by Aastrup-Grimstrup- Nest s noncommutative space of connections is weakly θ-summable if the sequence {a j } in its definition diverges sufficiently fast (see Section 6.2 for detailed analysis). The following Lemma was proved in [21] in the type I case. Lemma If (ρ, N, D) is p-summable for any finite p, then it is also θ-summable, and τ(e td2 ) = O(t p/2 ) as t. Proof. We can write e td2 = (1 + D 2 ) p/2 e td2 (1 + D 2 ) p/2 with τ((1 + D 2 ) p/2 ) < by hypothesis, and (1 + D 2 ) p/2 e td2 bounded by (1 + x 2 ) p/2 e tx2 = ( p p/2 2e) t p/2 e t by functional calculus. Hence as a consequence of Proposition and Proposition 2.1.2, we obtain ( p ) p/2 τ(e td2 ) t p/2 e t τ((1 + D 2 ) p/2 ), 2e which proves the lemma. To make the JLO character formula and other useful formulas easier to write down, let us define the JLO character formula in two steps. We start with the following definition.

26 Chapter 3. JLO Theory 2 Let n := {(t 1,..., t n ) R n : t 1 t n 1} denote the standard n-simplex and d n t = dt 1 dt n the standard Lesbeque measure on n with volume 1 n!. Definition Let (ρ, N, D) be a weakly θ-summable unbounded Breuer Fredholm module over A. Given F,..., F n operators affiliated with N, define F, F 1,..., F n n D := where χ = 1 when D is even. n τ ( χf e t 1D 2 F 1 e (t 2 t 1 )D 2... F n e (1 t n)d 2) d n t, Let T be an operator affiliated with N, and denote by T χ the degree of T with respect to χ. Any operators that will be considered are either even or odd. From here on, the commutator [, ] is always graded with respect to χ. Lemma Let F,..., F n be operators affiliated with N that are either even or odd, then 1. F,..., F n n D = ( 1)( F χ + + F j 1 χ )( F j χ + + F n χ ) F j,..., F n, F,..., F j 1 n D ; F,..., F n n D = n j= F,..., 1, F j,..., F n n+1 D ; n ( 1) F χ + + F j 1 χ F,..., [D, F j ],..., F n n D = ; j= 4. F,..., [D 2, F j ],..., F n n D = F,..., F j 1 F j, F j+1,..., F n n 1 D F,..., F j 1, F j F j+1,..., F n n 1 D. Proof.

27 Chapter 3. JLO Theory The statement follows from τ(χ[x, Y ]) = for X, Y operators affiliated with N. 2. The left hand side can be regarded as 1 F,..., F n n D du by introducing a trivial extra integration; the polyhedron n [, 1] can be subdivided by the inequalities t j u t j+1 into n + 1 simplices, each of which is a copy of n+1 ; integration over these simplices yields the terms on the right hand side. 3. By observing the Leibniz property of [D, ] and = τ ( ) χ[d, F e t 1D 2 F 1 e (t 2 t 1 )D 2 F n e (1 tn)d2 ], equality follows. 4. Let us first prove that It comes from = [e D2, X] [e D2, X] = e sd2 Xe (1 s)d2 = = 1 = 1 e sd2 [D 2, X]e (1 s)d2 ds. 1 d ds (e sd2 Xe (1 s)d2 )ds e sd2 ( D 2 )Xe (1 s)d2 + e sd2 XD 2 e (1 s)d2 ds e sd2 [D 2, X]e (1 s)d2 ds. Replacing D 2 by (t j+1 t j )D 2 and using the substitution u = (t j+1 t j )s + t j, we obtain = [e (t j+1 t j )D 2, X] + tj+1 t j (tj+1 u)d2 e [D 2, X]e (u t j)d 2 du. Inserting this into the definition of F,..., [D 2, F j ],..., F n n D gives the formula. Definition

