Convergence of Spectral Triples on Fuzzy Tori to Spectral Triples on Quantum Tori
aa r X i v : . [ m a t h . OA ] F e b CONVERGENCE OF SPECTRAL TRIPLES ON FUZZY TORI TOSPECTRAL TRIPLES ON QUANTUM TORI
FRÉDÉRIC LATRÉMOLIÈRE
Abstract.
Fuzzy tori are finite dimensional C*-algebras endowed with an ap-propriate notion of noncommutative geometry inherited from an ergodic actionof a finite closed subgroup of the torus, which are meant as finite dimensionalapproximations of tori and more generally, quantum tori. A mean to specifythe geometry of a noncommutative space is by constructing over it a spectraltriple. We prove in this paper that we can construct spectral triples on fuzzytori which, as the dimension grow to infinity and under other natural condi-tions, converge to a natural spectral triple on quantum tori, in the sense ofthe spectral propinquity. This provides a formal assertion that indeed, fuzzytori approximate quantum tori, not only as quantum metric spaces, but asnoncommutative differentiable manifolds — including convergence of the statespaces as metric spaces and of the quantum dynamics generated by the Diracoperators of the spectral triples, in an appropriate sense.
Contents
1. Introduction 11.1. The Problem 11.2. The Spectral Propinquity 72. The Geometry of Quantum and Fuzzy Tori 172.1. Background: the quantum and fuzzy tori 172.2. Spectral Triples on Fuzzy and Quantum Tori 272.3. Self-adjointness of the proposed Dirac operators 333. Metric Properties of the Spectral Triples 403.1. A Mean Value Theorem for our Spectral Triples 413.2. Convergence of the Quantum Metrics 513.3. Convergence of the Spectral Triples 58References 651.
Introduction
The Problem.
Matrix models for quantum field theories and string theo-ries (in particular, involving the geometry of so-called compactified dimensions,
Date : February 11, 2021.2020
Mathematics Subject Classification.
Primary: 46L89, 46L87, 46L30, 58B34, Secondary:34L40, 47D06, 47L30, 47L90, 81Q10, 81R05, 81R15, 81R60, 81T75.
Key words and phrases.
Noncommutative metric geometry, Gromov-Hausdorff convergence,Spectral Triples, Monge-Kantorovich distance, Quantum Metric Spaces, quantum tori, fuzzy tori,spectral propinquity, matrix approximations of continuum. or the geometry on D -branes) have become an interesting tool for the study ofsuch fundamental questions as the search for a quantum theory of gravitation, e.g.[27, 68, 3, 11, 52]. The asymptotic behaviors of such models, as the dimension ofthe matrix algebras involved grow to infinity, is of central interest. We have led aresearch program where the study of these asymptotic behaviors uses the formalismof convergence for certain distance functions on quantum spaces, starting with a dis-tance on the class of quantum compact metric spaces [9, 58, 59, 61, 64, 33, 38, 36, 39]which is a noncommutative analogue of the Gromov-Hausdorff distance . In partic-ular, we have recently introduced in [42] a generalization of the Gromov-Hausdorffdistance on the class of metric spectral triples , which are the structures introducedby Connes [9, 10] to describe the noncommutative analogues of Riemannian mani-folds. Our distance on metric spectral triples is called the spectral propinquity [42].In this paper, we construct metric spectral triples on fuzzy tori , which are a partic-ularly relevant family of matrix models used in physics [27, 68, 3, 11], and we provetheir convergences to metric spectral triples on quantum tori , in the sense of thespectral propinquity. Our construction is inspired by discussions in the literaturein mathematical physics, cited above, of what a spectral triple on a fuzzy torusshould be, in analogy with the construction of certain natural Dirac operators onclassical tori.Fuzzy tori are twisted convolution C*-algebras of finite products of finite cyclicgroups. In general, a fuzzy d -torus associated to a real d × d antisymmetric matrix ( θ jk ) j,k d and natural numbers k , . . . , k d , with the condition that gcd( k j , k k ) θ jk ∈ Z for all j, s ∈ { , . . . , d } , is the universal C*-algebra generated by d unitaries U , . . . , U d such that(1.1) ∀ j, s ∈ { , . . . , d } U j U s = exp(2 iπθ js ) U s U j and ∀ j ∈ { , . . . , d } U k j j = 1 .Fuzzy tori, or, at least, families of unitaries in finite dimension with the above com-mutation relation, can for instance be found in the discussion of quantum mechanicsin finite dimension in [72, pp. 272–280].An example of a fuzzy torus is given by the C*-algebra generated by the so-calledclock ( C n ) and shift ( S n ) matrices, defined by:(1.2) S n = and C n = (cid:0) iπn (cid:1) exp (cid:16) i ( n − πn (cid:17) ,for any n ∈ N \ { } . These matrices appear in many physically-motivated work(see, e.g., [67, 68, 27, 70]), as well as in such work as t’Hooft work [69] on using anunderlying dynamics to provide a model for quantum physics, where the clock andshift matrices are of course associated with the C*-crossed-product of the rotationon the cyclic group of n -roots of unity by itself via translation (again, see [72]).Fuzzy tori are often seen as finite dimensional approximations of quantum tori— including approximations of the classical tori. Quantum tori are the twisted convolution C*-algebras for the groups Z d . Equivalently, the quantum d -torusassociated with some real, d × d , antisymmetric matrix θ = ( θ jk ) j,k d is theuniversal C*-algebra generated by d unitaries U ,. . . , U d such that(1.3) ∀ j, k ∈ { , . . . , d } U j U k = exp(2 iπθ jk ) U k U j .Classical tori are particular cases of quantum tori, when θ is the zero matrix. Theanalogy between Equation (1.1) and Equation (1.3) is obvious. However, fuzzy toriare finite dimensional (their unitary generators have finite orders) while quantumtori are always infinite dimensional (and not approximately finite C*-algebras either[19, 5]). As a finite C*-algebra is seen as a sort of quantum analogue of a finiteset, the idea of approximating quantum tori with fuzzy tori has been a commonheuristics. The question for us is: how do we make these heuristics formal?Fuzzy tori, as finite dimensional C*-algebras, are finite products of full matrixalgebras. In particular, C ∗ ( S n , C n ) is simply the C*-algebra of n × n matrices.Thus, one may ask what makes the algebra of n × n matrices a finite dimensionalquantum analogue of a torus rather than, say, the finite dimensional quantum ana-logue of a sphere [62, 63, 64, 66]. The answer is, informally, given by introducingsome noncommutative, or quantum, geometry on these algebras. Quantum tori,in particular, have a long history as prototypes for noncommutative Riemannianmanifolds, starting with Connes’ first proposal for an operator-algebra based formof noncommutative geometry in [8]. Since fuzzy tori should be geometric approxi-mations of quantum tori, we thus have some guidance on what the geometry of afuzzy torus should be. A core proposal of noncommutative geometry is that theanalogue of a Riemannian geometry of a quantum space is encoded in a structurecalled a spectral triple [8, 10].A spectral triple is a far–reaching generalization of the Dirac operator acting onthe spinor bundle of a compact connected spin manifold [9], given by the followingdata: Definition 1.1. A spectral triple ( A , H , /D ) over a unital C*-algebra A is given bya Hilbert space H and, on a dense subspace dom (cid:0) /D (cid:1) of H , a self–adjoint operator /D with compact resolvent, such that:(1) there exists a *-representation π of A as bounded operators on H ,(2) there exists a dense *-subalgebra A ⊆ A such that ∀ a ∈ A π ( a )dom (cid:0) /D (cid:1) ⊆ dom (cid:0) /D (cid:1) and ∀ a ∈ A [ D, π ( a )] is a bounded operator on dom (cid:0) /D (cid:1) .The operator /D is referred to as the Dirac operator of the spectral triple ( A , H , /D ) .Various spectral triples have been constructed on quantum tori (e.g. [8, 58, 62,12, 13, 18]). Most of these constructions employ, as a starting point, the dualaction of the tori on quantum tori, and the Lie group structure of the tori. Ourown presentation will follow this path as well.The formalism of spectral triples is flexible enough that it is perfectly reasonableto define spectral triples on finite dimensional C*-algebras — i.e. quantum theanalogues of finite sets, thus providing a unified framework for differential structuresand their discrete analogues. In fact, any self-adjoint operator acting on the Hilbert FRÉDÉRIC LATRÉMOLIÈRE space of some finite dimensional *-representation of a finite dimensional C*-algebraautomatically gives us a spectral triple! Various spectral triples have been discussedon fuzzy tori [27, 68, 3], usually constructed in analogy with a classical Diracoperator on a torus. Once more, the heuristic behind these constructions is thatthe spectral triples on fuzzy tori should converge to some spectral triple on a torus(these papers only consider a commutative limit, though we actually want to allowthe limits to be any quantum torus), but no formalism of what such a statementcould mean is provided. It is, indeed, not a trivial matter.The main justifications for our construction of spectral triples on fuzzy toribelow are that, at once, they are analogous to natural spectral triples on classicaland quantum tori (especially, quite similar to suggestions found in [68, 3]), andthat indeed, our scheme will give convergent sequences of spectral triples. Whilethe complete description of our spectral triples on arbitrary fuzzy and quantumtori requires that we first lay down some notations regarding these C*-algebras, wenow give a general idea of the form of our spectral triples on fuzzy and quantumtori (up to a unitary equivalence), to provide an informal introduction to our mainresult.The C*-algebra C ∗ ( C n , S n ) is a fuzzy torus, where C n S n = z n S n C n and z n =exp (cid:0) iπn (cid:1) . Heuristically, and