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1 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P.-23 Journal of Functional Analysis Ihara s zeta function for periodic graphs and its approximation in the amenable case Daniele Guido a,, Tommaso Isola a, Michel L. Lapidus b 4 a Dipartimento di Matematica, Università di Roma Tor Vergata, I-0033 Roma, Italy 4 5 b Department of Mathematics, University of California, Riverside, CA , USA 5 6 Received 5 August 2006; accepted 7 July Communicated by J. Bourgain 9 Abstract In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi [B. Clair, S. Mokhtari-Sharghi, Zeta functions of discrete groups acting on trees, J. Algebra ] on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this context and examine its consequences for the Ihara zeta function. Moreover, we answer in the affirmative one of the questions raised in [R.I. Grigorchuk, A. Żuk, The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps, in: V.A. Kaimanovich, et al. Eds., Proc. Workshop, Random Walks and Geometry, Vienna, 200, de Gruyter, Berlin, 2004, pp. 4 80] by Grigorchuk and Żuk. Accordingly, we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs Elsevier Inc. All rights reserved Keywords: Periodic graphs; Ihara zeta function; Analytic determinant; Determinant formula; Functional equations; Amenable groups; Amenable graphs; Approximation by finite graphs 35 The first and second authors were partially supported by MIUR, GNAMPA and by the European Network Quantum Spaces Noncommutative Geometry HPRN-CT The third author was partially supported by the National 42 Science Foundation, the Academic Senate of the University of California, and GNAMPA * Corresponding author addresses: guido@mat.uniroma2.it D. Guido, isola@mat.uniroma2.it T. Isola, lapidus@math.ucr.edu M.L. Lapidus /$ see front matter 2008 Elsevier Inc. All rights reserved doi:0.06/j.jfa the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

2 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P D. Guido et al. / Journal of Functional Analysis 0. Introduction The zeta functions associated to finite graphs by Ihara [20], Hashimoto [5,8], Bass [4] and others, combine features of Riemann s zeta function, Artin L-functions, and Selberg s zeta function, and may be viewed as analogues of the Dedekind zeta functions of a number field. They are defined by an Euler product and have an analytic continuation to a meromorphic function satisfying a functional equation. They can be expressed as the determinant of a perturbation of the graph Laplacian and, for Ramanujan graphs, satisfy a counterpart of the Riemann hypothesis [28]. Other relevant papers are [3,0,6,7,9,2,22,25,27,29 3]. In differential geometry, researchers have first studied compact manifolds, then infinite covers of those, and finally, noncompact manifolds with greater complexity. Likewise, in the graph setting, one passes from finite graphs to infinite periodic graphs, and then possibly to other types of infinite graphs. In fact, the definition of the Ihara zeta function was extended to countable periodic graphs by Clair and Mokhtari-Sharghi [8], and a corresponding determinant formula was proved. They deduce this result as a specialization of the treatment of group actions on trees the so-called theory of tree lattices, as developed by Bass, Lubotzky and others, see [5]. We mention [3] for a recent review of some results on zeta functions for finite or periodic simple graphs, and [7 9,2] for the computation of the Ihara zeta function of several periodic simple graphs. In [2], Grigorchuk and Żuk defined zeta functions of infinite discrete groups, and of some class of infinite periodic graphs which they call residually finite, and asked how to obtain the zeta function of a periodic graph by means of the zeta functions of approximating finite subgraphs, in the case of amenable or residually finite group actions. The purpose of the present work is twofold: first, to give a different proof of the main result obtained by Clair and Mokhtari-Sharghi in [8]; second, to answer in the affirmative one of the questions raised by Grigorchuk and Żukin[2]. As for the first point, some combinatorial results in Section give a more direct proof of the determinant formula in Theorem 4.. Moreover, the theory of analytic determinants developed in Section 3 allows us to use analytic functions instead of formal power series in that formula, as well as to establish functional equations for suitable completions of the Ihara zeta function, generalizing results contained in [3]. As for the second point, we take advantage of the technical framework developed in this paper to show, in the case of amenable group actions, that the Ihara zeta function is indeed the limit of the zeta functions of a suitable sequence of approximating finite graphs. For the sake of completeness, we mention that, in [9], Clair and Mokhtari-Sharghi have given a positive answer in the case of residually finite group actions. This paper is organized as follows. We start in Section by recalling some notions from graph theory and prove all the combinatorial results we need in the following sections. In Section 2, we then define the analogue of the Ihara zeta function and show that it is a holomorphic function in a suitable disc, while, in Section 4, we prove a corresponding determinant formula, which relates the zeta function with the Laplacian of the graph. The formulation and proof of this formula requires some care because it involves the definition and properties of a determinant for bounded operators acting on an infinite-dimensional Hilbert space and belonging to a von Neumann algebra with a finite trace. This issue is addressed in Section 3. In Section 5, we establish several functional equations for various possible completions of the zeta function. In the final section, we prove the approximation result mentioned above the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

