Abstract

0. Introduction
JID:YJFAN AID:5356 /FLA D. Guido et al. / Journal of Functional Analysis ●●● (●● 2 3 2 3 The zeta functions associated to finite graphs by Ihara [20], Hashimoto [15,18], Bass [4] and others, combine features of Riemann's zeta function, Artin L-functions, and Selberg's zeta func- 5 tion, and may be viewed as analogues of the Dedekind zeta functions of a number field. They 6 are defined by an Euler product and have an analytic continuation to á 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 hypothe- 9 sis [28]. Other relevant papers are [3,10,16,17,19,21,22,25,27,29–31]. 10 In differential geometry, researchers have first studied compact manifolds, then infinite covers 11 of those, and finally, noncompact manifolds with greater complexity. Likewise, in the graph 12 setting, one passes from finite graphs to infinite periodic graphs, and then possibly to other types 13 of infinite graphs. In fact, the definition of the Ihara zeta function was extended to (countable) 14 periodic graphs by Clair and Mokhtari-Sharghi [8], and a corresponding determinant formula 15 was proved. They deduce this result as a specialization of the treatment of group actions on trees 16 (the so-called theory of tree lattices, as developed by Bass, Lubotzky and others, see [5]). We 17 mention [13] for a recent review of some results on zeta functions for finite or periodic simple 18 graphs, and [7–9,12] for the computation of the Ihara zeta function of several periodic simple 19 graphs. 20 In [12], Grigorchuk and Żuk defined zeta functions of infinite discrete groups, and of some 21 class of infinite periodic graphs (which they call residually finite), and asked how to obtain the 22 zeta function of a periodic graph by means of the zeta functions of approximating finite sub- 23 graphs, in the case of amenable or residually finite group actions. 24 The purpose of the present work is twofold: first, to give a different proof of the main result 25 obtained by Clair and Mokhtari-Sharghi in [8]; second, to answer in the affirmative one of the 28 questions raised by Grigorchuk and Żuk in [12]. 27 As for the first point, some combinatorial results in Section 1 give a more direct proof of the 28 determinant formula in Theorem 4.1. Moreover, the theory of analytic determinants developed 29 in Section 3 allows us to use analytic functions instead of formal power series in that formula, 30 as well as to establish functional equations for suitable completions of the Ihara zeta function, 31 generalizing results contained in [13]. 32 As for the second point, we take advantage of the technical framework developed in this 33 paper to show, in the case of amenable group actions, that the Ihara zeta function is indeed the 34 limit of the zeta functions of a suitable sequence of approximating finite graphs. For the sake of 35 completeness, we mention that, in [9], Clair and Mokhtari-Sharghi have given a positive answer 38 in the case of residually finite group actions. 37 This paper is organized as follows. We start in Section 1 by recalling some notions from graph 38 theory and prove all the combinatorial results we need in the following sections. In Section 2, we 39 then define the analogue of the Ihara zeta function and show that it is a holomorphic function in 40 a suitable disc, while, in Section 4, we prove a corresponding determinant formula, which relates 41 the zeta function with the Laplacian of the graph. The formulation and proof of this formula 42 requires some care because it involves the definition and properties of a determinant for bounded 43 operators (acting on an infinite-dimensional Hilbert space and) belonging to a von Neumann 44 algebra with a finite trace. This issue is addressed in Section 3. In Section 5, we establish several 45 functional equations for various possible completions of the zeta function. In the final section, 46 47 we prove the approximation result mentioned above. Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011 2 AID:5356 /FLA [m1+; v 1.95; Prn:30/07/2008; 22:22] P.3(1-23) JID:YJFAN D. Guido et al. / Journal of Functional Analysis ●●● (●● 3 2 3 4 5 8 7
1. Preliminary results
8 g 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 We note that the above bounded covolume property is equivalent to