in fact, formally for certain quantum metrics [31, 34], C ∗ ( C n , S n ) approximates C ( T ) .The fuzzy torus carries a natural action of (cid:0) Z / n Z (cid:1) , by setting, for all z ∈ Z / n Z : α ( z, n ( a ) = C zn aC − zn and α (1 ,z ) n ( a ) = S − zn aS zn noting that C zn is defined as C mn for any m whose class in Z / n Z is z , since C nn = 1 ;moreover we also note that C − zn = ( C ∗ n ) z . The same comment applies to S n . Since α z, n and α ,wn commute as *-automorphisms of C ∗ ( C n , S n ) , for all z, w ∈ Z / n Z ,we define an action of (cid:0) Z / n Z (cid:1) on C ∗ ( C n , S n ) , by setting ( z, w ) ∈ (cid:0) Z / n Z (cid:1) α z,wn = α z, n α ,wn . The action α n is called the dual action on C ∗ ( C n , S n ) . The dualaction is what gives C ∗ ( C n , S n ) its quantum geometry.The dual action of (cid:0) Z / n Z (cid:1) on C ∗ ( C n , S n ) converges, in a way which can beformalized (see [47, 44]), to the action by translation of T on the C*-algebra C ( T ) of C -valued continuous functions over the -torus T . This action is againcalled the dual action of T on C ( T ) . The C*-algebra C ( T ) is the universalC*-algebra generated by two commuting unitaries U and V , and the dual actionis uniquely characterized as the action of T on C ( T ) by *-automorphisms, suchthat ( z, w ) ∈ T is sent to the unique *-automorphism α ( z,w ) ∞ of C ( U, V ) such that: α ( z,w ) ∞ ( U ) = zU and α ( z,w ) ∞ ( V ) = wV .Since T is a Lie group, a general consequence of the existence of this dual action α ∞ implies the existence of a dense *-subalgebra C ( T ) of C ∗ ( U, V ) which carriesan action of the Lie algebra of T in a natural way; for our purpose, we focus onthe two canonical derivations on C ∗ ( U, V ) : ∀ a ∈ C ( T ) ∂ ∞ , ( a ) = lim t → α (exp( it ) , ∞ ( a ) − at and ∂ ∞ , ( a ) = lim t → α (1 , exp( it )) ∞ ( a ) − at .These two derivations are, of course, the usual vector fields on T used as thecanonical moving frame for T as a differential manifold. It is thus natural, at first glance, to consider, for n ∈ N , that the maps ∂ n, : a ∈ C ∗ ( C n , S n ) C n aC ∗ n − a πn and ∂ n, : a ∈ C ∗ ( C n , S n ) S ∗ n aS n − a πn are analogous to the derivations ∂ ∞ , and ∂ ∞ , of the torus — the normalizationby πn chosen to give the desired asymptotic behavior, and it could be replaced byany other sequence with the same asymptotic behavior.Unfortunately, thus defined, ∂ n, and ∂ n, are not derivations of C ∗ ( C n , S n ) . Infact, all derivations of C ∗ ( C n , S n ) are given as commutators. Nonetheless, we thenobserve that n π [ C n , a ] = α ( z, n ( a ) − a πn C n = ∂ n, ( a ) C n and n π [ S ∗ n , a ] = α (1 ,z ) n ( a ) − a πn S ∗ n = ∂ n, ( a ) S ∗ n ,thus connecting our initial guess and a more formally appropriate approach to adiscrete quantized calculus for C ∗ ( C n , S n ) . Similar computations hold for [ C ∗ n , · ] and [ S n , · ] .The C*-algebra of n × n matrices C ∗ ( C n , S n ) naturally acts on the Hilbert space C n on both the left (by multiplication) and the right (by ξ ∈ C n ( a ∗ ξ ∗ ) ∗ for all a ∈ C ∗ ( C n , S n ) , where ξ ∗ = ( ξ n , . . . , ξ ) whenever ξ = ( ξ , . . . , ξ n ) ∈ C n ).This leads us to propose the following self-adjoint operator as a Dirac operatorfor the fuzzy torus C ∗ ( C n , S n ) — the self-adjointness is why we use the real partand imaginary part of C n and S n : /D n = n π (cid:18) (cid:20) C n + C ∗ n , · (cid:21) ⊗ γ + (cid:20) C n − C ∗ n i , · (cid:21) ⊗ γ + (cid:20) S n + S ∗ n , · (cid:21) ⊗ γ + (cid:20) S ∗ n − S n i , · (cid:21) ⊗ γ (cid:19) ,where γ , γ , γ , γ are the canonical generators of the Clifford algebra of C , actingas Gamma matrices on C . Indeed, we easily compute that [ /D n , a ] = n π (cid:18) (cid:20) C n + C ∗ n , a (cid:21) ⊗ γ + (cid:20) C n − C ∗ n i , a (cid:21) ⊗ γ + (cid:20) S n + S ∗ n , a (cid:21) ⊗ γ + (cid:20) S ∗ n − S n i , a (cid:21) ⊗ γ (cid:19) .With this in mind, our heuristics suggests that the limit Dirac operator on C ( T ) should be given by the following operator acting on a dense subspace of L ( T ) ⊗ C , /D ∞ = U + U ∗ ∂ V ⊗ γ + U − U ∗ i ∂ V ⊗ γ + V + V ∗ ∂ U ⊗ γ + V ∗ − V i ∂ U ⊗ γ ,where ∂ U and ∂ V are the Sobolev derivatives on L ( T ) which are the closures,respectively, of the operators ∂ ∞ , and ∂ ∞ , , seeing C ( T ) canonically as a densesubspace of L ( T ) .The operator /D ∞ , as we shall prove, is indeed a densely defined self-adjointoperator with compact resolvent on L ( T , C ) , and ( C ( T ) , L ( T , C ) , /D ∞ ) isindeed a spectral triple on C ( T ) . In this paper, we will prove, in particular, that ( C ( C n , S n ) , C n ⊗ C , /D n ) converges, in the sense of the spectral propinquity, to FRÉDÉRIC LATRÉMOLIÈRE ( C ( T ) , L ( T , C ) , /D ∞ ) . This Dirac operator is natural, though it involves a sortof rotating frame of spinors, which is the cost of using commutators in defining ourspectral triples on fuzzy tori.Our work goes well beyond the special case discussed above. We allow for anyreasonable approximation of any quantum torus by fuzzy torus (and even by mix-tures of quantum and fuzzy tori) in our work. To this end, the scheme explainedabove needs some adjustment. For instance, C ( T ) can also be seen as the limit of C ( U n ) , with U n = { z ∈ C : z n = 1 } , yet the only derivation of C ( U n ) is the zerofunction. On the other extreme, if A is a simple quantum torus, which is basicallythe generic case, then we can find any nontrivial central unitaries, and thus, ourapproximation scheme would again not be possible as is. Both these situationsillustrate that, in general, we can not find, as in the case of C ∗ ( C n , S n ) , all thebasic ingredients to carry our scheme within the algebras of interest. But this canbe remedied.The idea is that we can always embed a fuzzy torus or a quantum torus in a C*-crossed product which contains the unitaries needed to perform our task above. Forinstance, if we embed C ( U n ) in C ( U n ) ⋊ α n U n , where α n is the action by translation(again, the canonical dual action), then there are by constructions two unitaries U n, and U n, in C ( U n ) ⋊ α n U n such that U n, aU ∗ n, = α ( z, n ( a ) and U n, aU ∗ n, = α (1 ,z ) n ( a ) for all a ∈ C ( U n ) . We are then able to follow a path analogous to ourwork on C ∗ ( C n , S n ) , constructing a spectral triple on C ( U n ) which converges, forthe propinquity, to a spectral triple on C ( T ) — constructed using some elementsfrom C ( T ) ⋊ Z for the trivial action.Thus, in this paper, we will start with a sequence ( A n ) n ∈ N of fuzzy or quantumtori, and a quantum torus A ∞ , subject to the following natural condition. For all n ∈ N ∪{∞} , let θ n = ( θ jsn ) j,k d be some d × d antisymmetric, real, matrix chosenso that the canonical, generating, unitaries U n, ,. . . , U n,d of A n satisfy U n,j U n,k =exp(2 iπθ jsn ) U n,k U n,j for all j, k ∈ { , . . . , d } . We assume that ( θ n ) n ∈ N converges to θ ∞ .For each n ∈ N , we will embed A n in a fuzzy or quantum d ′ -torus B n with d ′ > d , in order to carry out the scheme described above for the clock and shiftmatrix fuzzy tori — we will explain this embedding later in the paper, and we notethat in general, several choices are possible. We denote by H n the Hilbert space ofthe Gelfand-Naimark-Segal representation of B n for the canonical trace.We fix some gamma matrices γ ,. . . , γ d + d ′ acting on some finite dimensionalspaces — i.e. we fix some *-representation of a Clifford algebra on a finite dimen-sional space C . We will prove that natural spectral triples on fuzzy tori, whoseDirac operators are the form /D n = d X j =1 [ X n,j , · ] ⊗ γ j + d X j =1 [ Y n,j , · ] ⊗ γ d + j + term ,acting on some dense subspace of a Hilbert space H n ⊗ C + ′ , converge to spectraltriples on tori of the form /D ∞ = d X j =1 X ∞ ,j ∂ j ⊗ γ j + d X j =1 Y ∞ ,j ∂ j ⊗ γ d + j + term , acting on a dense subspace of H ∞ ⊗ C , where, for all n ∈ N ∪ {∞} , and for all j ∈ { , . . . , d } , the operators X n,j and Y n,j are, respectively, the real part andthe imaginary part of some of the generating unitaries in B n , with the additionalcondition, when n = ∞ , that X ∞ ,j and Y ∞ ,j commute with A ∞ . The additional“term”, which we leave for the main formal description, is needed in order to ensurethat /D ∞ has a compact resolvent — indeed, the operator /D ∞ is defined on H ∞ ⊗ C d where H ∞ carries a *-representation of B ∞ , not just A ∞ . In turn, this termmust have a discrete form, so that the desired convergence occurs. The formaldescription of these spectral triples will be given once we have the needed notationabout quantum tori.Our spectral triples are different, though related, to other constructions for spec-tral triples on fuzzy and quantum tori. There are obvious similarities with the spec-tral triples in [27, 68, 3] when working on C ∗ ( C n , S n ) — though our techniques inthis paper are applicable to a much wider family of examples. Our spectral triplesalso share some commonalities with the spectral triples in [12, 13], since they in-volve modifying the usual, “flat” spectral triple on quantum tori by elements whichcommute with the quantum torus. However, our approach requires, in general,an additional term to compensate for the introduction of these elements, which inturn, is caused by the form of the finite dimensional spectral triples on fuzzy tori— a matter foreign to the discussion in [12, 13].While our spectral triples have nice properties — their square give a familiarLaplacian, and the dual actions act by Lipschitz functions, and in fact, isometriesin many cases (we can always choose, with our scheme, that the dual action shouldact by quantum isometries) — their primary value is in the fact that they formconvergent sequence for the spectral propinquity.1.2. The Spectral Propinquity.