3 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P.3-23 D. Guido et al. / Journal of Functional Analysis 3 In closing this introduction, we note that in [4] we define and study the Ihara zeta functions attached to a new class of infinite graphs, called self-similar fractal graphs, which have greater complexity than the periodic ones. The contents of this paper have been presented at the 2st Conference on Operator Theory in Timisoara Romania in July Preliminary results We recall some notions from graph theory, following [26]. A graph X = VX, EX consists of a collection VX of objects, called vertices, and a collection EX of objects called oriented edges, together with two maps e EX oe, te VX VX and e EX e EX, satisfying the following conditions: e = e, oe = te, e EX. The vertex oe is called the origin of e, while te is called the terminus of e. The edge e is said to join the vertices u := oe, v := te, while u and v are said to be adjacent, which is denoted u v. The edge e is called a loop if oe = te.thedegree of a vertex v is degv := {e EX: oe = v}, where denotes the cardinality. A path of length m in X from u = oe VX to v = te m VX is a sequence of m edges e,...,e m, where oe i+ = te i,fori =,...,m. In the following, the length of a path C is denoted by C. A path is closed if u = v. A graph is said to be connected if there is a path between any pair of distinct vertices. The couple {e,e} is called a geometric edge.anorientation of X is the choice of one oriented edge for each couple, which is called positively oriented. Denote by E + X the set of positively oriented edges. Then the other edge of each couple will be called negatively oriented, and denoted e, if e E + X. The set of negatively oriented edges is denoted E X. Then EX = E + X E X. In this paper, we assume that the graph X = VX, EX is connected, countable i.e. VX and EX are countable sets and with bounded degree i.e. d := sup v VX degv <. We also choose, once and for all, an orientation of X. Let Γ be a countable discrete subgroup of automorphisms of X, which acts without inversions, i.e. γe e, γ Γ, e EX, 2 discretely, i.e. Γ v := {γ Γ : γv= v} is finite, v VX, 3 with bounded covolume, i.e. volx/γ := v F 0 Γ v <, where F 0 VX contains exactly one representative for each equivalence class in VX/Γ. 33 We note that the above bounded covolume property is equivalent to 40 Γ e <, e F 40 volex/γ := where F EX contains exactly one representative for each equivalence class in EX/Γ. Let us now define two useful unitary representations of Γ. Denote by l 2 VX the Hilbert space of functions f : VX C such that f 2 := v VX fv 2 <. A unitary representation of Γ on l 2 VX is given by λ 0 γ f x := fγ x, forγ Γ, f l 2 VX, x VX. Then the von Neumann algebra N 0 X, Γ := 44 the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

4 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P D. Guido et al. / Journal of Functional Analysis {λ 0 γ : γ Γ } of all the bounded operators on l 2 VX commuting with the action of Γ, inherits a trace given by Tr Γ A := 5 Γ x Ax,x, A N 0X, Γ.. 5 x F 0 Analogously, denote by l 2 EX the Hilbert space of functions ω : EX C such that ω 2 := e EX ωe 2 <. A unitary representation of Γ on l 2 EX is given by λ γ ωe := ωγ e, forγ Γ, ω l 2 EX, e EX. Then the von Neumann algebra N X, Γ := {λ γ : γ Γ } of all the bounded operators on l 2 EX commuting with the action of Γ, inherits a trace given by Γ 5 e Ae,e, A N X, Γ..2 e F 5 Tr Γ A := At this stage, we need to introduce some additional terminology from graph theory. Definition. Reduced paths. i A path e,...,e m has backtracking if e i+ = e i,forsomei {,...,m }. A path with no backtracking is also called proper. ii A closed path is called primitive if it is not obtained by going n 2 times around some other closed path. iii A proper closed path C = e,...,e m has a tail if there is k N such that e m j+ = e j, for j =,...,k. Denote by C the set of proper tail-less closed paths, also called reduced closed paths. Definition.2 Cycles. Given closed paths C = e,...,e m, D = e,...,e m, we say that C and D are equivalent, and write C o D, if there is k N such that e j = e j+k, for all j, where e m+i := e i, that is, the origin of D is shifted k steps with respect to the origin of C. The equivalence class of C is denoted [C] o. An equivalence class is also called a cycle. Therefore, a closed path is just a cycle with a specified origin. Denote by R the set of reduced cycles, and by P R the subset of primitive reduced cycles, also called prime cycles. 33 Definition.3 Equivalence relation. i Given C,D C, we say that C and D are Γ -equivalent, and write C Γ D, if there is an isomorphism γ Γ such that D = γc. We denote by [C] Γ the set of Γ -equivalence classes of reduced closed paths. ii Similarly, given C, D R, we say that C and D are Γ -equivalent, and write C Γ D, if there is an isomorphism γ Γ such that D = γc. We denote by [R] Γ the set of Γ -equivalence classes of reduced cycles, and analogously for the subset P. 44 Remark.4. In the rest of the paper, we denote by C m the subset of C consisting of closed paths of length m. An analogous meaning is attached to R m and P m. the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