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 ↦ (o(e), t(e)) ∈ VX × VX and e ∈ EX ↦ ē ∈ EX , satisfying the following conditions: ē = e, o(ē) = t(e). Ve ∈ EX. The vertex o(e) is called the origin of e, while t (e) is called the terminus of e . The edge e is said to join the vertices u := o(e), v := t(e), while u and v are said to be adjacent , which is denoted u ～ v . The edge e is called a loop if o(e) = t(e). The degree of a vertex v is deg(v) := |{e ∈ EX: o(e) = v}|, where | · | denotes the cardinality. A path of length m in X from u = o(e 1 ) ∈ VX to ν = t(e m ) ∈ VX is a sequence of m edges ( e 1 , . . . , e m ), where o( e i +1 ) = t ( e i ), for i = 1 , . . , m − 1 . In the following, the length of a path C is denoted by | C | . A path is closed if if = ν . A graph is said to be connected if there is a path between any pair of distinct vertices. The couple { e, ē} is called a geometric edge. An orientation 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 ori- ented edges. Then the other edge of each couple will be called negatively oriented, and denoted ē, 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 ∈ V χ deg( v ) < ∞ ). We also choose, once and for all, an orientation of X. Let Γ be a countable discrete subgroup of automorphisms of X , which acts 29 30 31 32 33 34 35 37 38 39 40 41 40 41 42 42 43 44 45 where F 1 ⊂ 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 bEVX lf (v)l 2 < 00. A unitary representation of Γ on ℓ 2 (VX) is given by (λ 0 (γ)f)(x) := x), for γ ∈ Γ , f ∈ ℓ 2 (VX), x ∈ VX . Then the von Neumann algebra N 0 (X, Γ ) := 46 47 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011 (1)  without inversions, i.e. γ(e) ≠ e, ∀γ ∈ Γ, e ∈ EX, (2)  discretely, i.e. Γν := {γ ∈ Γ: γ ν = ν} is finite, ∀ν ∈ VX, (3) with bounded covolume, i.e. vol(X/L) := ∑ v ∈ F o [ F i ] < ∞ , where F o ⊂ VX contains ex- actly one representative for each equivalence class in VX/ Γ . 36 37 38 39 In closing this introduction, we note that in [14] 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 21st Conference on Operator Theory in Timisoara (Romania) in July 2006. JID:YJFAN
4 D. Guido et al. / Journal of Functional Analysis ●●● (●● {λρ(γ): γ ∈ Γ} ′ of all the bounded operators on ℓ 2 (VX) commuting with the action of Γ , inherits a trace given by
8 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 25 26 27 21 23 Definition 1.1 (Reduced paths). Analogously, denote by ℓ 2 ( EX ) the Hilbert space of functions ω : EX → C such that ‖ ω‖ 2 := Eeex (w(e)| 2 < ∞. A unitary representation of Γ on ℓ 2 (EX) is given by (λ 1 (γ)ω)(e) := ω(γ− l e), for γ ∈ Γ, ω ∈ ℓ 2 (EX), e ∈ EX. Then the von Neumann algebra N 1 (X, Γ) := {λ 1 (γ): γ ∈ Γ } ′ of all the bounded operators on ℓ 2 ( EX ) commuting with the action of Γ ′ inherits a trace given by 11 12
15 16 At this stage, we need to introduce some additional terminology from graph theory. 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 38 37 38 39 40 41 42 43 44 45 46 47 40 41 42 43 44 45 Remark 1.4. 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 . 46 47 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011 (i)  A path ( e 1 , . . . , e m ) has backtracking if e i + i = e i , for some i ∈ { 1 , . . , m − 1 } . 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 1 , . . . , e m ) has a tail if there is k ∈ N such that e m − j + l = e j , for j = 1, . . . , k . Denote by C the set of proper tail-less closed paths, also called reduced closed paths. 28 29 30 31 32 33 34 35 36 37 38 39 (i) Given C, D E 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 = γ(C) . 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 = γ(C) . We denote by [R] Γ the set of Γ -equivalence classes of reduced cycles, and analogously for the subset P . Definition 1.2 (Cycles). Given closed paths C = (e 1 , . . . , e m ), D = (e 1 , . . . , e m ), we say that C and D are equivalent , and write C ～ o D , if there is k ∈ N such that e', = 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.
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Definition 1.3 (Equivalence relation).
JID:YJFAN AID:5356 /FLA D. Guido et al. / Journal of Functional Analysis ●●● (●● [m1+; v 1.95; Prn:30/07/2008; 22:22] P.5(1-23) 5 Our proof of formula (iv) in Theorem 2.2 requires a generalization of a result by Kotani and Sunada [21] to infinite covering graphs. This is done in Proposition 1.6, whose proof depends on a new combinatorial result contained in Lemma 1.5. Define the effective length of a cycle C , denoted by ℓ(C) , as the length of the prime cycle underlying C , and observe that ℓ( C ) is constant on the Γ -equivalence class of C . Therefore, if ξ ∈ [R] Γ , we can define ℓ(ξ) := ℓ( C ), for any representative C ∈ ξ . Recall that, for any cycle C , the stabilizer of C in Γ is the subgroup Γ C := {γ ∈ Γ : γ(C) = C}. Moreover, if C 1 , C 2 ∈ ξ , then the stabilizers Γ C 1 , Γ C 2 are conjugate subgroups in Γ , and we denote by S(ξ) their common cardinality. For the purposes of the next few results, for any closed path D = (eo, . . . , em – 1 ), we also denote e j by e j ( D ). 9 10 11 12 13 Lemma 1.5. Let ξ ∈ [Rm] Γ . Then 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 38 38
and, finally, 46
Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011