28 Chapter 3. JLO Theory The odd JLO character formula Ch JLO(D) C (A) of a weakly θ-summable odd unbounded Breuer Fredholm module (ρ, N, D) is defined to be Ch JLO(D) := k= Ch 2k+1 JLO (D). 2. The even JLO character formula Ch + JLO(D) C + (A) of a weakly θ-summable even unbounded Breuer Fredholm module (ρ, N, D) is defined to be Ch + JLO(D) := Ch 2k JLO(D), where k= (Ch n JLO(D), (a,..., a n ) n ) := a, [D, a 1 ],..., [D, a n ] n D. Theorem The JLO character formula Ch JLO(D) defines an entire cyclic cocycle in HE (A). More specifically, Ch JLO(D) C ω(a) and (b + B)Ch JLO(D) =. The following norm estimate will show that Ch JLO(D) is an entire cochain. Whenever we have an operator affiliated with N, let us implicitly assume it to be either even or odd with respect to χ. Lemma Let (ρ, N, D) be a weakly θ-summable unbounded Breuer Fredholm module over A. If F j and R j are operators in N for j =,..., n, and at most k of the operators F j are non-zero, then for ε [, 1), ) ( ) F D 1+ε + R,..., F n D 1+ε n k + R n 2 τ (e (1 δ)d2 D (1 ε)δe (n k)! where < δ < 1 2e. n ( F j + R j ) For the purpose of future applications, Lemma is a slight strengthening of the version in [25] in which ε is taken to be zero. The proof in [25] carries through to the present setting with minor modications. j=

29 Chapter 3. JLO Theory 23 Proof. From the generalized Hölder inequality, Theorem 2.1.4(1), the following estimate holds: τ(χt... T n ) τ( χt... T n ) = χt... T n 1 T s 1 if s + + s n = 1. Therefore, For each F D 1+ε + R,..., F n D 1+ε + R n (F D 1+ε + R )e s D 2 s 1 n (F n D 1+ε + R n )e s nd 2 s 1 n... T n s 1 n d n s. (F D 1+ε + R)e sd2 s 1, observe that by using Proposition and functional calculus, F D 1+ε e sd2 s 1 F D 1+ε e e δsd2 s(1 δ)d 2 s 1 F sup x R ( x 1+ε e δsx2) e s(1 δ)d2 s 1 and Re sd2 s 1 e R e sδd2 s(1 δ)d 2 s 1 R sup x R ( e sδx2) e s(1 δ)d2 s 1. Since the function x 1+ε e δsx2 is bounded by ( ) 1+ε 1+ε 2 and e sδx2 is bounded by 1, we 2δes can put together the above terms using Theorem 2.1.4(ii) and get that (F D 1+ε + R)e sd2 s 1 ( (1 ) 1+ε + ε 2 ( s F + R ) τ(e )) (1 δ)d2. 2δes Keeping in mind that at most k of the F j s are non-zero, we get F D 1+ε + R,..., F n D 1+ε n + R n D n ( ) 1+ε 1 + ε 2 k τ(e (1 δ)d2 ) ( F j + R j ) (s... s k 1 ) 1+ε 2 d n s. 2δe n j= Along with the estimates ( ) 1+ε 1 + ε 2 k ( ) k 1 2δe δe

30 Chapter 3. JLO Theory 24 and the proof is complete. n (s... s k 1 ) ( ) k 1+ε 2 2 d n 1 s 1 ε (n k)!, The above norm estimate immediately implies that Ch n JLO(D) < 1 n! τ(e (1 δ)d2 )C n. Therefore, Ch JLO(D) is an entire cochain when τ(e (1 δ)d2 ) <, which is exactly the weak θ-summability condition. Proof of Theorem Lemma guarantees that Ch JLO(D) is entire. What remains to check is that Ch JLO (D) is (b + B) closed. Let us adapt the computation in [28] to the type II case. Let us compute Ch n JLO(D) paired with b(a,..., a n+1 ) n+1 : (Ch n JLO(D), b(a,..., a n+1 ) n+1 ) = a a 1, [D, a 2 ],..., [D, a n+1 ] n D n + ( 1) j a,..., [D, a j a j+1 ],... n D j=1 +( 1) n+1 a n+1 a, [D, a 1 ],..., [D, a n ] n D = a a 1, [D, a 2 ],... n D a, a 1 [D, a 2 ],... n D n + ( 1) ( j 1 a,..., [D, a j 1 ]a j,... n D j=2 + a,..., a j [D, a j+1 ],... n ) D 3.1.2(4) = +( 1) n ( a, [D, a 1 ],..., [D, a n ]a n+1 n D a n+1 a, [D, a 1 ],..., [D, a n ] n D ) n+1 ( 1) j 1 a,..., [D 2, a j ],... n+1 j=1 The last term forms a telescoping sum and reduces to D. a D, [D, a 1 ],... n+1 D +( 1)n a, [D, a 1 ],..., [D, a n+1 ]D n+1 D = [D, a ],..., [D, a n+1 ] n+1 D. Now apply Lemma 3.1.2(1)(2); one checks that [D, a ],..., [D, a n+1 ] n+1 D = ( Ch n+2 JLO (D), B(a,..., a n+1 ) n+1 ).