5 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P.5-23 D. Guido et al. / Journal of Functional Analysis 5 Our proof of formula iv in Theorem 2.2 requires a generalization of a result by Kotani and Sunada [2] to infinite covering graphs. This is done in Proposition.6, whose proof depends on a new combinatorial result contained in Lemma.5. Define the effective length of a cycle C, denoted by lc, as the length of the prime cycle underlying C, and observe that lc is constant on the Γ -equivalence class of C. Therefore, if ξ [R] Γ, we can define lξ := lc, for any representative C ξ. Recall that, for any cycle C, the stabilizer of C in Γ is the subgroup Γ C := {γ Γ : γc= C}. Moreover, if C,C 2 ξ, then the stabilizers Γ C,Γ C2 are conjugate subgroups in Γ, and we denote by Sξ their common cardinality. For the purposes of the next few results, for any closed path D = e 0,...,e m,wealso denote e j by e j D. Lemma.5. Let ξ [R m ] Γ. Then 6 6 { D C 7 m : [D] o,γ = ξ, e 0 D = e } = lξ Γ 7 e Sξ. e F 8 8 Proof. Let us first observe that, if C,C 2 ξ, then e EC Γ e is conjugate in Γ to e EC 2 Γ e, and we denote by Iξ their common cardinality. Let C R m be such that [C] Γ = ξ. By choosing each time a different starting edge, we obtain l := lc lξ closed paths from C. Denote them by D,...,D l, and observe that any two of them can be Γ -equivalent, i.e. D i = γd j,forsomeγ Γ, if and only if γ Γ C. Moreover, if γ e EC Γ e Γ C, then γd i = D i,fori =,...,l. Therefore, there are only k kξ := lξiξ distinct Γ -classes of closed paths generated by the D i s, and we denote them by π,...,π k. Let π be one of them, and observe that, for any e F, there are either no closed paths D representing π and such that e 0 D = e, or there are Γ e distinct closed paths D representing π and such that e 0 D = e. Indeed, if there is a closed path D representing π and such that e 0 D = e, then any γ Γ e generates a closed path γd representing π and such that e 0 γ D = e, but, if γ e ED Γ e, then γd= D. Hence, the claim is established. Let us now introduce a discrete measure on F. Let us say that a Γ -class of closed paths π starts at e F if there is D π such that e 0 D = e. Let us set, for e F, μ ξ e =, if e is visited by some π i, i =,...,k, and μ ξ e = 0, otherwise. It is easy to see that μ ξ depends only on ξ and is in particular independent of the representative C. Observe that μ ξ F = kξ. Therefore, for any e F, we get 26 Sξ Iξ { D C m : [D] o,γ = ξ, e 0 D = e } = μ ξ e Γ e, Iξ and, finally, { 46 D C m : [D] o,γ = ξ, e 0 D = e } = μ ξ e = kξ 47 Γ e Iξ Iξ = lξ Sξ. 47 e F e F the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

6 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P D. Guido et al. / Journal of Functional Analysis Define, for ω l 2 EX, e EX, T ωe = ωe. 4 te =oe 4 5 e e 5 Then, we have Proposition.6. i T N X, Γ, T d, ii for m N, T m e = me,e,...,e m e m,fore EX, proper path iii Tr Γ T m = Nm Γ := l[c] Γ [C] Γ [R m ] Γ, the number of Γ -equivalence classes of reduced cycles of length m. Here,Tr Γ is the trace on N X, Γ introduced in.2. 4 S[C] Γ 4 Proof. i, ii are easy to check. iii Using Lemma.5, we obtain 2 Γ e T m ẽ,ẽ e F 2 23 = 23 Γ 24 e e F e,e,...,e m,e reduced path 25 = { C C 27 m : e 0 C = e } 27 Γ e 28 e F 28 = { D C 30 m : [D] 0 Γ C, e 0 D = e } Γ 30 e [C] 3 Γ [R] Γ e F 3 Tr Γ T m = The Zeta function = N Γ m. Before introducing the zeta function of an infinite periodic graph, we recall its definition for a finite q + -regular graph X i.e. such that degv = q +, for all v VX. In that case, the Ihara zeta function Z X is defined by an Euler product of the form Z X u := u C, for u < q, 2. C P where P is the set of prime cycles of X. By way of comparison, recall that the Riemann zeta function is given by the Euler product 44 ζs:= p p s, for Re s>, 2.2 the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

7 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P.7-23 D. Guido et al. / Journal of Functional Analysis 7 where p ranges over all the rational primes. To see the correspondence between Z X and ζ,simply let u := q s and observe that u C = q C s. Also note that u < q if and only if Re s>. Let us now return to the case of periodic graphs and introduce the Ihara zeta function via its Euler product as well as show that this defines a holomorphic function in a suitable disc. Definition 2. Zeta function. Let Zu = Z X,Γ u be given by u C 9 Z X,Γ u := Γ C, 9 0 [C] Γ [P] Γ 0 for u C sufficiently small so that the infinite product converges. In the following proposition we let det Γ B := exp Tr Γ logb, for B N X, Γ. We refer to Section 3 for more details. Formula iv in the following theorem was first established in [8], although with a different proof. Theorem 2.2. i Zu := [C] Γ [P] Γ u C Γ C defines a holomorphic function in the open disc {u C: 25 u < d 25 ii u Z u 27 Zu = m= Nm Γ um,for u < d 27 iii Zu = exp m= N Γ m m u m,for u < d. 30 iv Zu = det Γ I ut,for u < d. 30 Proof. Observe that it follows from Proposition.6 that N Γ m= m um defines a function which is holomorphic in {u C: u < }. Moreover, for any u C such that u < 33 d d, Nm Γ um = l[c] Γ S[C] Γ u C [C] Γ [R] Γ m= = C Γ C u Cm 4 [C] Γ [P] Γ m= 4 43 C u C m 43 Γ 44 C [C] Γ [P] Γ m= 44 = = 47 Γ C u d u C m du m 47 [C] Γ [P] Γ m= the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

8 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P D. Guido et al. / Journal of Functional Analysis Fig.. A periodic graph. 25 Fig. 2. A cycle with Γ C = Fig. 3. A cycle with Γ C =2. 38 = 4 Γ C u d du log u C 4 [C] Γ [P] Γ = u d log Zu, du 44 where, in the last equality, we have used uniform convergence on compact subsets of {u C: u < }. From what has already been proved, i iii follow. Finally, for u < d,wehave d the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