Proof. Let us first observe that, if C 1 , C 2 ∈ ξ 3 then f N e ∈ E C 1 Γ e is conjugate in Γ to ∩ e ∈ E C 2 Γ e , and we denote by I ( ξ ) their common cardiñality. Let C ∈ R m be such that [C] r = ξ . By choosing each time a different starting edge, we obtain ℓ := ℓ(C) = ℓ(ξ) closed paths from C . Denote them by D 1 , . . , D ℓ , and observe that any two of them can be Γ -equivalent, i.e. Di = γ(Dj) , for some γ ∈ Γ , if and only if γ ∈ Γ C . Moreover, if γ ∈ ∩ e∈EC Γ e ⊂ Γ c , then γ ( Di ) = Di, for i = 1, . . . , ℓ . Therefore, there are only k = k(ξ) := ℓ(ξ)ﾉ(ξ) distinct Γ -classes of closed paths generated by the D i 's, and we denoted S(5) them by π 1 , . . , π k . Let π be one of them, and observe that, for any e ∈ F 1 , there are either no closed paths D rep- resenting π and such that e 0 ( D ) = e , or there are ティニュ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 ε generates a closed path γ (D) representing π and such that e 0 (γ (D)) = e , but, if γ ∈ ∩ e ∈ ED Γ e , then γ ( D ) = D . Hence, the claim is established. Let us now introduce a discrete measure on F 1 . Let us say that a Γ -class of closed paths π starts at e ∈ F 1 if there is D ∈ π such that e 0 ( D ) = e . Let us set, for e ∈ F 1 , μ ξ ( e ) = 1, if e is visited by some π i , i = 1, . . . k , and μ ε ( e ) = 0, otherwise. It is easy to see that μ ε depends only on ξ and is in particular independent of the representative C . Observe that μ ξ ( F 1 ) = k ( ξ ). Therefore, for any e E F 1 , we get
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6 Define, for ω ∈ ℓ 2 (EX), e ∈ EX, 2 3 5 6 Then, we have 7 8 8 Proposition 1.6. 9 9 10 11 11 12 (i)  T ∈ N 1 (X, Γ), || T || ≤ d (ii)  for m ∈ N, Tm e 10 12 13 13 , the number of Γ -equivalence classes of reduced (iii) Trr(Tm) = Nm := E[C]re[Rm]r S([C]r) cycles of length m. Here, Trr is the trace on N 1 (X, Γ) introduced in (1.2). 15 16 16 18 Proof.  (i), (ii) are easy to check. (iii) Using Lemma 1.5, we obtain 17 18 19 20 21 22 23 23 24
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2.  The Zeta function
34 35 Before introducing the zeta function of an infinite periodic graph, we recall its definition for a finite (q + 1)-regular graph X (i.e. such that deg( v ) = q + 1, for all v ∈ VX). In that case, the Ihara zeta function Z x is defined by an Euler product of the form 38 37 38 39 40 42
41 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 43 44 45
46 47 47 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011
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1.95; Prn:30/07/2008; 22:22] P.7(1-23) D. Guido et al. / Journal of Functional Analysis ●●● (●● 7 1 where p ranges over all the rational primes. To see the correspondence between Z χ and ζ , simply 2 let u := q − s and observe that u l Cl = (q l Cl )− s . Also note that | u | < g if and only if Re s > 1. 3 Let us now return to the case of periodic graphs and introduce the Ihara zeta function via its 4 Euler product as well as show that this defines a holomorphic function in a suitable disc. 5 6 Definition 2.1 (Zeta function). Let Z(u) = Zχ,r(u) be given by 7 7 8 8 g 9 10 12 11 for u ∈ C sufficiently small so that the infinite product converges. 10 11 12 13 13 14 In the following proposition we let 14 15 16
16 15 17 18 17 18 19 19 20 We refer to Section 3 for more details. Formula (iv) in the following theorem was first established in [8], although with a different proof. 21 Theorem 2.2. 22 23 24 l cl  defines a holomorphic function in the open disc {u ∈ C: 24 26 (11) 27 28 (iii) Z(u) 29 30 (iv)  Z(u) = det Γ(I − 29 30 31 32 N 32 Proof.  Observe that it follows from Proposition 1.6 that ∑ ∞ u” defines a function which 33 is holomorphic in { u ∈ C:  | u | < a_」}. Moreover, for any u ∈ C such that | u | 34 34 35 36 37 36 38 37 38 39 40 41 42 43 44 45 46
46 47 47 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011 40 41 44 45 21 26 27 28 31 33 25 35 22 23 25 20 39 43 42 6
8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 44 45 46 47 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 31 32 33 34 35 38 37 38 39 40 41 42 43 44 45 46 where, in the last equality, we have used uniform convergence on compact subsets of { u ∈ C : т, we have = }. From what has already been proved, (i)–(iii) follow. Finally, for | μ | < π | u | < Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011 30 47 29 43 JID:YJFAN AID:5356 /FLA [m1+; v 1.95; Prn:30/07/2008; 22:22] P.9(1-23) D. Guido et al. / Journal of Functional Analysis ●●● (●●● 9
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12 Example 2.3. Some examples of cycles with different stabilizers are shown in Figs. 2, 3. They 13 refer to the graph in Fig. 1 which is the standard lattice graph X = Z 2 endowed with the action 14 of the group Γ generated by the reflection along the x-axis and the translations by elements 15 (m,n) ∈ Z 2 , acting as (m,n)(v 1 , v 2 ) := (v 1 + 4m,v 2 + 4n), for v = (v 1 ,v 2 ) ∈ VX = Z 2 . 16 17
3. An analytic determinant for von Neumann algebras with a finite trace