We now must of course explain what the con-vergence of metric spectral triples actually means, since it is the very reason forthe present work. Such a notion is certainly involved: for instance, convergenceof the spectra of the Dirac operators of spectral triples is vastly insufficient, sincethe spectrum of the Dirac operator of a spectral triple can not distinguish evenbetween non-homeomorphic spaces. Our notion of convergence will now take us tothe realm of noncommutative metric geometry.Our idea for convergence for spectral triples begins, as a first step, by exploitingthe crucial observation by Connes that a spectral triple induces an extended metricon the state space of its underlying C*-algebra. Indeed, it may help to motivateour idea with a simple observation. Informally, it is natural to look upon the sets U n = { z ∈ C : z n = 1 } as approximations of the circle T = { z ∈ C : | z | = 1 } .However, U n is, topologically, just a set of n elements, and thus it is homeomorphicto S n = { n , . . . , nn = 1 } . Yet the latter set is, informally, an approximation of [0 , .Underlying our intuition here is not topology, but metric geometry. Indeed, as n tends to ∞ , the sequence ( U n ) n ∈ N , where U n is endowed with its usual metric as asubspace of C , does converges to T for the Hausdorff distance [22] induced by theEuclidean metric on C , whereas ( S n ) n ∈ N converges, for the same metric, to [0 , .We seek a similar framework to formalize that some sequences of “fuzzy tori”converge to quantum tori: at a minimum, we want to endow these spaces with aquantum analogue of a metric, and prove the convergence of the resulting struc-ture for an analogue of the Hausdorff distance, adapted to our noncommutative FRÉDÉRIC LATRÉMOLIÈRE geometric setting. Since many reasonable spectral triples do give rise to metricson state spaces [9], we have the starting point for our definition of a convergenceof spectral triples: metrics they induce should converge in a generalized Hausdorffdistance. This certainly seems a reasonably physical concept. The generalizedHausdorff distance we shall use is the Gromov-Hausdorff propinquity on the classof quantum compact metric spaces, which we introduced in [38, 36, 40], and wenow describe. As we shall see shortly, convergence of spectral triples involve morethan the convergence of the underlying metric, but this is the first step.The idea of a quantum metric begins with the well-known and profound (con-travariant) equivalence of category between the category of compact Hausdorffspaces and the category of unital
Abelian
C*-algebras, established by Gelfand andNaimark. In general, a category of quantum spaces consists of algebras which gen-eralizes (and include) some algebras of functions over certain types of spaces, withmorphisms the reversed arrows from the natural morphisms between these alge-bras. For instance, a quantum compact Hausdorff space is described by a unitalC*-algebra, which is no longer assumed to be commutative, and seen as an objectin the dual category of C*-algebras.Quantum compact metric spaces, in turn, are described by generalizations of thealgebras of
Lipschitz functions over a compact metric space. If ( X, d ) is a compactmetric space, then the Lipschitz constant L d ( f ) of an R -valued function f : X → R is given by(1.4) L d ( f ) = sup (cid:26) | f ( x ) − f ( y ) | d ( x, y ) : x, y ∈ X, x = y (cid:27) ,allowing L d ( f ) = ∞ .A function from X to C with a finite Lipschitz constant is called a Lipschitzfunction over ( X, d ) — and it is always an element of C ( X ) . This definition givesa seminorm L d defined on a dense subalgebra of C ( X ) . Convention 1.2. If L is a seminorm defined on a dense subspace dom ( L ) of anormed vector space E , then we set L ( x ) = ∞ whenever x / ∈ dom ( L ) , so that dom ( L ) = { x ∈ E : L ( x ) < ∞} . Within this convention, we will use ∞ = 0 , t ∞ , and ∞ + t = ∞ for all t ∈ R .Kantorovich, motivated by his work on Monge’s transportation problem, defined,in [24, 25], a distance mk L d on the state space S ( C ( X )) of C ( X ) , i.e. the set ofintegrals against Radon probability measures over X , by setting, for all µ, ν ∈ S ( C ( X )) : mk L d ( µ, ν ) = sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)Z X f dµ − Z X f dν (cid:12)(cid:12)(cid:12)(cid:12) : f ∈ C ( X ) , f = f , L d ( f ) (cid:27) .This metric, known as the Monge-Kantorovich metric, induces the weak* topologyon S ( C ( X )) , making it a fundamental tool not only in optimal transport theory,but also probability, statistics, and even fractal theory, where it has been oftenrenamed (for instance, Dobrushin [15] called this metric the Wasserstein metric,following works in probability by Wasserstein [71]). Moreover, the map whichsends points of X to their corresponding Dirac point measures over X becomes anisometry from ( X, d ) to ( S ( C ( X )) , mk L d ) ; therefore, L d allows us to recover themetric over X . Rieffel [58, 59], motivated by Connes’ original proposal about quantum metricspaces [9], proposed that quantum metrics could be metrics on state spaces in-duced by duality from seminorms on (no longer necessarily commutative) unitalC*-algebras which share basic properties with the Lipschitz seminorms. The exactlist of which properties of the Lipschitz seminorms should be kept has evolved intime. We settle on the definition which we used in [38, 39].
Notation . If A is a normed vector space, the norm of A is denoted by k·k A ,unless otherwise specified. Definition 1.4. A quantum compact metric space ( A , L ) is an ordered pair of aunital C*-algebra and a seminorm L , defined on a dense subspace dom ( L ) ⊆ sa ( A ) of the space sa ( A ) = { a ∈ A : a ∗ = a } of self-adjoint elements in A , such that:(1) ∀ a ∈ dom ( L ) L ( a ) = 0 ⇐⇒ a ∈ R A , where A is the unit of A ,(2) the Monge-Kantorovich metric mk L , defined between any two states ϕ , ψ of A by mk L ( ϕ, ψ ) = sup {| ϕ ( a ) − ψ ( a ) | : a ∈ dom ( L ) and L ( a ) } ,is a metric on the state space S ( A ) of A which induces the weak* topologyon S ( A ) ,(3) for all a, b ∈ dom ( L ) , the Jordan product ab + ba of a with b , and the Lieproduct ab − ba i of a with b , are both elements of dom ( L ) , and moreover: max (cid:26) L (cid:18) ab + ba (cid:19) , L (cid:18) ab − ba i (cid:19)(cid:27) k a k A L ( b ) + L ( a ) k b k A ,(4) { a ∈ dom ( L ) : L ( a ) } is closed in A .It can indeed be checked that if ( X, d ) is a compact metric space, then ( C ( X ) , L d ) is a quantum compact metric space. Noncommutative examples include C*-algebrasendowed with a strongly continuous ergodic action of a compact group equippedwith a continuous length functions [58] — which includes quantum tori, fuzzy tori[58], and noncommutative solenoids [17]; AF algebras [2], C*-algebras of word hy-perbolic group [53], C*-algebras of nilpotent discrete groups [65], Podles spheres[1], certain C*-crossed-products [23], certain quantum groups [4], and more (e.g.,[60, 50, 49, 7]). A generalization of the theory of quantum compact metric spacesto locally compact quantum metric spaces can be found in [32, 33].The relationship between spectral triples and quantum compact metric spacesis captured in the following definition, which owes to the original observation byConnes [9] that a spectral triple induces an extended metric on the state space ofits underlying C*-algebras. Notation . If T : E → F is a linear map between two normed spaces, we writeits norm as ||| T ||| EF (allowing for ∞ ). When E = F , we simply write ||| T ||| F . Definition 1.6. A metric spectral triple ( A , H , /D ) is a spectral triple such that,if we set, for all a ∈ sa ( A ) : L /D ( a ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ /D, a ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H ,then ( A , L /D ) is a quantum compact metric space. An example of metric spectral triple is given on quantum tori, and on certainquantum homogeneous spaces, by Rieffel in [58]. Another family of examples isgiven by the curved quantum tori [12, 13], as proven in [35]. Other examplesinclude spectral triples from length functions on certain groups [53, 60, 6], spectraltriples over Podles spheres [1], or spectral triples over certain fractals [7, 30], forinstance.In this paper, we will prove that our proposed spectral triples on fuzzy andquantum tori are all metric spectral triples. Therefore, it becomes possible to applyto them our theory of quantum compact metric spaces and their convergence, whichwe now turn to.The class of quantum compact metric spaces form a category for an appropriatenotion of Lipschitz morphisms [37]. We focus here on the subcategory whose objectsare quantum compact metric spaces and morphisms are quantum isometries, in thefollowing sense (note that strictly speaking, quantum isometries should be given bythe opposite arrows, but it will be clearer to keep our arrows in the same direction asthe actual morphisms). The idea behind the following notion of quantum isometriesis due to Rieffel, and is motivated by McShane’s theorem for the extension of real-valued
Lipschitz functions.