9 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P.9-23 D. Guido et al. / Journal of Functional Analysis 9 Nm Γ log Zu = 2 m um 2 m= = m Tr Γ T u m T u m = Tr Γ 6 m= 6 m 9 m= 9 = Tr Γ logi ut. Example 2.3. Some examples of cycles with different stabilizers are shown in Figs. 2, 3. They refer to the graph in Fig. which is the standard lattice graph X = Z 2 endowed with the action of the group Γ generated by the reflection along the x-axis and the translations by elements m, n Z 2, acting as m, nv,v 2 := v + 4m, v 2 + 4n,forv = v,v 2 VX = Z An analytic determinant for von Neumann algebras with a finite trace In this section, we define a determinant for a suitable class of not necessarily normal operators in a von Neumann algebra with a finite trace. The results obtained are used in Section 4 to prove a determinant formula for the zeta function. In a celebrated paper [], Fuglede and Kadison defined a positive-valued determinant for finite factors i.e. von Neumann algebras with trivial center and finite trace. Such a determinant is defined on all invertible elements and enjoys the main properties of a determinant function, but it is positive-valued. Indeed, for an invertible operator A with polar decomposition A = UH, where U is a unitary operator and H := A A is a positive self-adjoint operator, the Fuglede Kadison determinant is defined by deta = exp τ log H, where log H may be defined via the functional calculus. Note, however, that the original definition was only given for a normalized trace. For the purposes of the present paper, we need a determinant which is an analytic function. As we shall see, this can be achieved, but corresponds to a restriction of the domain of the determinant function and implies the loss of some important properties. In particular, the product formula of the Fuglede Kadison determinant only holds under certain restrictions in our case; see Propositions 3.4, 3.6, 3.7 and 3.8. Let A,τ be a von Neumann algebra endowed with a finite trace. Then, a natural way to obtain an analytic function is to define, for A A, det τ A = exp τ log A, where 33 loga := log λλ A dλ, 2πi 43 Γ and Γ is the boundary of a connected, simply connected region Ω containing the spectrum of A. Clearly, once the branch of the logarithm is chosen, the integral above does not depend on Γ, provided Γ is given as above. the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

10 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P D. Guido et al. / Journal of Functional Analysis Then a naïve way of defining det is to allow all elements A for which there exist an Ω as above, and a branch of the logarithm whose domain contains Ω. Indeed, the following holds. Lemma 3.. Let A, Ω, Γ be as above, and ϕ, ψ two branches of the logarithm such that both domains contain Ω. Then exp τ ϕa = exp τ ψa. Proof. The function ϕλ ψλ is continuous and everywhere defined on Γ. Since it takes its values in 2πiZ, it should be constant on Γ. Therefore, exp τ ϕa = exp τ 2πin 0 λ A dλ exp τ ψa 2πi 4 Γ 4 = exp τ ψa. The problem with the previous definition is its dependence on the choice of Ω. Indeed, it is easy to see that when A = 0 and we choose Ω containing {e iϑ,ϑ [0,π/2]} and any suitable branch of the logarithm, we get deta = e iπ/4, if we use the normalized trace on 2 2 matrices. By contrast, if we choose Ω containing {e iϑ,ϑ [π/2, 2π]} and a corresponding branch of the logarithm, we get deta = e 5iπ/4. Therefore, we make the following choice. 9 0 i 9 Definition 3.2. Let A,τ be a von Neumann algebra endowed with a finite trace, and consider the subset A 0 ={A A: 0/ conv σa}, where σa denotes the spectrum of A. For any A A 0 we set det τ A = exp τ log λλ A dλ, 2πi 30 Γ 30 where Γ is the boundary of a connected, simply connected region Ω containing conv σa, and log is a branch of the logarithm whose domain contains Ω. 33 Corollary 3.3. The determinant function defined above is well defined and analytic on A 0. We collect several properties of our determinant in the following result. Proposition 3.4. Let A,τ be a von Neumann algebra endowed with a finite trace, and let A A 0. Then i det τ za = z τi det τ A, for any z C \{0}, ii if A is normal, and A = UH is its polar decomposition, 44 det τ A = det τ U det τ H, iii if A is positive, det τ A = DetA, where the latter is the Fuglede Kadison determinant. the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

11 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P.-23 D. Guido et al. / Journal of Functional Analysis Proof. i If, for a given ϑ 0 [0, 2π, the half-line {ρe iϑ 0 C: ρ>0} does not intersect conv σa, then the half-line {ρe iϑ0+t C: ρ>0} does not intersect conv σza, where z = re it. If log is the branch of the logarithm defined on the complement of the real negative half-line, then ϕx = iϑ 0 π + loge iϑ0 π x is suitable for defining det τ A, while ψx = iϑ 0 + t π+ loge iϑ0+t π x is suitable for defining det τ za. Moreover, if Γ is the boundary of a connected, simply connected region Ω containing conv σa, then zγ is the boundary of a connected, simply connected region zω containing conv σza. Therefore, det τ za = exp τ ψλλ za dλ 2πi zγ = exp τ iϑ0 + t π+ log e iϑ0+t π re it μ μ A dμ 2πi 4 Γ 4 = exp τ log r + iti + ϕμμ A dμ 2πi 7 Γ 7 = z τi det τ A. ii When A = UH is normal, U = [0,2π] eiϑ duϑ, H = [0, rdhr, then A = [0, [0,2π] reiϑ dhr uϑ. The property 0 / conv σa is equivalent to the fact that the support of the measure dhr uϑ is compactly contained in some open half-plane { ρe iϑ : ρ>0, ϑ ϑ 0 π/2,ϑ 0 + π/2 }, or, equivalently, that the support of the measure dhr is compactly contained in 0,, and the support of the measure duϑ is compactly contained in ϑ 0 π/2,ϑ 0 + π/2. Therefore, A A 0 is equivalent to U,H A 0. Then log A = log r + iϑd hr uϑ, 32 [0, ϑ 0 π/2,ϑ 0 +π/ which implies that 35 ϑ 0 +π/2 35 det τ A = exp τ log rdhr ϑ 0 π/2 38 = det τ U det τ H. iϑ duϑ iii This follows by the argument given in ii. Remark 3.5. We note that the above defined determinant function strongly violates the product property det τ AB = det τ A det τ B. Indeed, the fact that A,B A 0 does not imply AB A 0, as is seen e.g. by taking A = B = 0. Moreover, even if A,B,AB A0 and A and B commute, the product property may be violated, as is shown by choosing A = B = 0 0 e, and using the 3iπ/4 normalized trace on 2 2 matrices i 45 the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