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43 44 45 and Γ is the boundary of a connected, simply connected region Ω containing the spectrum of A . 46 Clearly, once the branch of the logarithm is chosen, the integral above does not depend on Γ , 47 provided Γ is given as above. 45 46 47 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 In this section, we define a determinant for a suitable class of not necessarily normal operators 20 in a von Neumann algebra with a finite trace. The results obtained are used in Section 4 to prove 21 a determinant formula for the zeta function. 22 In a celebrated paper [11], Fuglede and Kadison defined a positive-valued determinant for 23 finite factors (i.e. von Neumann algebras with trivial center and finite trace). Such a determinant 24 is defined on all invertible elements and enjoys the main properties of a determinant function, 25 but it is positive-valued. Indeed, for an invertible operator A with polar decomposition A = U H , 26 where U is a unitary operator and H := √ A*A is a positive self-adjoint operator, the Fuglede– 27 Kadison determinant is defined by 28 where log H may be defined via the functional calculus. Note, however, that the original defini- tion 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 r (A) = exp o τ o log A, where 8 JID:YJFAN
10 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. 2 3 4 5 Lemma 3.1. Let A, Ω, Γ be as above, and φ, ψ two branches of the logarithm such that both domains contain Ω. Then 6 7 8 9
9 Proof.  The function ϕ(λ) − ψ(λ) is continuous and everywhere defined on Γ . Since it takes its values in 2π i L , it should be constant on Γ . Therefore, 10 10 11 12 13 11 12 13 14 14 15
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 where Γ is the boundary of a connected, simply connected region Ω containing conv σ( A ), and log is a branch of the logarithm whose domain contains Ω . 32 33 34 35 Corollary 3.3. The determinant function defined above is well defined and analytic on Ao. 34 We collect several properties of our determinant in the following result. 35 36 38 37 38 39 Proposition 3.4. Let (A, τ) be a von Neumann algebra endowed with a finite trace, and let A ∈ A0. Then 38 39 40 41 42 (i)  det r (zA) = z r (1) det r (A), for any z ∈ C \ {0}, (ii)  if A is normal, and A = U H is its polar decomposition, 42 43 44 45 46 (iii)  if A is positive, det r (A) = Det(A), where the latter is the Fuglede–Kadison determinant. 16 17 18 19 20 21 22 24 27 28 31 37
45 47 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011 The problem with the previous definition is its dependence on the choice of Ω . Indeed, it is easy to see that when A = ( 1 0 ) and we choose Ω containing { e i θ , θ ∈ [0 , π/2]} and any suitable branch of the logarithm, we get det( A ) = e i π / 4 , if we use the normalized trace on 2 × 2 matrices. By contrast, if we choose $2 containing { e i $, θ ∈ [π/2, 2π]} and a corresponding branch of the logarithm, we get det(A) = e 5ix/4 . Therefore, we make the following choice. Definition 3.2. Let ( A, τ) be a von Neumann algebra endowed with a finite trace, and con- sider the subset A 0 = {A ∈ A : 0 ℓ conν σ (A)} , where σ (A) denotes the spectrum of A . For any A ∈ A0 we set
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D. Guido et al. / Journal of Functional Analysis ●●● (●●●) 11 1 Proof. (i) If, for a given θ 0 ∈ [0,2π), the half-line {ρ e iθ 0 ∈ C: ρ > 0} does not intersect conv σ(A), then the half-line {ρei(ρo+t) ∈ C: ρ > 0} does not intersect conv σ(zA), where 2 z = re it . If log is the branch of the logarithm defined on the complement of the real nega- 3 tive half-line, then ϕ(x) = i(θ 0 − π) + log(e − i(θ 0 − π)x) is suitable for defining det r (A), while 4 ψ(x) = i(θο + t − π) + log(e−i(θο+t−π)x) is suitable for defining det τ (zA). Moreover, if Γ is 5 the boundary of a connected, simply connected region Ω containing conv σ ( A ), then z Γ is the 8 7 boundary of a connected, simply connected region z Ω containing conv σ ( z A ). Therefore, 8 g 10 11 10 12 13 12 13 14 14 16 16 17 18
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24 25 26 26 or, equivalently, that the support of the measure dh ( r ) is compactly contained in (0, ∞ ), and the support of the measure du ( θ ) is compactly contained in ( θ 0 − π/2, θ 0 + π/2). Therefore, A ∈ A 0 is equivalent to U , H ∈ A 0 . Then 27 28 29 27 28 29 30 31
30 32 31 33 which implies that 32 34
36 37 37 36 34 35 39 = det τ ( U ) · det τ ( H ) . 39 40 38 41 (iii) This follows by the argument given in (ii). □ 41 42 Remark 3.5. We note that the above defined determinant function strongly violates the product 43 property det τ ( AB ) = det τ ( A ) det τ ( B ). Indeed, the fact that A , B ∈ A 0 does not imply AB ∈ A 0 , 44 as is seen e.g. by taking A = B = ( 0 j 0 ). Moreover, even if A , B , AB ∈ A 0 and A and B commute, 45 the product property may be violated, as is shown by choosing A = B = 46 3iπ/4 ), and using the 47 normalized trace on 2 × 2 matrices. Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011 (ii) When A = UH is normal, U = J[0,∞) r dh(r), then A = [0,2π] eig du(θ), H = ʃ(0,00)×[0,2π] r e i 0 d(h(r) 8 u(i)). The property 0 f conv σ(A) is equivalent to the fact that the support of the measure d(h(r) & u( θ )) is compactly contained in some open half-plane.