Definition 1.7 ([59, 61, 38]) . A quantum isometry π : ( A , L A ) → ( B , L B ) is asurjective *-morphism π : A ։ B such that ∀ b ∈ sa ( B ) L B ( b ) = inf { L A ( a ) : π ( a ) = b } .In particular, we will consider two quantum compact metric spaces to be iso-morphic when they are fully quantum isometric, in the following sense. Definition 1.8 ([38]) . A full quantum isometry π : ( A , L A ) → ( B , L B ) is a *-isomorphism π : A → B such that L B ◦ π = L A .Our principal interest is the construction of a distance function on the class ofquantum compact metric spaces, called the Gromov-Hausdorff propinquity , whichgeneralizes the Gromov-Hausdorff distance [21, 20, 16], itself a generalization of theHausdorff distance [22].
Notation . The
Hausdorff distance [22] induced on the class of closed subsets ofa compact metric space ( X, d ) is denoted by Haus d . We recall that if A , A ⊆ X are two closed subsets of ( X, d ) , then Haus d ( A , A ) = max { j,s } = { , } sup x ∈ A j inf y ∈ A k d ( x, y ) .The construction of the propinquity, upon which all of our work is built, goes asfollows. Motivated by Edwards [16] and Gromov [20] constructions, we define thepropinquity by first considering any quantum analogues of an isometric embeddingof two quantum compact metric spaces into a third one. Definition 1.10. A tunnel τ = ( D , L D , π , π ) between two quantum compactmetric spaces ( A , L ) and ( A , L ) is given by a quantum compact metric space ( D , L D ) and, for each j ∈ { , } , a quantum isometry from ( D , L D ) onto ( A j , L j ) .Our metric is then defined as follows. ( D , T ) π A (cid:1) (cid:1) ✄✄✄✄✄✄✄ π B (cid:30) (cid:30) ❁❁❁❁❁❁❁ ( A , L A ) ( B , L B ) Figure 1.
A tunnel
Definition 1.11.
Let ( A , L A ) and ( B , L B ) be two quantum compact metric spaces.If τ is a tunnel from ( A , L A ) to ( B , L B ) , then the extent χ ( τ ) of τ is the non-negativereal number χ ( τ ) = max n Haus mk L D ( S ( D ) , S ( A )) , Haus mk L D ( S ( D ) , S ( B )) o .The propinquity Λ ∗ (( A , L A ) , ( B , L B )) is the non-negative number Λ ∗ (( A , L A ) , ( B , L B )) = inf { χ ( τ ) : τ is a tunnel from ( A , L A ) to ( B , L B ) } .We were able to prove that the dual propinquity satisfies the following set ofdesirable properties, which, in particular, extend the properties of the Gromov-Hausdorff distance from the class of compact metric spaces to the class of quantumcompact metric spaces: Theorem 1.12 ([38, 36, 40]) . The propinquity Λ ∗ is a complete distance (class)function over the class of quantum compact metric spaces, up to full quantum isom-etry. Moreover, the (class) function which sends any compact metric space ( X, d ) tothe quantum compact metric space ( C ( X ) , L d ) , with L d defined by Expression (1.4),is an homeomorphism from the class of compact metric spaces, endowed with thetopology of the Gromov-Hausdorff distance, to its range, with the topology inducedby the Gromov-Hausdorff propinquity. Since, in particular, a sequence ( X n , d n ) n ∈ N of compact metric spaces convergesto a compact metric space ( X, d ) for the Gromov-Hausdorff distance if, and onlyif, the sequence ( C ( X n ) , L d n ) n ∈ N converges to ( C ( X ) , L d ) for the propinquity Λ ∗ ,the propinquity allows us to extend the topology of the Gromov-Hausdorff distanceto noncommutative geometry. The propinquity allows us, for instance, to prove ananalogue of Gromov’s compactness theorem in noncommutative geometry [39].Our main application for the propinquity, so far, is to obtain continuous familiesof quantum compact metric spaces, and in particular, to construct rigorous finitedimensional approximations of quantum compact metric spaces, often motivated bymathematical physics. The physical relevance of our metric is that it indeed impliesthe convergence of the state space of a C*-algebra. Examples of convergence ofquantum compact metric spaces for the propinquity include any convergence in thesense of the Gromov-Hausdorff distance for classical compact metric spaces (suchas convergence of some fractals for their geodesic distances [30]), finite dimensionalapproximation of AF algebras and continuous families of AF algebras (such asUHF algebras and Effros-Shen algebras) [2], convergence of matrix algebras to thesphere [66], finite dimensional approximation of quantum compact metric spacesbuilt over nuclear quasi-diagonal C*-algebras, and, notably for our purpose, theconvergence of sequences of fuzzy tori to quantum tori [31, 34], for the quantummetrics introduced by Rieffel in [58]. In this work, we will prove, as a first step toward the convergence of spectraltriples, that the quantum metrics induced by our spectral triples on fuzzy tori con-verge to quantum metrics on quantum tori under natural conditions . The quantummetrics induced by the spectral triples in this paper are not the same as the quan-tum metrics used in [34] (which were not constructed using a spectral triple), sothis result is new to the present work; however, our strategy consists in establishingproperties for the quantum metrics of our spectral triples which then enables usto apply the techniques of [34], which we will not repeat here. We will, however,record the main ingredients of the proof of [34, Theorem 5.2.5] which we need tofurther study the convergence of spectral triples, beyond the metric aspects.Now, a spectral triples contain more information than the metric it induces. Aproper notion of convergence for spectral triples should also account for this extrainformation. Our idea, which we developed in [42], begins with the observation thatif ( A , H , /D ) is a spectral triple, then t ∈ R S t = exp(2 iπt /D ) is a continuousgroup action by unitaries of R on H , i.e. it is a quantum dynamics (since /D isself-adjoint). This quantum dynamics is, of course, tightly related to /D , which isits infinitesimal generator. Moreover, for any operator a in a dense *-subalgebraof A , we also note that [ /D, a ] = lim t → S t aS − t − at . Thus, we propose to define theconvergence of a spectral triple by adding, to the convergence of the underlyingquantum metrics, the convergence of the associated quantum dynamics, seen as thenatural physical object associated with a spectral triple. To this end, we proceedin two steps. First, since the quantum dynamics for two different spectral triplestypically act on different Hilbert spaces (in the present work, spectral triples forfuzzy tori act on finite dimensional Hilbert spaces, while spectral triples on quantumtori act on infinite dimensional Hilbert spaces), and since spectral triples involve *-representations as well, we need a mean to compare these Hilbert spaces, as modulesover quantum compact metric spaces. Second, we will indeed use a covariant versionof the propinquity generalized to modules over quantum compact metric spaces.The construction of the spectral propinquity, which we now describe, is thusbased on various new distances, based upon the propinquity, but defined for variousclasses of quantum objects: we shall use the ideas of [45, 48, 46, 41], where wedefine the modular propinquity — a metric between appropriately defined modulesover quantum compact metric spaces, as defined below. We also use the ideasof [43, 47, 44, 42], where we define a covariant version of the propinquity and themodular propinquity, which enables us to discuss convergences of group actions. Wepresent a synthesis of the tools from these works which need to define the spectralpropinquity below.Let ( A , H , /D ) be a metric spectral triple. By [42], if we set: ∀ ξ ∈ H DN ( ξ ) = k ξ k H + (cid:13)(cid:13) /Dξ (cid:13)(cid:13) H and ∀ a ∈ { b ∈ sa ( A ) : b dom ( D ) ⊆ dom (cid:0) /D (cid:1) } L /D ( a ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ /D, a ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H allowing for the value ∞ , the tuple(1.5) qvb (cid:0) A , H , /D (cid:1) = ( H , DN , C , , A , L /D ) is an example of a metrical C ∗ -correspondence in the following sense: Definition 1.13 ([45, 41]) . A K -metrical C ∗ -correspondence ( M , DN , B , L B , A , L A ) ,for K > , is given by two quantum compact metric spaces ( A , L A ) and ( B , L B ) ,a A - B -bimodule M , which also carries a right B -Hilbert module structure, and anorm DN defined on a dense A -submodule dom ( DN ) of M such that:(1) ∀ ω ∈ dom ( DN ) k ω k M DN ( ω ) ,(2) { ω ∈ dom ( DN ) : DN ( ω ) } is compact in k·k M ,(3) for all ω, η ∈ M , denoting b = h ω, η i M ∈ B , we have: max (cid:26) L B (cid:18) b + b ∗ (cid:19) , L B (cid:18) b − b ∗ i (cid:19)(cid:27) DN ( ω ) DN ( η ) ,(we refer to this inequality as the inner Leibniz inequality),(4) for all a ∈ dom ( L A ) and ω ∈ dom ( DN ) we have: DN ( aω ) K ( k a k A + L A ( a )) DN ( ω ) .(we refer to this inequality as the modular Leibniz inequality).The norm DN is called a D-norm .When ( M , DN , B , L B , C , is a metrical C ∗ -correspondence, ( M , DN , B , L B ) iscalled a metrized quantum vector bundle .Examples of metrized quantum vector bundles include Hermitian bundles overRiemannian manifolds, with the D-norm constructed from any metric connection[45], and Heisenberg modules over quantum tori [48], where the D-norms also arisenaturally from a noncommutative analogue of a metric connection. For our purpose,the main example is given by metric spectral triples [42], using Expression (1.13).We thus will apply this construction to the spectral triples on fuzzy and quantumtori we introduce in this paper.Thus, the question is: how do we define the convergence of metrical quantumvector bundles? The idea, actually, is rather natural. For each j ∈ { , } , let: (cid:0) M j , DN j , B j , L j , A j , L ′ j (cid:1) be a metrical C ∗ -correspondence. Our first task, of course, is to determine howclose ( A , L ′ ) and ( A , L ′ ) are, and how close ( B , L ) and ( B , L ) are. Thismeans that we will need two tunnels, τ A and τ B (see Definition (1.10) above). Todeal with the modules M and M , we then mimic our idea from the propinquity.Remarkably, once the proper notion of tunnel between metrical C*-correspondenceis introduced, the tools introduced for the propinquity can be applied directly.Formally, it is convenient to proceed in two steps: first, we work with the “modularquantum vector bundle” part. Definition 1.14.