12 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P D. Guido et al. / Journal of Functional Analysis Proposition 3.6. Let A,τ be a von Neumann algebra endowed with a finite trace, and let A,B A. Then, for sufficiently small u C, we have det τ I + uai + ub = detτ I + ua det τ I + ub. Proof. The proof is inspired by that of Lemma 3 in []. Let us write a := logi + ua, b := logi + ub A, and let ct := e ta e b, t [0, ]. As a log u A, and b log u B, we get ct = e ta e b e b e b e a + e b 2 u B u A + u B 2 <, for all t [0, ], if we choose u sufficiently small; hence, ct A 0 for all t [0, ]. Now apply Lemma2in[]whichgives d τ log ct = τ ct c t = τ e b e ta ae ta e b = τa. dt Therefore, after integration for t [0, ], we obtain τlog c τlog c0 = τa, which means and hence implies the claim. τ log I + uai + ub = τ log c = τa+ τb = τ logi + ua + τ logi + ub, Proposition 3.7. Let A,τ be a von Neumann algebra endowed with a finite trace. Further, let A A have a bounded inverse, and let T A 0. Then 33 det τ AT A = det τ T. Proof. Indeed, for any polynomial p, wehavepat A = ApT A. Applying the Stone Weierstrass theorem on the compact set σata = σt, we obtain logat A = A logt A, from which the result follows. Proposition 3.8. Let A,τ be a von Neumann algebra endowed with a finite trace, and let T = T T 2 0 T 22 Mat2 A, with T ii A such that σt ii B := {z C: z < }, for i =, 2. Then 44 det τ T = det τ T det τ T 22. the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

13 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P.3-23 D. Guido et al. / Journal of Functional Analysis 3 Proof. Indeed, for any k N {0}, T k T k = B k 0 T22 k, for some B k A, so that, for any polynomial p, pt B pt =, 0 pt 22 for some B A. It is easy to see that σt σt σt 22 B. Hence, applying the Stone Weierstrass theorem on the compact set σt, we obtain for some C A. Therefore, as desired. logt C logt =, 0 logt 22 det τ T = exp τ logt = exp τ logt + τ logt 22 = det τ T det τ T 22, Corollary 3.9. Let Γ be a discrete group, π, π 2 unitary representations of Γ, and τ, τ 2 finite traces on π Γ and π 2 Γ, respectively. Let π := π π 2, τ := τ + τ 2, T = T T 2 0 T 22 πγ, with σt ii B ={z C: z < },fori =, 2. Then det τ T = det τ T det τ2 T 22. Proof. It is similar to the proof of Proposition The determinant formula In this section, we prove the main result in the theory of the Ihara zeta functions, which says that Z is the reciprocal of a holomorphic function, which, up to a factor, is the determinant of a deformed Laplacian on the graph. We first need some technical results. Let us denote by A the adjacency matrix of X,i.e.Af v = w v fw, f l2 VX. Then by [23,24] A d := sup v VX degv <, and it is easy to see that A N 0 X, Γ. Introduce Qf v := degv f v, v VX, f l 2 VX, and Δu := I ua + u 2 Q N 0 X, Γ, for u C. Let us recall that d := sup v VX degv, and set α := d+ 2. Then d 2 +4d 39 Theorem 4. Determinant formula. We have Z X,Γ u = u 2 χ 2 X detγ Δu, for u < α, where χ 2 X := 46 v F 0 Γ v 2 e F Γ e is the L2 -Euler characteristic of X, Γ,asintro- duced in [6]. the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

14 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P D. Guido et al. / Journal of Functional Analysis This theorem was first proved in [8] and is based on formula iv in Theorem 2.2 and the equality det Γ I ut = u 2 χ 2X det Γ Δu, for u < α. The main difference with their proof is that we use an analytic determinant and operator-valued analytic functions instead of Bass noncommutative determinant [4] and formal power series of operators. We first prove two lemmas. Define, for f l 2 VX, ω l 2 EX, 0 f e := f oe, e EX, f e := f te, e EX, σ ωv := ωe, v VX, 2 oe=v 2 J ωe := ωe, e EX, and use the short-hand notation I V := Id l 2 VX and I E := Id l 2 EX. Lemma 4.2. i J = 0, ii σλ γ = λ 0 γ σ, i λ 0 γ = λ γ i, i = 0,, γ Γ, iii σ 0 = I + Q, iv σ = A, v 0 σ = JT + I E, vi σ = T + J, vii I E uj I E ut = u 2 I E u σ + u 2 0 σ. Proof. Let f l 2 VX, v VX. Then σ 0 f v = 0 f e = f oe = + Qv,v fv, 3 oe=v oe=v 3 σ f v = f e = f te = Af v oe=v oe=v 34 Moreover, for ω l 2 EX, e EX,wehave 39 oe =te 39 The rest of the proof is clear. σ ωe = σ ω te = 0 σ = J σ = JT + J= JT + I E. ωe = T ωe + J ωe, 44 Let us now consider the direct sum of the unitary representations λ 0 and λ, namely λγ := λ 0 γ λ γ Bl 2 VX l 2 EX. Then, the von Neumann algebra λγ := {S Bl 2 VX l 2 EX: Sλγ = λγ S, γ Γ } consists of operators S = S 00 S 0 S 0 S, where the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