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[m1+; v 1.95; Prn:30/07/2008; 22:22] P.12(1-23) 12 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 2 5
8 7 8 9 Proof. The proof is inspired by that of Lemma 3 in [11]. Let us write a := log(1 + uA), b := log(I + uB) ∈ A, and let c(t) := e¹αeb, t ∈ [0,1]. As ∥ a‖ ≪ − log(1 − ∥ |||A‖), and ∥ b‖ ≪ −log(1 − |u||| B ||), we get 10 11 12 12 13 14 13 14 15 16 17 18 for all t ∈ [0, 1], if we choose | u | sufficiently small; hence, c ( t ) ∈ A 0 for all t ∈ [0, 1]. Now apply Lemma 2 in [11] which gives 17 18 19 25 26 27 28 29 30 31 and hence implies the claim.  □ 30 31 32 33 32 34 35 36
38 37 39 40 Proof. Indeed, for any polynomial p , we have p(ATA-1) = Ap(T)A-1. Applying the Stone– Weierstrass theorem on, the compact set σ( ATA − 1 ) = σ( T ), we obtain log( ATA − 1 ) = A log(T)A-1, from which the result follows. □ 38 40 41 42 43 44 43 Proposition 3.8. Let (A, τ) be a von Neumann algebra endowed with a finite trace, and let 1 Ti2) ∈ Mat2(A), with T ii ∈ A such that σ(T ii ) ⊂ B1(1) := {z ∈ C: |z − 1| < 1}, for i = 1, 2. Then 45 46 47
15 16 19 20 20 21 22 23 Therefore, after integration for t ∈ [0,1], we obtain τ(logc(1)) − τ(logc(0)) = τ(α), which means 24 25 26
29 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 34 35 38 det r (T) = det r (T11) det r (T22). 45 46 47 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011
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13 Proof.  Indeed, for any k ∈ N ∪ {0}, 2 5 for some B k ∈ A , so that, for any polynomial p , 8 6 7 7 8 9 8 9 10 10 11 12 13 12 for some B ∈ A . It is easy to see that σ(T) ⊂ σ(T 11 ) ∪ σ(T 22 ) ⊂ B 1 (1). Hence, applying the Stone–Weierstrass theorem on the compact set σ(T) , we obtain 11 14 13 14 15 16 for some C ∈ A. Therefore, 16 17 18 det r (T) = exp o τ o log(T) = exp(τ(log(T 11 )) + τ(log(T 22 ))) = det r (T 11 ) det r (T 22 ), 18 19 19 20 as desired. □ 20 21 22 23 24 Corollary 3.9. Let Γ be a discrete group, π 1 , π 2 unitary representations of Γ , and τ 1 , τ 2 traces on π 1 (Γ) ′ and π 2 (Γ) ′ , respectively. Let π := π 1 ⊕ π 2 , τ := τ 1 + τ 2 , T = π(Γ)′, with σ(Τ ii) ⊂ Β 1 (1) = {z ∈ C: | z − 1| < 1}, for i = 1, 2. Then 25 26 det r (T) = det r 1 (T11) det r 2 (T22). 27 28 28 27 Proof.  It is similar to the proof of Proposition 3.8. 29 30 30 29 31
4. The determinant formula
31 32 33 34 32 36 35 36 37 38 39 39 40 41 Theorem 4.1 (Determinant formula). We have 42 43 43 45 46 47
45 1 is the L 2 -Euler characteristic of (X, Γ), as intro- where χ (2) := duced in [6]. 46 47 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011
15 17 44 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 f ( w ), f ∈ ℓ 2 ( VX ). Then (by [23,24]) || A || ≪ d := sup c ∈ v x deg( v ) ≪ 00, and it is easy to see that A ∈ N 0 ( X , Γ ). Introduce (Qf)(v) := (deg(v) - I)f(v), v ∈ VX, f ∈ ℓ 2 (VX), and Δ(u) := I − uA + u 2 Q ∈ No(X, Γ), d²+4d . Then for u ∈ C. Let us recall that d := sup v ∈ v x deg( v ), and set α := d + 26 21
14 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 r ( I − u T ) = (1 − u 2 )− x (2) (X) det r ( ∆ ( u )), for | u | < ½ . 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), 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 and use the short-hand notation Iv := Id ℓ 2 (VX) and I E := Id ℓ 2 (EX) 16 17 18 19 20 21 22 23 24 25 28 28 29
31 32 32 33 34 35 36 Moreover, for ω ∈ ℓ 2 ( 37 38 39 40
37 38 39 40 41 42 43 44 45 46 47 47 Let us now consider the direct sum of the unitary representations λ 0 and λ 1 , namely λ(γ) := λο(γ) ⊕ λι(γ) ∈ Β(ℓ 2 (VX) ⊕ ℓ 2 (EX)). Then, the von Neumann algebra λ(Γ) ′ {S ∈ B(ℓ 2 (VX) ⊕ ℓ 2 (EX)):  Sλ(γ) = λ(γ)S, γ ∈ Γ} consists of operators S = ( The rest of the proof is clear. 44 45 46 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011
11 12 13 14 15 Lemma 4.2. 18 19 20 21 23 24 (v)  ∂ρσ = JT + IE (vi) ∂ισ = T + 25 26 27 Proof. Let f ∈ ℓ 2 (VX), v ∈ VX. Then 28 30 31 33 34 35 38 10
(i) J91 = 90, (ii) σλ 1 (γ) = λ 0 (γ)σ, ∂ iλ 0 (γ) = λ 1 (γ)∂ i, i (iii) σ ∂ = I + Q, (iv) σθι = A, 27 30 9 8
43 41 42 22 29 17