Let A , B be two unital C*-algebras. A module morphism (Π , π ) from a left A -module M to a right B -module N is given by the following data: • a unital *-morphism π : A → B , • a linear map Π : M → N such that ∀ a ∈ A , ∀ ω ∈ M , Π( aω ) = π ( a )Π( ω ) . The module morphism (Π , π ) is said to be surjective when both Π and π aresurjective maps, and it is said to be an isomorphism when both Π and π arebijections.A Hilbert module morphism (Π , π ) from a left A -Hilbert module M to a right B -Hilbert module N is a module morphism when ∀ ω, ξ ∈ M , h Π( ω ) , Π( ξ ) i N = h ω, ξ i M .We refer to [45] for examples of such a structure and for its motivations.We defined the modular propinquity in [45, 41] by extending the notion of atunnel between quantum compact metric spaces to the notion of a tunnel betweenmetrized quantum vector bundles. Definition 1.15 ([41]) . Let ( M j , DN j , A j , L j ) be a metrized quantum vector bun-dle, for j ∈ { , } . A modular tunnel ( D , (Π , π ) , (Π , π )) is given by(1) a metrized quantum vector bundle D = ( P , DN ′ , D , L D ) ,(2) a tunnel ( D , L D , π , π ) from ( A , L ) to ( A , L ) ,(3) for each j ∈ { , } , (Π j , π j ) is a surjective Hilbert module morphism from P (over A ) to M j such that ∀ ω ∈ M j , DN j ( ω ) = inf (cid:8) DN ′ ( η ) : Π j ( η ) = ω (cid:9) .The extent of a modular tunnel is computed just like the extent of the under-lying tunnel between quantum compact metric spaces, as we see in the followingdefinition. Definition 1.16 ([41]) . The extent , χ ( τ ) , of a modular tunnel τ = ( D , (Π , π ) , (Π , π )) ,with D = ( P , DN ′ , D , L D ) , is the extent of the tunnel ( D , L D , π , π ) .The modular propinquity is then defined along the same lines as the propinquity. Definition 1.17 ([45, 41]) . The modular propinquity , Λ mod ( A , B ) , between twometrized quantum vector bundles A and B is the nonnegative number given by Λ ∗ mod ( A , B ) = inf { χ ( τ ) : τ is a modular tunnel from A to B } .We now record a few fundamental properties of the modular propinquity. Theorem 1.18 ([45, 41]) . The modular propinquity is a complete metric, up to fullisometric isomorphism, where two metrized quantum vector bundles ( M , DN , A , L A ) and ( N , DN ′ , B , L B ) are fully isometrically isomorphic if and only if there exists aHilbert module isomorphism (Π , π ) such that • L B ◦ π = L A , • DN ′ ◦ Π = DN . We extend a tunnel between metrized quantum vector bundles to a tunnel be-tween metrical C*-correspondences as follows.
Definition 1.19 ([41]) . Let A j = ( M j , DN M j , A j , L j , B j , L j ) , for j ∈ { , } .A K -metrical tunnel ( τ, τ ′ ) from A to A , for some K > , is given by thefollowing data:(1) a modular tunnel τ = ( D , ( θ , Θ ) , ( θ , Θ )) from ( M , DN M , A , L ) to ( M , DN M , A , L ) , where we write D = ( P , DN , D , L D ) , (2) a tunnel τ ′ = ( D ′ , L ′ , π , π ) from ( B , L ) to ( B , L ) ,(3) P is also a D ′ -left module,(4) ∀ ω ∈ P , ∀ d ∈ D ′ , DN ( dω ) K ( L ′ ( d ) + k d k D ′ ) DN ( ω ) ,(5) for all j ∈ { , } , the pair ( π j , Θ j ) is a left module morphism from the left D ′ -module P to the left A j -module M j . Definition 1.20 ([41]) . The extent , χ ( τ, τ ′ ) , of a metrical tunnel ( τ, τ ′ ) is givenby χ ( τ, τ ′ ) = max { χ ( τ ) , χ ( τ ′ ) } . Definition 1.21 ([41]) . Fix K > . The metrical K -propinquity , Λ ∗ met K ( A , B ) ,between two metrized quantum vector bundles A and B is the nonnegative numbergiven by Λ ∗ met ( A , B ) = inf { χ ( τ ) : τ is a metrical K -tunnel from A to B } . Theorem 1.22 ([41]) . Fix K > . The metrical K -propinquity is a completemetric, up to full isometry, on the class of K -metrical C*-correspondence, wheretwo metrical C*-correspondences (cid:0) M j , DN j , B j , L j , A j , L ′ j (cid:1) are fully isometric when: • the underlying metrized quantum vector bundles ( M , DN , B , L ) and ( M , DN , B , L ) are isometrically isomorphic, • the quantum compact metric spaces ( A , L ′ ) and ( A , L ′ ) are fully quantumisometric, • we can choose the above quantum isometries so that the actions of A on M and A on M are intertwined by these morphisms. We should address the constant K which is a part of the definition of the metricalpropinquity. In general, the properties of our various propinquity metrics dependon the choice of some uniform Leibniz properties, common to all the spaces underconsiderations. There is some useful flexibility as to what this Leibniz propertyneeds to be. For instance, in this work, we will see that K = 5 (or anythinglarger) is an appropriate choice to accommodate our examples. The value of K is not important, as long as we can find some value which works for the entiresequence we wish to prove converge in the sense of the propinquity. Moreover,there is an obvious relation between metrical propinquities for different values of K — the propinquity for a larger K dominates all the ones with a smaller choice of K , making the picture easy to understand.We now can describe the last step in the definition of the convergence of spectraltriples, i.e., the convergence of the associated quantum dynamics, which occur ondifferent metrical C*-correspondences. Let us assume that we are given two metricspectral triples ( A , H , /D ) and ( A , H , /D ) . In Equation (1.13), we have definedthe metrical quantum vector bundles qvb (cid:0) A , H , /D (cid:1) and qvb (cid:0) A , H , /D (cid:1) , as-sociated respectively with the metric spectral triples ( A , H , /D ) and ( A , H , /D ) .Let P = ( τ, τ ′ ) be a metrical tunnel between qvb (cid:0) A , H , /D (cid:1) and qvb (cid:0) A , H , /D (cid:1) .In particular, let us write τ = ( P , (Φ , φ ) , (Φ , φ )) and note that by Defini-tion 1.19 for metrical tunnels, τ is a modular tunnel between ( H , DN , C , and ( H , DN , C , , where DN and DN are, respectively, the graph norms of /D and /D . Furthermore, let us write P = ( P , DN D , D , L D ) . Since /D and /D are self-adjoint, we can define two strongly continuous actionsof R by unitaries on H and H by letting ∀ j ∈ { , } , ∀ t ∈ R , S tj = exp( itD j ) .The spectral propinquity is defined by extending the metrical propinquity ofDefinition 1.21 to include the actions S and S of R .With this in mind, let ε > and assume that χ ( τ ) ε . Let us call a pair ( ς , ς ) of maps from R to R an ε - iso-iso , for some ε > , whenever ∀ j ∈ { , } , ∀ x, y, z ∈ (cid:20) − ε , ε (cid:21) , || ς j ( x ) ς j ( y ) − z | − | xy − ς k ( z ) || ε ,and ς (0) = ς (0) = 0 .As discussed in [47], such maps can be used to define a distance on the class ofproper monoids, but for our purpose, as we only work with the proper group R , thedefinition simplifies somewhat. In fact, we only recall the definition so that we mayproperly define the spectral propinquity below: for our purposes, the only iso-isomap we will work with is simply the identity of R .Thus, suppose that we are given an ε -iso-iso ( ς , ς ) from R to R , as above. Wecall ( τ, τ ′ , ς , ς ) an ε -covariant metrical tunnel.The ε -covariant reach of ( τ, ς , ς ) is then defined as follows: max j ∈{ , } sup ξ ∈ H j DN j ( ξ ) inf ξ ′ ∈ H k DN k ( ξ ′ ) sup | t | ε sup ω ∈ P DN D ( ω ) (cid:12)(cid:12)(cid:12)(cid:10) S tj ξ, Π j ( ω ) (cid:11) H j − D S ς k ( t ) k ξ ′ , Π k ( ω ) E H k (cid:12)(cid:12)(cid:12) .We define the ε -magnitude µ ( τ, τ ′ , ς , ς | ε ) of an ε -covariant tunnel ( τ, ς , ς ) tobe the maximum of the extent of τ , the extent of τ ′ , and the ε -covariant reach of ( τ, ς , ς ) .The spectral propinquity Λ spec (( A , H , /D ) , ( A , H , D )) is the nonnegative number min ( √ , inf { ε > ∃ ε -covariant metrical tunnel τ , µ ( τ | ε ) ε } ) .We refer to [47, 41] for a discussion of the fundamental properties of this met-ric, as a special case of the covariant metrical propinquity, including a discussionof sufficient conditions for completeness (on certain classes). We record here thefollowing property of this metric. Theorem 1.23 ([42]) . The spectral propinquity Λ spec is a metric on the class ofmetric spectral triples, up to the following coincidence property: for any metricspectral triples ( A , H , /D ) and ( A , H , /D ) , Λ spec (( A , H , D ) , ( A , H , D )) = 0 if and only if there exists a unitary map V : H → H such that V D V ∗ = D and Ad V = V ( · ) V ∗ is a *-isomorphism from A onto A , where (as is customary) we identify A and A with their images by their repre-sentations on H and H , respectively. In particular, the *-isomorphism from A onto A implemented by the adjoint action of V is a full quantum isometry from ( A , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ D , · ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H onto ( A , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ D , · ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H ) . An interesting example of convergence of spectral triples for the spectral propin-quity is given by approximations of spectral triples on fractals by spectral tripleson their natural approximating, finite, graphs [30].In this paper, we will prove that sequences of spectral triples on fuzzy tori con-verge, under very natural assumptions, to spectral triples on quantum tori, in thesense of the spectral propinquity.