31 Chapter 3. JLO Theory 25 Therefore, bch n JLO(D) = BCh n+2 JLO (D) and (b + B)Ch JLO(D) =. The proof is complete. As a result, the JLO character formula defines an entire cyclic cohomology. Let us call this class the (type II) JLO class. 3.2 Homotopy Invariance of the JLO Class In this section, it will be shown that the cohomology class given by the JLO character formula is homotopy invariant. As a consequence, the JLO character formula descends to a well-defined map from (semi-finite) K-homology to entire cyclic cohomology. We follow closely the work of Getzler and Szenes [25]. Definition Let V be an operator affiliated with N. Define the contraction ι(v ) by V to be ι(v ) F,..., F n n D := n k= F,..., F k, V, F k+1,..., F n n+1 D. Definition Let V be an operator affiliated with N which it has the same degree as D, i.e., with D χ = V χ. Define Ch JLO(D, V ) to be given by the equation (Ch n JLO(D, V ), (a,..., a n ) n ) n+1 := ( 1) j a, [D, a 1 ],..., [D, a j 1 ], V,..., [D, a n ] n+1 j=1 D. Theorem Let (ρ, N, D) be a weakly θ-summable unbounded Breuer Fredholm module. 1. Ch JLO(D, V ) is an entire cochain if V = F D 1+ε + R where ε < 1, F and R are operators in N. 2. Let V be an operator affiliated with N with the same degree as D, i.e., such that D χ = V χ. Then bch n 1 JLO (D, V ) + BCh n+1 JLO (D, V ) = ι(dv + V D)Ch n JLO(D) + α n (D, V ), (3.1)

32 Chapter 3. JLO Theory 26 where α n (D, V ) is defined to be (α n (D, V ), (a,..., a n )) := n a, [D, a 1 ],..., [V, a j ],..., [D, a n ] n D. j=1 Proof. 1. From Lemma 3.1.4, we have that Therefore, the sum ( ) Ch n 2 (n + 1) JLO(D, V ) τ(e (1 ε)d2 )C n. (1 ε)δe n! Γ( n 2 ) Chn JLO(D, V ) z n n= is an entire function of z and so Ch JLO(D, V ) is entire. 2. Recall that ( ) Ch n 1 JLO (D, V ), (b,..., b n 1 ) n 1 n = ( 1) j 1 b,..., [D, b j 1 ], V, [D, b j ],..., [D, b n 1 ] n D. j=1 Denote by E j the cochain (E j, (b,..., b n 1 ) n 1 ) := b,..., [D, b j 1 ], V, [D, b j ],..., [D, b n 1 ] n D, so that Ch n 1 JLO (D, V ) = n ( 1) j E j. j=1 First let us compute E j paired with b(a,..., a n ) n : (E j, b(a,..., a n ) n ) = a a 1,..., [D, a j ], V, [D, a j+1 ],..., [D, a n ] n D j 1 + ( 1) k a,..., [D, a k a k+1 ],..., [D, a j ], V,..., [D, a n ] n D k=1 n 1 + ( 1) k a,..., V, [D, a j ],..., [D, a k a k+1 ],..., [D, a n ] n D k=j +( 1) n a n a,..., V, [D, a j ],..., [D, a n 1 ] n D.