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D. Guido et al. / Journal of Functional Analysis ●●● (●● 15 Sijλ j (γ) = λ i (γ) Sij, γ ∈ Γ, i, j = 0, 1, so that Sij ∈ λ i (Γ) ′ ≡ λ i (Χ, Γ), i = 0, 1. Hence λ (Γ) ′ inherits a trace given by
Introduce 9
10 10 11 which both belong to λ(Γ) ′ . Then, we have 12 12 13 13 14 Lemma 4.3. 15 15 16 (i) 17 18
19 19 20 Moreover, for |u| sufficiently small, 21 20 22 (iii)  L(u), M(u) are invertible, with a bounded inverse, (iv) detr(M(u)L(u)) = (1 — u2)Trr(IE) detr(A(u)), 23 (v) detr(L(u)M(u)) = (1 - u2)Trr(Iv)- 2 Trr(Ie) detr(Ie - uT). 24 25 24 23 26 26 Proof.  The formulas for M(u)L(u) and L(u)M(u) follow from the previous lemma. Moreover, 27 for | u | sufficiently small, σ(Δ(u)), σ((1 − u 2 ) I E ), σ((1 − u 2 ) I V ) and σ((I E − u J )(I E − u T )) ⊂ 28 B1(1) = {z ∈ C:  |z − 1| < 1}, hence σ(M(u)L(u)) and σ(L(u),M(u)) ⊂ B1(1), as in the proof 29 of Proposition 3.8. Therefore, L ( u ) and M ( u ) are invertible, with a bounded inverse, for | u | 30 sufficiently small. By Propositions 3.4(i), 3.6 and Corollary 3.9, we obtain 31 32
32 33 34
35 36 35 37 and 36 38 39 38
39 41 42 43 44 Moreover, we have det r ( I E − u J ) = (1 − u 2 ) ⇒ Tr r ( I E ) . Indeed, using J to identify ℓ 2 ( E − X ) with ℓ 2 ( E + X ), we obtain a representation ρ of B (ℓ 2 ( EX )) onto Mat 2 ( B (ℓ 2 ( E + X ))), under which ρ(IE) = ( 1 0 ). Hence, by Propositions 3.6 and 3.8, 45 46 47 detr(IE — uJ) = detr(p(IE — uJ)) 46 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011
16 17 14 JID:YJFAN 16 AID:5356 /FLA [m1+; v 1.95; Prn:30/07/2008; 22:22] P.16(1-23) D. Guido et al. / Journal of Functional Analysis ●●● (●●●)
10 11 12 Proof of Theorem 4.1. Let us observe that, for sufficiently small | u |, we have 13 12 11 10 13 14 15
14 15 16 17 18 16 so that, by Proposition 3.7, we get det Γ ( L ( u ) M ( u )) = det Γ ( M ( u ) L ( u )). Therefore, the claim follows from Lemma 4.3(iv) and (v), Eqs. (1.1) and (1.2) and Theorem 2.2.     □ 17 18 19
5.  Functional equations
19 20 21 22 23 24 25 26 27 27 28 29 30 31 Lemma 5.1. Let X be a (q + 1)-regular graph, on which Γ acts freely and with finite quotient B := X / Γ. Let Δ(u) := (1 + qu²)1 − u A. Then 28 29 30 31 32 (i) χ (2) (X) = χ (B) = |V (B)|(1 − q)/2 ∈ Z, (ii)  Zχ, Γ(u) = (1 − u 2 )χ(B) det Γ((1 + qu 2 )Ι − u A)− 1 , for |u| < (iii)  by using the determinant formula in (ii), Ζχ , Γ can be extended to a function holomorphic at least in the open set 33 34 35 35 36 38 37 38 39 40 41 42 43 44 45 47 σ(Δ(u)) = {1 + qu² − uλ: λ ∈ σ(A)} ⊂ {1 + qu² − uλ: λ ∈ [−d,d]}. 47 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011 46 21 22 23 24 25 26 In this section, we obtain several functional equations for the Ihara zeta functions of (q + 1)- regular graphs, i.e. graphs with deg( v ) = q + 1 , for any v ∈ VX , on which Γ acts freely (i.e. Γ ν is trivial, for ν ∈ 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 [13]. See Fig. 4. )) = (qu 2 )-\VBI detr(Δ(u)), for u ∈ Ω \ {0}. (iv) det r ( △ ( ¬ 41 42 43 44 45 46 20 39 Proof.  (i) This follows by a simple computation. (ii) This follows from (i). (iii) Let us observe that 37 38 33 34 40 32 AID:5356 /FLA JID:YJFAN D. Guido et al. / Journal of Functional Analysis ●●● (●●●) [m1+; v 1.95; Prn:30/07/2008; 22:22] P.17(1-23) 17 6 9 10 11 12 13 It follows that 0 & conv σ( Δ( u )) at least for u ∈ C such that 1 + qu 2 − u λ̸= 0 for λ ∈ [−d,d ], 14 that is for u = 0 or 1±9n2 ₫ [−d,d] , or equivalently, at least for u ∈ Ω . The rest of the proof 15 follows from Corollary 3.3. 16 17 (iv) This follows from Proposition 3.4(i) and the fact that Tr r ( Iv ) = |VB | .    □ 18 15 16 17 19 The question whether the extension of the domain of Z x r by means of the determinant 20 formula is compatible with an analytic extension from the defining domain is a non-trivial issue, 21 see the recent paper by Clair [7]. 22 23 Theorem 5.2 (Functional equations). Let X be a (q + 1)-regular graph, on which Γ acts freely 24 and with finite quotient B := X/ Γ. Then, for all u ∈ Ω, we have 25 27 28 29 30 Proof.  They all follow from Lemma 5.1(iv) by a straightforward computation. We prove (i) as 31 32 31 32 33 34 34 33 35 36 37 38 39 40