Physically, this implies that the state spaces ofthe physical models described by fuzzy tori converge, as metric spaces, to the statespaces of the quantum tori, for the metric induced by the spectral triples, and thatthe quantum dynamics associated with these spectral triples converge as well. Werefer to [38, 36, 45, 47, 42] for a detailed discussion of this metric, but an informalunderstanding can be glanced with the following construction from these papers;we simply state that, informally, if two spectral triples are close, then, in particu-lar, there exists a compact-set valued map from the Hilbert space of one of thesespectral triples to the other, which behaves as a set-valued version of a unitary, aswell as almost intertwining the actions by unitaries induced by the Dirac operatorsof the spectral triples.The structure of the paper follows the strategy described above. We first set ournotation by describing the formal background on quantum and fuzzy tori. We thenintroduce our spectral triples — which include the proof that indeed, our proposedDirac operators are self-adjoint with compact resolvent. We then establish coretechnical results about the metric properties of these spectral triples, includingan analogue of the mean value theorem. This allows us to prove that our spectraltriples are indeed, metric spectral triples. We can then prove the convergence of thequantum metrics induced by these spectral triples on fuzzy tori, for the propinquity.We then prove that the underlying Hilbert spaces of our spectral triples, seen asmodules over fuzzy and quantum tori, converge as metrical C*-correspondence forthe metrical propinquity. We conclude by proving the convergence of the quantumdynamics associated with our spectral triples.2.
The Geometry of Quantum and Fuzzy Tori
We begin with a description of fuzzy and quantum tori, and then introduce thespectral triples on fuzzy tori which we will consider, as well as the spectral tripleson quantum tori which will be their limit for the spectral propinquity.2.1.
Background: the quantum and fuzzy tori.
A torus is the Gelfand spec-trum of the convolution C*-algebra of a finite product of infinite cyclic groups.A quantum torus is a deformation of a classical torus, defined as a twisted con-volution algebra of a finite product of infinite cyclic groups, or equivalently, as adeformation-quantization of the torus for certain Poisson structures [56, 57]. Weadopt the former description. A fuzzy torus is then a finite dimensional versionof a quantum torus, namely, a twisted convolution C*-algebra for a finite productof finite cyclic groups. Since our work in this paper focuses on spectral triples, we will give a presentation of quantum and fuzzy tori which stresses a particular*-representation for these C*-algebras.
Notation . Let N ∗ = N \ { , } . Let N ∗ = N ∗ ∪ {∞} be the one point compact-ification of N ∗ .Let d ∈ N ∗ . For all k = ( k (1) , . . . , k ( d )) ∈ N d ∗ , we set: k Z d = d Y j =1 k ( j ) Z and Z dk = Z d / k Z d = d Y j =1 Z . k ( j ) Z ,with the convention that ∞ Z = { } , so that Z d ( ∞ ,..., ∞ ) = Z d . We also write ∞ d for ( ∞ , . . . , ∞ ) ∈ N d ∗ .A -cocycle of a discrete group G , with values in the group T = { z ∈ C : | z | = 1 } [51], is a map σ : G × G T such that(2.1) ∀ x, y, z ∈ G σ ( x, y ) σ ( xy, z ) = σ ( x, yz ) σ ( y, z ) ,and σ ( e, x ) = σ ( x, e ) = 1 for all x ∈ G , and e ∈ G the unit of G . Two -cocycles σ and σ ′ are cohomologous when there exists a function f : G → T such that ∀ x, y ∈ G σ ( x, y ) = f ( x ) f ( y ) f ( x, y ) σ ′ ( x, y ) .In particular, a -cocycle is trivial when it is cohomologous to the constant function . Now, any -cocycle of G is cohomologous to a normalized -cocycle of G [29],where σ is a normalized -cocycle when ∀ x ∈ G σ ( x, − x ) = 1 .Since two cohomologous -cocycles of a discrete group G give rise to *-isomorphictwisted convolution C*-algebras of G , we will only work with specific choices of -cocycles of the groups Z dk , all normalized, and representing every possible coho-mology class. Notation . Let d ∈ N ∗ . By [28, Theorem 7.1], any -cocycle of Z d is cohomolo-gous to a skew bicharacter, i.e. a -cocycle of the form(2.2) ς Θ : ( x, y ) ∈ Z d × Z d exp ( iπ y Θ x ⊺ ) for some real d × d antisymmetric matrix Θ ∈ M d ( R ) over R d , with the conventionthat we write elements in Z d as × d matrices, and the transpose of a matrix A by A ⊺ . Skew bicharaters are, in particular, normalized.Let us now fix k = ( k (1) , . . . , k ( d )) ∈ N ∗ . We let AS d ( k ) be the set of realantisymmetric d × d -matrices Θ = ( θ js ) j,s d such that gcd( k ( j ) , k ( s )) θ j,s ∈ Z forall s, j ∈ { , . . . , d } , where, for any n, m ∈ N ∗ , define gcd( n, m ) = if n = ∞ and m = ∞ , n if n < ∞ and m = ∞ , m if n = ∞ and m < ∞ ,the greatest common divisor of n and m otherwise.With this notation, if Θ ∈ AS d ( k ) , if n, m ∈ Z dk , and if w, w ′ , p, p ′ ∈ Z d such that n = w + k Z d = w ′ + k Z d and m = p + k Z d = p ′ + k Z d , then ς Θ ( w, p ) = ς Θ ( w ′ , p ′ ) .We simply write ς Θ ( n, m ) for this common value, as a slight abuse of notation. Any -cocycle of Z dk can be lifted to a -cocycle of Z d . Therefore, any -cocycleof Z dk is cohomologous to a -cocycle: ( n, m ) ∈ Z dk ς Θ ( n, m ) for some Θ ∈ AS d ( k ) — where the conditions on Θ follow from the fact that x, y ∈ Z dk σ ( x, y ) σ ( y, x ) must be a skew bicharacter [28]. We note that ς Θ is anormalized -cocycle, but it is not a bicharacter, in general. For our purpose, it isimportant that our -cocycles are normalized, while we do not need the bicharacterproperty. We do note, however, that ∀ n, m ∈ Z dk ς Θ ( n, m ) = ς Θ ( m, n ) .We then define: Ξ d = n ( k, ς Θ ) ∈ N d ∗ × (cid:16) T ( Z d ) (cid:17) : Θ ∈ AS ( k ) o .We endow the set Ξ d with the subspace topology from N d ∗ × T ( Z d ) , where the setof T ( Z d ) of T -valued functions from Z d is endowed with the topology of pointwiseconvergence.The space Ξ d is actually compact, and its topology can also be described as thequotient topology for the surjective map: ( k, Θ) ∈ Υ d ( k, ς Θ ) with Υ d = { ( k, Θ) ∈ N d ∗ × M d ( R ) : Θ ∈ AS ( k ) } , where the algebra M d ( R ) of d × d real matrices is endowed with its usual topology, given by any norm. Remark . We will rely on the proofs in [34] later in this paper, and we choose ournotations to match this reference. However, there is a small correction which weneed to point out: in [34, Notation 3.1.3], the definition of Ξ d is stated improperly:it should be defined as we did here, and not be restricted to pairs of the form ( k, σ ) with σ a bicharacter of Z dk . Indeed, the entire work in [34] only uses the -cocycleproperty and the fact it is normalized; in fact the very form we use in Notation(2.2) is indeed given in [34, Notation 3.1.3]. This unfortunate misprint of ours hasno impact on [34], with the single exception of one computation in the proof of [34,Theorem 3.1.7], which we reprove below as Lemma (2.5) — and [34, Theorem 3.1.7]is true and well-known anyway. Thus, past this unfortunate terminology error ofours in [34], we will use [34] unchanged.Quantum and fuzzy tori are twisted convolution C*-algebras of the groups Z dk ,for any d ∈ N ∗ , for any k ∈ N d ∗ and for any -cocycle σ of Z dk — and indeed, up to a*-isomorphism, all fuzzy and quantum tori are obtained using the -cocycle σ with ( k, σ ) ∈ Ξ d . We begin our presentation of these C*-algebras by introducing certainnatural projective unitary representations of Z dk on the Hilbert space ℓ ( Z dk ) , wherewe use the following notation. Notation . For any (nonempty) set E and any p ∈ [1 , ∞ ) , the set ℓ p ( E ) is theset of all absolutely p -summable complex valued functions over E , endowed withthe norm: k ξ k ℓ p ( E ) = X x ∈ E | ξ ( x ) | p ! p for all ξ ∈ ℓ p ( E ) .For all e ∈ E , we define δ e by δ e : x ∈ E ( if x = e , otherwise;of course δ e ∈ ℓ p ( Z dk ) .Moreover, if p = 2 then ( ℓ ( E ) , k · k ℓ ( E ) ) is a Hilbert space, where the innerproduct is given by ∀ ξ, η ∈ ℓ ( E ) h ξ, η i ℓ ( E ) = X x ∈ E ξ ( x ) η ( x ) Lemma 2.5.