15 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P.5-23 D. Guido et al. / Journal of Functional Analysis 5 S ij λ j γ = λ i γ S ij, γ Γ, i, j = 0,, so that S ii λ i Γ N i X, Γ, i = 0,. Hence λγ inherits a trace given by Introduce S00 S Tr 0 Γ := Tr S 0 S Γ S 00 + Tr Γ S. 4. u 2 I V 0 9 I 9 Lu := and Mu := V uσ 0 u 0 I E u 0 u 2, I E 0 which both belong to λγ. Then, we have Lemma 4.3. Δu uσ i MuLu = 0 u 2, I E u ii LuMu = 2 I V u 2 uσ. 0 I E uj I E ut Moreover, for u sufficiently small, iii Lu, Mu are invertible, with a bounded inverse, iv det Γ MuLu = u 2 Tr Γ I E det Γ Δu, v det Γ LuMu = u 2 Tr Γ I V 2 Tr Γ I E det Γ I E ut. Proof. The formulas for MuLu and LuMu follow from the previous lemma. Moreover, for u sufficiently small, σδu, σ u 2 I E, σ u 2 I V and σ I E uj I E ut B ={z C: z < }, hence σmulu and σlumu B, as in the proof of Proposition 3.8. Therefore, Lu and Mu are invertible, with a bounded inverse, for u sufficiently small. By Propositions 3.4i, 3.6 and Corollary 3.9, we obtain 33 and det Γ MuLu = detγ Δu detγ u 2 I E I 0 0 I 45 = u 2 Tr Γ I E detγ Δu det Γ LuMu = detγ u 2 I V detγ I E uj det Γ I E ut = u 2 Tr Γ I V detγ I E uj det Γ I E ut. Moreover, we have det Γ I E uj = u 2 2 Tr Γ I E. Indeed, using J to identify l 2 E X with l 2 E + X, we obtain a representation ρ of Bl 2 EX onto Mat 2 Bl 2 E + X, under which ρj = 0 I, ρie = I 0. Hence, by Propositions 3.6 and 3.8, det Γ I E uj = det Γ ρie uj the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

16 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P D. Guido et al. / Journal of Functional Analysis I ui = det Γ ui I I ui I ui = det Γ 0 I ui I u = det 2 I 0 Γ ui I = u 2 Tr Γ I = u 2 2 Tr Γ I E. Proof of Theorem 4.. Let us observe that, for sufficiently small u, wehave MuLu = MuLuMuMu, so that, by Proposition 3.7, we get det Γ LuMu = det Γ MuLu. Therefore, the claim follows from Lemma 4.3iv and v, Eqs.. and.2 and Theorem Functional equations In this section, we obtain several functional equations for the Ihara zeta functions of q + - regular graphs, i.e. graphs with degv = q +, for any v VX, on which Γ acts freely i.e. Γ v is trivial, for v VX and with finite quotient i.e. B := X/Γ is a finite graph. The various functional equations correspond to different ways of completing the zeta functions, as is done in [28] for finite graphs. We extend here to non-necessarily simple graphs the results contained in [3]. Lemma 5.. Let X be a q + -regular graph, on which Γ acts freely and with finite quotient B := X/Γ. Let Δu := + qu 2 I ua. Then i χ 2 X = χb= VB q/2 Z, ii Z X,Γ u = u 2 χb det Γ + qu 2 I ua,for u < q, iii by using the determinant formula in ii, Z X,Γ can be extended to a function holomorphic at least in the open set { Ω := R 2 \ x, y R 2 : x 2 + y 2 = } {x, 0 R 2 : q } q x. 33 See Fig. 4. iv det Γ Δ qu = qu2 VB det Γ Δu, foru Ω \{0}. Proof. i This follows by a simple computation. ii This follows from i. iii Let us observe that 44 σ Δu = { + qu 2 uλ: λ σa } { + qu 2 uλ: λ [ d,d] }. the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