33 Chapter 3. JLO Theory 27 By expanding the [D, a k a k+1 ] terms using the Leibniz rule and re-ordering the sum, we get (E j, b(a,..., a n ) n ) = a a 1,..., [D, a j ], V,... n D a, a 1 [D, a 2 ],..., [D, a j ], V,... n D j + ( 1) k 1 (..., [D, a k 1 ]a k,..., V,... n D..., a k[d, a k+1 ],..., V,... n D ) k=2 +( 1) j 1..., a j V,... n D ( 1)j 1..., V a j,... n D n 1 + ( 1) k 1 (..., V,..., [D, a k 1 ]a k,... n D..., V,..., a k[d, a k+1 ],... n D ) k=j +( 1) n 1 (..., V,..., [D, a n 1 ]a n n D a na,..., V,..., [D, a n 1 ] n D ). We are now in the setting to apply Lemma 3.1.2(4) to obtain (E j, b(a,..., a n ) n ) = ( 1) j..., [V, a j ],... n D j + ( 1) k 1 a,..., [D 2, a k ],..., V,... n+1 D + k=1 n k=j ( 1) k 1 a,..., V,..., [D 2, a k ],... n+1 D. From the facts that [D 2, a k ] = D[D, a k ] + [D, a k ]D and D commutes with e s kd 2, one observes the above forms a telescoping sum and reduces to the following: (E j, b(a,..., a n ) n ) = ( 1) j..., [V, a j ],... n D + a D,..., [D, a j ], V,... n+1 D +( 1) j 1..., [D, a j ], DV,... n+1 D + ( 1)j 1..., V D, [D, a j ],... n+1 D +( 1) n 1..., V, [D, a j ],..., [D, a n ]D n+1 D = ( 1) j..., [V, a j ],... n D + a D,..., [D, a j ], V,... n+1 D + Da,..., V, [D, a j ],..., [D, a n ] n+1 D +( 1) j 1..., [D, a j ], DV,... n+1 D + ( 1)j 1..., V D, [D, a j ],... n+1 D.

34 Chapter 3. JLO Theory 28 Now let us sum over j with the appropriate sign: ( n j=1 = ( 1) j E j, b(a,..., a n ) n ) (3.2) n..., [V, a j ],... n D (3.3) j=1 n j= n j=..., [D, a j ], DV + V D,... n+1 D (3.4) ( 1) j [D, a ],..., [D, a j ], V,..., [D, a n ] n+1 D. (3.5) Equations (3.3) and (3.4) give α n (D, V ) and ι(dv + V D)Ch n JLO(D) respectively. By using Lemma 3.1.2(1)(2), each summand in Equation (3.5) can be written as ( 1) j [D, a ],..., [D, a j ], V,... n+1 D (3.6) j (2) = ( 1) j..., [D, a k 1 ], 1,..., V,... n+2 D (3.7) 3.1.2(1) = +( 1) j k= n k=j+1..., V,..., 1, [D, a k ],... n+2 D (3.8) j+1 ( ) ( 1) j+2 k E j+2 k, ( 1) nk (1, a k,..., a k 1 ) n+1 k= + n k=j+1 (3.9) ( ( 1) n k+j+3 E n k+j+3, ( 1) nk (1, a k,..., a k 1 ) n+1 ). (3.1) Now let us sum over j = to n, and introduce a change of indices of i = j k+1 for Equation (3.9) and i = n k + j + 1 for Equation (3.1). Then the expression (3.5) becomes ( ) n n i ( 1) i+1 E i+1, ( 1) nk (1, a k,..., a k 1 ) n+1 i= + k= ( n ( 1) i+2 E i+2, i= ) n ( 1) nk (1, a k,..., a k 1 ) n+1, k=n i which equals ( Ch n+1 JLO (D, V ), B(a,..., a n ) n ). Hence we have obtained bch n 1 JLO (D, V ) = ι(dv + V D)Ch n JLO(D) + α n (D, V ) BCh n+1 JLO (D, V ), which is the desired result.