Let d ∈ N ∗ and ( k, σ ) ∈ Ξ d . For all m ∈ Z dk , the operator W mk,σ defined, for all ξ ∈ ℓ ( Z dk ) , by: W mk,σ ξ : n ∈ Z dk σ ( n, m ) ξ ( n + m ) ,is a unitary operator on ℓ ( Z dk ) , and moreover, for all m, n ∈ Z dk , (2.3) W mk,σ W nk,σ = σ ( m, n ) W m + nk,σ .Proof. By Expression (2.1), if m, n, p ∈ Z dk , and if ξ ∈ ℓ ( Z dk ) , then (cid:0) W mk,σ W nk,σ ξ (cid:1) ( p ) = σ ( p, m ) (cid:0) W nk,σ ξ (cid:1) ( p + m )= σ ( p, m ) σ ( m + p, n ) ξ ( p + m + n )= σ ( m, n ) σ ( p, m + n ) ξ ( p + ( m + n ))= (cid:16) σ ( m, n ) W m + nk,σ ξ (cid:17) ( p ) .Hence, Expression (2.3) holds. An easy computation shows that W k,σ is the identity,and that (cid:16) W mk,σ (cid:17) ∗ = W − mk,σ ; our lemma is thus proven. (cid:3) The quantum and fuzzy tori are the C*-algebras generated by the projective*-representations of Z dk defined in Lemma (2.5). Definition 2.6.
Let d ∈ N ∗ and let ( k, σ ) ∈ Ξ d . The C*-algebra C ∗ (cid:0) Z dk , σ (cid:1) is thecompletion of the *-algebra generated by { W mk,σ : m ∈ Z dk } (using the notation ofLemma (2.5)), for the operator norm |||·||| ℓ ( Z dk ) .When k = ∞ d , the C*-algebra C ∗ (cid:0) Z d ∞ d , σ (cid:1) = C ∗ ( Z d , σ ) is called a quantumtorus . When k ∈ N d ∗ , the C*-algebra C ∗ (cid:0) Z dk , σ (cid:1) is called a fuzzy torus . Remark . We are not aware of a common term to name the C*-algebras C ∗ (cid:0) Z dk , σ (cid:1) when k ∈ N d ∗ , with k / ∈ N d ∗ and k = ∞ d , i.e. when k contains some finite and someinfinite values. Our work in this paper applies to these mixed objects just as wellas to quantum tori and fuzzy tori.The C*-algebra C ∗ (cid:0) Z dk , σ (cid:1) , for any ( k, σ ) ∈ Ξ d , is *-isomorphic to a C*-algebraconstructed as a completion of ℓ ( Z dk ) , for an appropriate product, adjoint, andnorm. This presentation of quantum and fuzzy tori will also be helpful. Lemma 2.8.
Let d ∈ N ∗ and ( k, σ ) ∈ Ξ d . We use the notation of Lemma (2.5).For all f ∈ ℓ ( Z dk ) , the operator π k,σ ( f ) = X m ∈ Z dk f ( m ) W mk,σ is bounded on ℓ ( Z dk ) , with ||| π k, Θ ( f ) ||| ℓ ( Z dk ) k f k ℓ ( Z dk ) , and moreover, π k,σ ( f ) ∈ C ∗ (cid:0) Z dk , σ (cid:1) . Moreover,(1) π k,σ is injective;(2) for all f, g ∈ ℓ ( Z dk ) , π k,σ ( f ) π k,σ ( g ) = π k,σ ( f ∗ k,σ g ) where (2.4) f ∗ k,σ g : n ∈ Z dk X m ∈ Z dk f ( m ) g ( n − m ) σ ( m, n − m ) ;(3) for all f ∈ ℓ ( Z dk ) , π k,σ ( f ) ∗ = π k,σ ( f ∗ ) where f ∗ : m ∈ Z dk f ( − m ) ;Proof. Since W mk,σ is unitary for all m ∈ Z dk , it is immediate that ||| π k,σ ( f ) ||| ℓ ( Z dk ) k f k ℓ ( Z dk ) ; moreover it then follows immediately that π k,σ ( f ) ∈ C ∗ (cid:0) Z dk , σ (cid:1) . If π k,σ ( f ) = 0 , then π k,σ ( f ) δ = X m ∈ Z dk f ( m ) σ ( m, n ) δ m so P m ∈ Z dk | f ( m ) | = k π k,σ ( f ) δ k ℓ ( Z dk ) = 0 , and thus f = 0 . So π k,σ is injective.The other assertions follow from direct computations. (cid:3) Remark . We note that, for any d ∈ N ∗ and k ∈ N d ∗ , the adjoint operation of ( ℓ ( Z dk ) , ∗ k,σ , · ∗ ) does not depend on σ . This will be a very helpful property for us,and it follows from our choice to work with normalized -cocycles. Lemma 2.10. [73]
Let d ∈ N ∗ , and let ( k, σ ) ∈ Ξ d . Using the notation of Lemma(2.8), the triple ( ℓ ( Z dk ) , ∗ k,σ , · ∗ ) is a Banach *-algebra. Moreover, the function f ∈ ℓ (cid:0) Z dk (cid:1)
7→ ||| π k,σ ( f ) ||| ℓ ( Z dk ) is a C*-norm on ( ℓ ( Z dk ) , ∗ k,σ , · ∗ ) ; the completion of ( ℓ ( Z dk ) , ∗ k,σ , · ∗ ) for thisnorm is a C*-algebra, and the unique extension of π k,σ to this completion is a*-isomorphism onto C ∗ (cid:0) Z dk , σ (cid:1) . For all d ∈ N ∗ , and for all ( k, σ ) ∈ Ξ d , the C*-algebra C ∗ (cid:0) Z dk , σ (cid:1) is finitelygenerated by a subset of the unitaries n W mk,σ : m ∈ Z dk o . In fact, C ∗ (cid:0) Z dk , σ (cid:1) isalways finitely generated. Notation . Let d ∈ N ∗ . We will use a common notation for elements of Z d , R d and Z dk throughout this work, as it will keep our notation simpler, without anyrisk of confusion. For all j ∈ { , . . . , d } , we set e j = , . . . , , |{z} index j , , . . . , ∈ Z d , and we identify e j with its class in Z dk for any k ∈ N d ∗ .Let d ∈ N ∗ and ( k, σ ) ∈ Ξ d . For each j ∈ { , . . . , d } , note that π k, Θ ( δ e j ) = W e j k,σ .By Expression (2.3), we note that for all j, s ∈ { , . . . , d } :(2.5) W e j k,σ W e s k,σ = σ ( e j , e s ) W e s k,σ W e j k,σ = exp(2 iπθ j,s ) W e s k,σ W e j k,σ ,where σ = ς Θ and Θ = (Θ j,s ) j,s d ∈ AS d ( k ) . By Expression (2.3), we also notethat C ∗ (cid:0) Z dk , σ (cid:1) is the closure of the *-algebra generated by { W e k,σ , . . . , W e d k,σ } . Wethus refer to the unitaries W e k,σ ,. . . , W e d k,σ as the canonical unitaries of C ∗ (cid:0) Z dk , σ (cid:1) .A consequence of Expression (2.5), for any ( k, σ ) ∈ Ξ d , is that the commuta-tion relations between the canonical unitaries of C ∗ (cid:0) Z dk , σ (cid:1) is sufficient to recoverthe cohomology class of the -cocycle σ . In fact, C ∗ (cid:0) Z dk , σ (cid:1) has the followinguniversal property: if V , . . . , V d are unitaries on a Hilbert space such that V j V s = σ ( e j , e s ) V s V j for all j, s ∈ { , . . . , d } , then there exists a unique *-morphism π from C ∗ (cid:0) Z dk , σ (cid:1) onto the C*-algebra generated by V ,. . . , V d such that π ( W e j k,σ ) = V j forall j ∈ { , . . . , d } . We will not need this characterization of quantum or fuzzy tori,but it is an important motivation for their study.In this paper, we will construct operators modeled after the Dirac operatorconstruction from Riemannian geometry, and thus, we will make use of Cliffordalgebras. These algebras are actually examples of fuzzy tori, and this will provehelpful. Example 2.12.
Let n ∈ N ∗ . Let r be the n × n matrix r = 12 ,and σ = ς r − r ⊺ . Let k = , . . . , | {z } n times . The fuzzy torus C ∗ ( Z nk , ς r ) is actually the Clifford algebra Cl ( C n ) of C n . Indeed: ∀ j, s ∈ { , . . . , n } W e j k,σ W e s k,σ + W e s k,σ W e j k,σ = ( if j = s , if j = s .Thus, W e k,σ ,. . . , W e n k,σ satisfy the universal conditions of the generators of Cl ( C n ) .By the universal properties of both C ∗ (cid:0) Z dk , σ (cid:1) and Cl ( C n ) , the two algebras arethus isomorphic as associative algebras.We will make use of this identification. We will however use the notation ∀ j ∈ { , . . . , n } γ j = iW e j k,σ ,so that ∀ j, s ∈ { , . . . , n } γ j γ s + γ s γ j = ( − if j = s , if j = s .Of course, C ∗ ( Z nk , σ ) = C ∗ ( γ , . . . , γ n ) , but this choice will somewhat lighten ournotation. We note that γ ∗ j = − γ j for all j ∈ { , . . . , n } .Quantum and fuzzy tori include classical examples. Example 2.13. If Θ = 0 (as a d × d matrix), then ς Θ = 1 , and, for all k ∈ N d ∗ , theC*-algebra C ∗ (cid:0) Z dk , (cid:1) is the C*-algebra C ( U dk ) of C -valued continuous functionsover the compact group U dk ; this includes the case where k = ∞ d and C ∗ (cid:0) Z d ∞ d , (cid:1) is the C*-algebra C ( T d ) .On the other hand, quantum and fuzzy tori can be simple. Example 2.14.