38 39 41 Remark 5.3. Recall that a key property of the Riemann zeta function ζ is that its mero- 42 morphic continuation satisfies a functional equation ξ(s) = ξ(1 − s), for all s ∈ C, where 43 ξ(s) := π − s /2 Γ(s /2)ζ(s) denotes the completion of ζ and Γ is the usual Gamma function. Like- 44 wise, in Theorem 5.2, any of the functional equations relates the values of the corresponding 45 completed Ihara zeta function at s and 1 − s , provided we set u = q −s , as was explained at the 46 beginning of Section 2. Note that 47 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011 10 11 12 13 14 19 20 21 22 23 24 25 26 26 27 28 29 30 (i) Ax,r(u):=(1 - u 2 )-x(B)(1 - u 2 )|vB|/2(1 - q 2 u 2 )|vB|/2 Zx,r(u)==Ax,r(u) (ii) ξχ, Γ(u) := (1 − u 2 ) − χ( B ) (1 − u) VBl (1 − qu) lVBl Zχ, Γ(u) = ξχ, Γ(qu) (iii) ミχ, Γ (u) := (1 − u²)− χ(B)(1 + qu²)|VB|Zχ, Γ (u) = Sχ, Γ (qu 35 36 9
6.  Approximation by finite graphs in the amenable case

D. Guido et al. / Journal of Functional Analysis ●●● (●●●) 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 Żuk in [12]. 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 1 , . . . , e k } , where the edges have been ordered in such a way that, for each i ∈ { 1 , . . . , k } , e i has at least a vertex in common with some e j , with j < i . Choose ē 1 ∈ EX such that p (ē 1 ) = e 1 . Assume ē 1 , . . . , ē i have already been chosen in such a way that p(ē j ) = e j , for j = 1, . . . , i , and, for any such j , ē j has at least a vertex in common with some e h , with h < j . Let e i +1 ∈ EB have a vertex in common with e j , for some j ∈ {1, . . . , i} and choose ē j +1 ∈ VX such that p(ē j +1 ) = e j +1 and ē j +1 has a vertex in common withﺯē . This completes the induction. Let EF := { ē 1 , . . , ē k } and VF := {o(ē 1 ), . . . , o(ē k )} ∪ {t(ē 1 ), . . . , t(ē 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. 8 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 27 28 29 30 31 Definition 6.1. 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: 32 (i) Kn = Uy ∈ E n Y F , where E n ⊂ Γ , for all n ∈ N , (ii) Unen Kn = X, 33 33 34 34 (iii)  K n ⊂ K n +1, for all n ∈ N, 35 35 (iv)  if FK n := { v ∈ VK n : d(v, VX \ VK n ) = 1}, then lim n →∞  |VK n 36 38 37 37 Then X is called an amenable graph if it possesses an amenable exhaustion. 38 38 39 39 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. 40 41 42 43 40 41 42 43 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 : dist(γ F, F) ≪ 1} is symmetric (i.e. γ ∈ A ↔ γ − 1 ∈ A), generates Γ as a group, and contains the unit element. Introduce the Cayley graph C(Γ, Α) , whose vertices are the elements of Γ , and, by definition, there is one edge from γ 1 to γ 2 iff γ − 1 γ 2 ∈ A . A subset 44 44 45 46 47 45 46 47 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011 [m1+; v 1.95; Prn:30/07/2008; 22:22] P.18(1-23) 31 32 JID:YJFAN AID:5356 /FLA 1.95; Prn:30/07/2008; 22:22] P.19(1-23) D. Guido et al. / Journal of Functional Analysis ●●● (● 19 E ⊂ VC ( Γ, A ) is said to be connected if, for any pair of distinct vertices of E , there is a path in 2 C ( Γ , A ), joining those two vertices, and consisting only of vertices of E . From [1, Theorem 4], it follows that there is a sequence { E j } j ∈ N of connected finite subsets of Γ such that g 10 10 11 12 13 14 where, for any U 1 , U 2 ⊂ Γ , we set U 1 ⋅ U 2 ＝ {γιγ 2 : γι ∈ U 1 , i ＝ 1 , 2} . For each n ∈ N , let K n := ∪ γ ∈ E n γ F . Then { K 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 , we have 15 14 16 16 18 where U n := { γ ∈ E n : there is δ ∈ A such that γδ ∈ E n } . Indeed, let ν ∈ F K n and w ∈ V X V 19 VK n be such that d(v, w) = 1. Then, there are γ 0 , γ 1 ∈ Γ , v 0 , v 1 ∈ VF , such that v = γ 0 v 0 and 20 w = γ 1 ν 1 . Moreover, we have γ 0 ∈ E n and γ 1 ∈ E n . Let δ := γ 0 − 1 γ 1 , so that dist( F, δ F ) = 21 dist( γο F , γι F ) ≪ d(v, w) = 1, which implies that δ ∈ A . Hence, γι ∈ U n , and the claim follows. 22 Finally, 23 24 24 26 27 28 29 31 32 34 so condition (iii) of Definition 6.1 is satisfied, showing that { K n : n ∈ N } is an amenable exhaus- tion. Hence, X is amenable, as desired. □ 37 35 35 36 38 39 If Ω ⊂ VX, r ∈ N, we write Br(Ω) := { v ′ ∈ VX: ρ(v ′ , v) ≪ r }, where ρ is the geodesic metric on VX. 40 41 42 Lemma 6.3. Let (X, Γ, F) be as above. Let d := sup v ∈ ν χ deg( v ) < ∞ . Let { K n } be an amenable exhaustion of X, and ε n := Knl → 0. Then, for any r ∈ N, |Br(FKn)| ≤ (d + 1) r εn |VKn|. 43 44 Proof.  Since 45 46 47