35 Chapter 3. JLO Theory 29 Suppose that D t is a 1-parameter family of operators defining a differentiable family of weakly θ-summable unbounded Breuer Fredholm modules (ρ, N, D t ). By this, it is meant that D t is a 1-parameter family of self-adjoint operators on H with common domain of definition such that the following conditions are satisfied: ˆ D t is affiliated with N for all t, ˆ For all a A, [D t, ρ(a)] is a norm-differentiable family of operators in N, and there is a constant C for each compact interval such that [D t, ρ(a)] C a for t in this interval, ˆ (1 + D 2 t ) 1/2 is a norm-differentiable family of operators in K N, ˆ There exists a u (, 1) such that τ(e ud2 t ) is bounded for each compact interval. If (ρ, N, D t ) is equipped with a Z 2 grading χ N so that ρ(a) is even for all a A and D t is odd for all t, then analogously, let us call the family of Breuer Fredholm modules (ρ, N, D t ) even. The differentiable families of unbounded operators in our discussion will often be functions of D, and so we do not alter the spectral projections. For more general notions of differentiable family of unbounded operators, the reader may refer to [32]. Lemma ([12]). Let (ρ, N, D t ) be a differentiable family of weakly θ-summable unbounded Breuer Fredholm modules, and let F,..., F n be operators affiliated with N. Then d dt F,..., F n n D t = n j F,..., F j, D t D t + D t D t, F j+1,..., F n n+1 D t. Theorem If D t = F t D t 1+ε + R t for ε < 1 and F t, R t N are continuous families of operators that are uniformly bounded in t then Ch JLO(D t, D t ) is an entire

36 Chapter 3. JLO Theory 3 cochain and for every n, dch n JLO(D t ) dt = bch n 1 JLO (D t, D t ) + BCh n+1 JLO (D t, D t ). Proof. By applying the Leibniz rule on d dt Chn JLO(D t ), we will get terms containing d dt e (t j+1 t j )D 2 t and terms containing d dt [D t, a j ]. D t D t )Ch n JLO(D t ), while the latter collects into α n (D t, D t ). Hence together with Theorem 3.2.1(3), By Lemma 3.2.2, the former collects into ι(d t D t + dch n JLO(D t ) dt = ι(d D t + D t D)Ch n JLO(D t ) + α n (D t, D t ) = bch n 1 JLO (D t, D t ) + BCh n+1 JLO (D t, D t ). The fact that Ch JLO(D t, D t ) is entire follows from Lemma and the uniform boundedness of F t and R t. The result is obtained. The following Proposition gives a stability of bounded perturbation of weakly θ- summable unbounded Breuer Fredholm modules. It is the type II analogue of Theorem C of [25]. Proposition Let there be given a weakly θ-summable unbounded Breuer Fredholm module (ρ, N, D), and an operator V N such that V has the same degree as D, i.e., such that V χ = D χ. Then (ρ, N, D + V ) is again a weakly θ-summable unbounded Breuer Fredholm module and τ ( e (1 ε/2)(d+v )2) ( e (1+2/ε) V 2 τ e (1 ε)d2). Proof. It is obvious that [D + V, a] (C + 2 V ) a, ) and the first statement follows. It remains to prove that τ (e (1 ε/2)(d+v )2 ) τ (e (1 ε)d2. (1+2/ε) V e 2