17 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P.7-23 D. Guido et al. / Journal of Functional Analysis 7 Fig. 4. The open set Ω. It follows that 0 / conv σδu at least for u C such that + qu 2 uλ 0forλ [ d,d], that is for u = 0or +qu2 / [ d,d], or equivalently, at least for u Ω. The rest of the proof follows from Corollary 3.3. iv This follows from Proposition 3.4i and the fact that Tr Γ I V = VB. 5 u 5 The question whether the extension of the domain of Z X,Γ by means of the determinant formula is compatible with an analytic extension from the defining domain is a non-trivial issue, see the recent paper by Clair [7]. Theorem 5.2 Functional equations. Let X be a q + -regular graph, on which Γ acts freely and with finite quotient B := X/Γ. Then, for all u Ω, we have i Λ X,Γ u := u 2 χb u 2 VB /2 q 2 u 2 VB /2 Z X,Γ u = Λ X,Γ qu, ii ξ 28 X,Γ u := u 2 χb u VB qu VB Z X,Γ u = ξ X,Γ qu, iii Ξ X,Γ u := u 2 χb + qu 2 VB Z X,Γ u = Ξ X,Γ qu. 29 Proof. They all follow from Lemma 5.iv by a straightforward computation. We prove i as an example. 33 Λ X u = u 2 VB /2 q 2 u 2 VB /2 detγ Δu VB /2 36 VB /2 36 qu VB q = u VB 2 q 2 u 2 = Λ X. qu q 2 u 2 qu 2 VB det Γ Δ qu Remark 5.3. Recall that a key property of the Riemann zeta function ζ is that its meromorphic continuation satisfies a functional equation ξs = ξ s, for all s C, where ξs := π s/2 Ɣs/2ζs denotes the completion of ζ and Ɣ is the usual Gamma function. Likewise, in Theorem 5.2, any of the functional equations relates the values of the corresponding completed Ihara zeta function at s and s, provided we set u = q s, as was explained at the 44 beginning of Section 2. Note that 47 qu =. q s 47 the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

18 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P D. Guido et al. / Journal of Functional Analysis 6. Approximation by finite graphs in the amenable case In this section, we show that the zeta function of a graph, endowed with a free and cofinite action of a discrete amenable group of automorphisms, is the limit of the zeta functions of a suitable sequence of finite subgraphs, thus answering in the affirmative a question raised by Grigorchuk and Żukin[2]. Before doing that, we establish a result which is considered folklore by specialists. Roughly speaking, it states that a Γ -space is amenable if Γ is an amenable group, where a space is said to be amenable if it possesses a regular exhaustion. Such a result was stated by Cheeger and Gromov in [6] for CW-complexes and was proved by Adachi and Sunada in [2] for covering manifolds. We give here a proof in the case of covering graphs. Throughout this section, X is a connected, countably infinite graph, and Γ is a countable discrete amenable group of automorphisms of X, which acts on X freely i.e., any γ id has no fixed-points, and cofinitely i.e., B := X/Γ is a finite graph. A fundamental domain for the action of Γ on X can be constructed as follows. Let B = VB, EB be the quotient graph, and p : X B the covering map. Let EB ={e,...,e k }, where the edges have been ordered in such a way that, for each i {,...,k}, e i has at least a vertex in common with some e j, with j<i. Choose ẽ EX such that pẽ = e. Assume ẽ,...,ẽ i have already been chosen in such a way that pẽ j = e j,forj =,...,i, and, for any such j, ẽ j has at least a vertex in common with some e h, with h<j.lete i+ EB have a vertex in common with e j,forsomej {,...,i} and choose ẽ i+ VX such that pẽ i+ = e i+ and ẽ i+ has a vertex in common with ẽ j. This completes the induction. Let EF := {ẽ,...,ẽ k } and VF := {oẽ,...,oẽ k } {tẽ,...,tẽ k }, so that F = VF, EF is a connected finite subgraph of X which does not contain any Γ -equivalent edges. Then, F is said to be a fundamental domain for the action of Γ on X. Definition 6.. Let X be a countably infinite graph and Γ a countable discrete amenable group of automorphisms of X, which acts on X freely and cofinitely; further, let F be a corresponding fundamental domain. A sequence {K n : n N} of finite subgraphs of X is called an amenable exhaustion of X if the following conditions hold: i K n = γ E n γf, where E n Γ, for all n N, 33 ii n N K n = X, iii K n K n+, for all n N, FK iv if FK n := {v VK n : dv,vx \ VK n = }, then lim n n VK n = 0. Then X is called an amenable graph if it possesses an amenable exhaustion. Theorem 6.2. Let X be a connected, countably infinite graph, Γ be a countable discrete amenable subgroup of automorphisms of X which acts on X freely and cofinitely and let F be a corresponding fundamental domain. Then X is an amenable graph. Proof. The proof is an adaptation of a proof by Adachi and Sunada in the manifold case, see [2]. The finite set A := {γ Γ :distγ F, F } is symmetric i.e. γ A γ A, generates Γ as a group, and contains the unit element. Introduce the Cayley graph CΓ, A, whose vertices are the elements of Γ, and, by definition, there is one edge from γ to γ 2 iff γ γ 2 A. A subset 44 the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

19 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P.9-23 D. Guido et al. / Journal of Functional Analysis 9 E V CΓ, A is said to be connected if, for any pair of distinct vertices of E, there is a path in CΓ, A, joining those two vertices, and consisting only of vertices of E. From [, Theorem 4], it follows that there is a sequence {E j } j N of connected finite subsets of Γ such that E j = Γ, E j E j+, j N, 7 j N 7 E j A \ E j, j N, E j j A where, for any U, U 2 Γ,wesetU U 2 ={γ γ 2 : γ i U i,i=, 2}. For each n N, letk n := γ E n γf. Then {K n : n N} satisfies the claim. Indeed, let b := VF and a := F 0, so that a E n VK n b E n, n N. Moreover, for any n N,wehave FK n γ U n γf, where U n := {γ E n : there is δ A such that γδ / E n }. Indeed, let v FK n and w VX \ VK n be such that dv,w =. Then, there are γ 0,γ Γ, v 0,v VF, such that v = γ 0 v 0 and w = γ v. Moreover, we have γ 0 E n and γ / E n.letδ:= γ 0 γ, so that distf, δf = distγ 0 F,γ F dv,w =, which implies that δ A. Hence, γ 0 U n, and the claim follows. Finally, FK n U n F b En \ E n δ δ A = b E n δ \ E n δ A b A E n A \ E n b n E n b an VK n, 33 so condition iii of Definition 6. is satisfied, showing that {K n : n N} is an amenable exhaustion. Hence, X is amenable, as desired. If Ω VX, r N, we write B r Ω := {v VX: ρv,v r}, where ρ is the geodesic metric on VX. Lemma 6.3. Let X,Γ,Fbe as above. Let d := sup v VX degv <. Let {K n } be an amenable exhaustion of X, and ε n := FK n VK n 0. Then, for any r N, B r FK n d + r ε n VK n. 44 Proof. Since B r+ v = v B r v B v, the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