46 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011
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[m1+; v 1.95; Prn:30/07/2008; 22:22] P.20(1-23) 20 D. Guido et al. / Journal of Functional Analysis ●●● (●● we have | Br+1 (v)| ≪ (d + 1)| Br (v)| , giving | Br (v)| ≪ (d + 1) r , ∀v ∈ VX, r ≥ 0. As a conse- quence, for any finite set Ω ⊂ VX , we have Br(Ω) = ∪ v ∈Ω Br(v'), giving 2 3 4 5 6
Therefore, | Br ( FK n ) | ≤ (d + 1) r | FK n | = (d + 1) r ε n | VK n | . 7 8 9 Lemma 6.4. Let (X, Γ, F) be as above. Let { K n } be an amenable exhaustion of X. Then, for any B ∈ No(X, Γ), we have 9 10 11 12 12 13 11 10
13 where P(K n ) is the orthogonal projection of ℓ 2 (VX) onto ℓ 2 (VK n ). 14 15 14 15 16 17 16 18 17 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 , γ F 0 ∪ Ω n , where ⊥ denotes “disjoint union” and Ω n ⊂ B δ ( F K n ). Indeed, if ν ∈ Ω n := VK n \ └ └ vEE, YFo, then there is a unique γ ∈ Γ such that ν ∈ γ F 0 , so that γ ∈ E n , which implies γ F ∩ (VX \ VK n ) ≠ ∅, and d(v, VX \ VK n ) ≪ δ , which is the claim. Therefore, 19 18 20 21 22 23 22 23 24 25 26 27 28 29 30 31 32 Moreover, 33 34
28 29 30 31 32 33 34 35 35 36 37 so that 38 39 38 39 40 41 42 Besides, 43 44 45
44 45 because |VK n | = |E n | · | Fo | + | S n | . The claim follows.口47 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011
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D. Guido et al. / Journal of Functional Analysis ●●● (● 21 1 Lemma 6.5. Let (X, Γ) be as above. Let A and Q be as in Section 4. Let f ( u ) := Au – Qu 2 for u ∈ C. Then ‖ f (u)‖ < 2 7, for | u | d2+2(d-1) 3 4 Proof.  This follows from the estimate 5 8 7 f (u) | ≪ | u || A || + | u | 2 || Q || ≪ d | u | + (d  — 8 which is valid for any u ∈ C. g g 10 11 10 Theorem 6.6 (Approximation by finite graphs). Let X be a connected, countably infinite graph, 11 and let Γ be a countable discrete amenable subgroup of automorphisms of X, which acts on X 12 freely and cofinitely, and let F be a corresponding fundamental domain. Let { K n : n ∈ N } be an 13 amenable exhaustion of X . Then 12 13 14 14 15 16 16 17 uniformly on compact subsets of {u ∈ Ω 18 17 19 19 20 21 22 23 24 24 25
25 26 26 27 Moreover, 27 28 29 28 30
30 31 31 33 Observe that, for k ≥ 2, 34 35 34 35 36 37 38 38 39 40 42 stands for P n , the projection onto the orthogonal complement of ℓ 2 (VK n ) in ℓ 2 (VX), where P n and 43 44 45 46 46 47 47 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011 15 20 Proof. For a finite subset N ⊂ VX, denote by P(N) ∈ B(ℓ 2 (VX)) the orthogonal projection 21 of ℓ 2 (VX) onto span( N ). Observe that, since N is an orthonormal basis for ℓ 2 ( N ), we have 22 Tr(P(N)) = |N|. 23 Let f ( u ) := Au − Qu 2 and P n := P ( VK n ) . Then 32 29 36 39 41 32 33
v 1.95; Prn:30/07/2008; 22:22] P.22(1-23) [m1+; D. Guido et al. / Journal of Functional Analysis ●●● (● 1 Moreover, with Ω n := B 1 (VK n ) \ VK n ⊂ B 1 (FK n ), we have 2 3 a 5
8 7
8 9 9 Therefore, we obtain 11 10 10 11 12 |Tr(Pnf (u) k Pn) — Tr((Pnf (u) Pn) k 13 14 15 so that 16 17 Tr log( P n ( I  - – Tr( P n log( I 16 18 19 20 21 22 23
≪ C (d + 1)ε n | V K , where the series converges for | u | by Lemma 6.5. Hence, d2+2(d-l) 28 27 27 28 28 29 30
30 31 32 33 and, by using Lemma 6.4, 32 34 lim
lim 37 38 37 39 40 41 41 42 43 44 from which the claim follows. □ 45 45 47 Remark 6.7. Observe that d+ d2+2(d 46 Please cite this article in press as: D. Guido et al., Ihara's zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.07.011
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43 44 46 22 24 29 35 38 36 39 | Fol 42 12 14 15 18 21 23 26 25 25 35 34 33 38 19 20 24 17 8 13 47 31
Acknowledgments
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