37 Chapter 3. JLO Theory 31 Observe that if A and B are positive operators, then τ(e A B ) τ(e A ). Consider the operators A = (1 ε)d 2, B = ε 2 D2 + (1 ε 2 ) ( DV + V D + V 2) + (1 + 2 ε ) V 2. A is a positive operator, and to see that B is also positive, we use the fact that (DV + V D) ε 2 D2 + 2 ε V 2 ε 2 D2 + 2 ε V 2. Therefore, (e (1 ε)d2) ( τ e (1 ε 2 )D2 (1 ε 2 )(DV +V D+V 2 ) (1+ 2 ε ) V 2) = τ ( e A B) τ ( e A) = τ ( τ e (1 ε/2)(d+v )2) ( e (1+2/ε) V 2 τ e (1 ε)d2), and the desired inequality follows immediately. 3.3 The JLO Character as an Index Formula This section will show that the JLO character formula for a weakly θ-summable even unbounded Breuer Fredholm module produces an index formula. For the odd case, we refer to the work of Carey and Phillips [12], who developed the JLO character in the type II setting. Theorem ([12]). Let (ρ, N, D) be an odd weakly θ-summable unbounded Breuer Fredholm module over A and u M N (A) be a unitary. Then [(ρ, N, D)], [u] = ( [Ch JLO(D)], [ch (u)] ), where the angle bracket on the left is the spectral flow pairing [12] and the round bracket on the right is the (co)homology pairing.

38 Chapter 3. JLO Theory 32 Theorem Let (ρ, N, D) be an even weakly θ-summable unbounded Breuer Fredholm module over A and p M N (A) be a projection. Then [(ρ, N, D)], [p] = ( [Ch + JLO(D)], [ch + (p)] ), where the angle bracket on the left is the index pairing and the round bracket on the right is the (co)homology pairing. Proof. It suffices to prove that Ind τ (p (D 1 N )p + ) = ( Ch + JLO(D), ch + (p) ). It follows from the definition of (co)homology that the above equality will descend to the result stated in the theorem. For any projection p A, one can deform D to (pdp + (1 p)d(1 p)) via the homotopy D t = D + t(2p 1)[D, p] where t [, 1]. As D t = (2p 1)[D, p] is odd and in N, by Proposition 3.2.4, (ρ, N, D t ) is a differentiable family of weakly θ-summable unbounded Breuer Fredholm modules. By Theorem 3.2.3, Ch + JLO(D) and Ch + JLO (pdp + (1 p)d(1 p)) are cohomologous. Specifically, Therefore, Ch + JLO (pdp + (1 p)d(1 p)) Ch + JLO(D) = Ch + JLO(D 1 ) Ch + JLO(D ) = (b + B) 1 Ch + JLO(D t, D t ). ( Ch + JLO(D), ch + (p) ) = ( Ch + JLO (pdp + (1 p)d(1 p)), ch + (p) ) ( 1 ) (b + B) Ch + JLO(D t, D t ), ch + (p) = ( Ch JLO (pdp + (1 p)d(1 p)), ch + (p) ) where the last equality follows from the fact that [D 1, p] = [pdp + (1 p)d(1 p), p] = and ch + (p) is closed. Hence the pairing ( Ch + JLO(D), ch + (p) ) yields the McKean-Singer

39 Chapter 3. JLO Theory 33 index formula τ(χpe D2 ). The fact that the McKean-Singer index formula produces the desired index Ind τ (p Dp + ) is proved in [16]. If p is a projection in M N (A), one extends D to D 1 N and τ to τ Tr, where Tr is the matrix trace from M N (C) C, and the result follows.

40 Chapter 4 The Cohomological Chern Character The Connes character formula assigns to each finitely summable bounded Fredholm module an entire cyclic cocycle, and it descends to define a cohomological Chern character. In analogy with the work in Chapter 3, a counterpart of the Connes character formula for p- summable bounded Breuer Fredholm modules will be developed. The Connes character formula descends to a map from K-homology classes to entire cyclic cohomology classes. While this appears to be a fairly standard fact, I am unaware of a published argument and hence a proof to Theorem is included. In Section 4.2, the JLO character formula is simplified when the unbounded Breuer Fredholm module is p-summable. Section 4.3 shows that a p-summable unbounded Breuer Fredholm module with a certain property gives rise to a p-summable bounded Breuer Fredholm module. Section 4.4 proves that the entire cyclic cohomology class of a p-summable unbounded Breuer Fredholm module coincides with its bounded counterpart. Much of this chapter is based on the work of Connes-Moscovici [21]. The chapter also includes some analysis concerning the entire cochains, which was left out in the setting of periodic cyclic cohomology used in [21]. 34