20 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P D. Guido et al. / Journal of Functional Analysis we have B r+ v d + B r v, giving B r v d + r, v VX, r 0. As a consequence, for any finite set Ω VX, wehaveb r Ω = v Ω B rv,giving Br Ω Ω d + r, r Therefore, B r FK n d + r FK n =d + r ε n VK n. Lemma 6.4. Let X,Γ,Fbe as above. Let {K n } be an amenable exhaustion of X. Then, for any B N 0 X, Γ, we have lim TrP K n BP K n = F 0 Tr Γ B, 2 n VK n 2 where PK n is the orthogonal projection of l 2 VX onto l 2 VK n. Proof. Denote by F 0 a subset of VF consisting of one representative vertex for each Γ -class, and let F := VF \ F 0 and δ := diam F. Then, for any n N, VK n = γ E n γ F 0 Ω n, where denotes disjoint union and Ω n B δ FK n. Indeed, if v Ω n := VK n \ γ E n γ F 0, then there is a unique γ Γ such that v γ F 0, so that γ/ E n, which implies γf VX \ VK n, and dv,vx \ VK n δ, which is the claim. Therefore, Tr PK n B = Bv,v v VK n = Bγv,γv + Bv,v v F 0 v Ω n = Bv,v + Bv,v v F 0 v Ω n 26 γ E n γ E n 28 Moreover, = E n Tr Γ B + 33 so that n VK 4 n 4 Besides, n 45 v Ω n Bv,v. Bv,v B Ω n B B δ FK n d + δ B ε n VK n, v Ω n lim v Ω n Bv,v = 0. lim E n VK n = F 0, because VK n = E n F 0 + Ω n. The claim follows. the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

21 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P.2-23 D. Guido et al. / Journal of Functional Analysis 2 Lemma 6.5. Let X, Γ be as above. Let A and Q be as in Section 4. Let fu:= Au Qu 2, for u C. Then fu < 2,for u <. d+ d 2 +2d Proof. This follows from the estimate fu u A + u 2 Q d u +d u 2, which is valid for any u C. Theorem 6.6 Approximation by finite graphs. Let X be a connected, countably infinite graph, and let Γ be a countable discrete amenable subgroup of automorphisms of X, which acts on X freely and cofinitely, and let F be a corresponding fundamental domain. Let {K n : n N} be an amenable exhaustion of X. Then 5 Z X,Γ u = lim Kn, 5 n Z K n u F0 uniformly on compact subsets of {u C: u < d+ d 2 +2d }. Proof. For a finite subset N VX, denote by PN Bl 2 VX the orthogonal projection of l 2 VX onto spann. Observe that, since N is an orthonormal basis for l 2 N, wehave TrP N = N. Let fu:= Au Qu 2 and P n := PVK n. Then log Z Kn u = 2 Tr P n Q IP n log u 2 Tr log P n I fu Pn. Moreover, Tr log P n I fu Pn = k Tr k P n fup n. 3 k= 3 Observe that, for k 2, 33 Tr P n fu k P n = Tr Pn fu Pn + Pn kpn 37 = Tr k k [ σ P n fup n + Tr P n fup j 37 n, 38 σ {,} k j= σ {,,...,} 39 4 n k Tr [ σ P n fup j ] n fupn = Tr...P n fupn j= ] fupn where P stands for Pn, the projection onto the orthogonal complement of l2 VK n in l 2 VX, and fu k Tr P n fupn. the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

22 JID:YJFAN AID:5356 /FLA [m+; v.95; Prn:30/07/2008; 22:22] P D. Guido et al. / Journal of Functional Analysis Moreover, with Ω n := B VK n \ VK n B FK n,wehave Tr P n fupn = Tr PK n f up Ω n fu Tr PΩ n = fu Ωn fu d + ε n VK n. Therefore, we obtain Tr Pn fu k k P n Tr Pn fup n 2 k fu k d + εn VK n, so that Tr log Pn I fu Pn Tr Pn log I fu P n = k Tr k P n fup n k Tr P n fu k P n 2 k fu k d + ε n VK n k 20 k= k= k= 23 Cd + ε n VK n, where the series converges for u <, by Lemma 6.5. Hence, 27 d+ d 2 +2d 27 Tr logp n I f up n TrP n logi f up n VK n VK n 0, and, by using Lemma 6.4, 33 log Z Kn u lim = VK n 2 lim TrP n Q IP n log u 2 VK n 35 n n 35 lim TrP n logi f up n 37 n VK n 37 n, = F 0 2 Tr Γ Q Ilog u 2 + Tr Γ log I fu = F 0 log Z X,Γ u, from which the claim follows Remark 6.7. Observe that 2α < < 47 d+ d 2 +2d α. 47 the amenable case, J. Funct. Anal. 2008, doi:0.06/j.jfa

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