Let n ∈ N ∗ . The C*-algebra A n = C ∗ (cid:16) Z n,n ) , ς c n (cid:17) , with c n = (cid:18) n − n (cid:19) , is the C*-algebra of all n × n matrices over C , and thus it is simple.Note that ς c n is not a bicharacter.The fuzzy torus A n is related to the clock and shift matrices, given by Expression(1.2), by universality, and since W e ( n,n ) ,ς cn W e ( n,n ) ,ς cn = exp (cid:18) iπn (cid:19) W e ( n,n ) ,ς cn W e ( n,n ) ,ς cn ,so there exists a *-isomorphism from A n to the C*-algebra generated by the clockand shift matrices C n and S n , sending W e ( n,n ) ,ς cn to C n , and W e ( n,n ) ,ς cn to S n .Another way to observe this relation is that the clock and shift matrices areoperators of the form given in Lemma (2.5), where the -cocycle is chosen as σ : x, y ∈ Z n,n ) exp(2 iπy Θ x ⊺ ) where Θ = (cid:18) n (cid:19) . The latter -cocycle isa bicharacter, and it is cohomologous to ς c n , but it is not normalized; thus weprefer to work with ς c n , without loss of generality. Example 2.15. If θ ∈ R \ Q and if Θ = (cid:18) θ − θ (cid:19) , then C ∗ ( Z d , ς Θ ) is a simpleC*-algebra [14].Quantum tori are prototypes of noncommutative manifolds, whose geometry [8]derives from a natural action of torus T d on quantum tori, called the dual action .The dual action of the Lie group T d induces an action of the Lie algebra R d of T d by *-derivations on quantum tori. Thus, the starting point for the geometricconsiderations in this paper are the following actions of the group U dk on fuzzy andquantum tori. As we will construct a spectral triple on both quantum and fuzzytori, we start with natural actions of U dk on the Hilbert spaces ℓ ( Z dk ) , which thendefines the dual action on our C*-algebras by conjugation. Notation . Let k = ( k (1) , . . . , k ( d )) ∈ N d ∗ . The Pontryagin dual c Z dk of Z dk isidentified with the closed subgroup U dk = n ( z , . . . , z d ) ∈ T d : ∀ j ∈ { , . . . , d } k ( j ) < ∞ = ⇒ z k ( j ) j = 1 o ,of the d -torus T d , where T = { z ∈ C : | z | = 1 } . To make our identificationexplicit, we will use a simple notation for the dual pairing between Z dk and U dk . If z = ( z , . . . , z d ) ∈ T d and n = ( n , . . . , n d ) ∈ Z d , then we set: z n = d Y j =1 z n j j . Now, if z ∈ U dk and n ∈ Z dk , then, for any w, w ′ ∈ Z d such that n = w + k Z d = w ′ + k Z d , we easily observe that z w = z w ′ , and we denote this element of U dk simplyby z n .Every character of Z dk is of the form χ z : n ∈ Z dk z n for a unique z ∈ U dk ,and the map z ∈ U dk χ z ∈ c Z dk is indeed a topological group isomorphism, as caneasily be checked. Notation . Let d ∈ N ∗ and k ∈ N d ∗ . For all z ∈ U dk and ξ ∈ ℓ ( Z dk ) , we define v zk ξ : m ∈ Z dk z − m ξ ( m ) .The map z ∈ U dk v zk thus defined is a strongly continuous action of the group U dk on ℓ ( Z dk ) by unitaries.The dual action of U dk on C ∗ (cid:0) Z dk , σ (cid:1) is defined by conjugation with the action v k . Lemma 2.18.
Let d ∈ N ∗ . Let ( k, σ ) ∈ Ξ d . For all f ∈ ℓ ( Z dk ) , v zk π k,σ ( f ) v zk = π k,σ ( α zk ( f )) where α zk ( f ) : m ∈ Z dk z m f ( z ) .The map α zk thus defined is a *-automorphism of ( ℓ ( Z dk ) , ∗ k,σ , · ∗ ) , and the map z ∈ U dk α zk is a strongly continuous action of U dk on ℓ ( Z dk ) .Proof. This lemma follows from a simple computation. (cid:3)
Notation . The unit of ℓ ( Z dk ) , for any d ∈ N ∗ and for any k ∈ N d ∗ , is denotedby k , or even, , if no confusion may arise. Corollary 2.20. [73]
Let d ∈ N ∗ . Let ( k, σ ) ∈ Ξ d . For all a ∈ C ∗ (cid:0) Z dk , σ (cid:1) , theelement α zk,σ ( a ) = v zk a v zk is in C ∗ (cid:0) Z dk , σ (cid:1) . Thus defined, z ∈ U dk α zk,σ is a strongly continuous action of U dk on C ∗ (cid:0) Z dk , σ (cid:1) by *-automorphisms, called the dual action of U dk on C ∗ (cid:0) Z dk , σ (cid:1) .The action α k,σ thus defined is ergodic, i.e. (cid:8) a ∈ C ∗ (cid:0) Z dk , σ (cid:1) : ∀ z ∈ U dk α zk,σ ( a ) = a (cid:9) = C k . Let now k = ( k (1) , . . . , k ( d )) ∈ N d ∗ , for d ∈ N ∗ . The action of the Lie group U dk on ℓ ( Z dk ) defines, in turn, a natural action by its Lie algebra. The Lie algebra u dk of the Lie group U dk is given by: u dk = (cid:8) ( x , . . . , x d ) ∈ R d : ∀ j ∈ { , . . . , d } k ( j ) < ∞ = ⇒ x j = 0 (cid:9) ,with the exponential function given by: exp U dk : ( x , . . . , x d ) ∈ u dk (exp( ix ) , . . . , exp( ix d )) .Using the actions defined in Theorem (2.20), we have actions of the Lie algebra u dk on ℓ ( Z dk ) , defined as follows. Notation . Let d ∈ N ∗ . Let k = ( k (1) , . . . , k ( d )) ∈ N d ∗ . For all n = ( n , . . . , n d ) ∈ Z dk , and for all X = ( X , . . . , X d ) ∈ u k , we set: h X, n i k = X j ∈{ ,...,d } k ( j )= ∞ X j n j . Notation . Let k ∈ N d ∗ . For all X ∈ u k , let: dom ( ∂ k,X ) = n f ∈ ℓ (cid:0) Z dk (cid:1) : ( h X, n i k f ( n )) n ∈ Z dk ∈ ℓ (cid:0) Z dk (cid:1)o and for all f ∈ dom ( ∂ k,X ) , we set:(2.6) ∂ k,X ( f ) : m ∈ ℓ ( Z dk ) i h X, m i k f ( m ) . Lemma 2.23.
Let k ∈ N d ∗ . The operator i∂ k,X is self-adjoint from dom ( ∂ k,X ) onto ℓ ( Z dk ) such that for any ξ ∈ dom ( ∂ k,X ) , we have ∀ t ∈ R α exp U dk ( tX ) k ξ = exp ( it∂ k,X ) ξ .Proof. It is immediate that i∂ k,X is symmetric. Moreover, the operator R defined,for all ξ ∈ ℓ ( Z dk ) , by Rξ : m ∈ Z dk ( −h X, m i k + i ) − ξ ( m ) ,is bounded (in fact, compact), and maps ℓ ( Z dk ) onto dom ( ∂ k,X ) . Furthermore, all ξ ∈ ℓ ( Z dk ) , an easy computation shows that ( i∂ k,X + i ) Rξ = ξ . Thus, i∂ k,X + i is surjective. Similarly, i∂ k,X − i is surjective as well. Since R i∂ k,X ξ = ξ for all ξ ∈ dom ( ∂ k,X ) as well, and R is bounded, we conclude that i∂ k,X is closed, andthus by [54, Theorem VIII.3], it is self-adjoint.It is an easy computation to check that ∂ k,X is indeed the generator of theunitary action t ∈ R α exp U dk ( tX ) k . (cid:3) The dual action of U dk also defines an action of the Lie algebra u dk of U dk on C ∗ (cid:0) Z dk , σ (cid:1) ; in fact the following observation holds. Definition 2.24.
For all d ∈ N ∗ , for all p ∈ [1 , ∞ ) , and for all k ∈ N d ∗ , an element f ∈ ℓ p ( Z dk ) is finitely supported when the support { m ∈ Z dk : f ( m ) = 0 } of f isfinite. Lemma 2.25.
Let d ∈ N ∗ . Let ( k, σ ) ∈ Ξ d . For all f ∈ ℓ ( Z dk ) such that ( h X, m i k f ( m )) m ∈ Z dk ∈ ℓ ( Z dk ) , we have [ ∂ k,X , π k,σ ( f )] = π k,σ ( ∂ Xk f ) where ∂ Xk f : m ∈ Z dk i h X, m i k f ( m ) .Moreover, for any finitely supported f ∈ ℓ ( Z dk ) , we have (2.7) ∂ Xk f = lim t → α exp U dk ( tX ) k,σ ( f ) − ft .Proof. This is a direct computation. (cid:3)
Remark . Expression (2.7) holds for elements besides finitely supported ones,but this is the degree of generality which we need.
Now, let ( k, σ ) ∈ Ξ d . If k ∈ N d ∗ , then the Lie algebra u dk of U dk is { } ; moregenerally, if k = ( k , . . . , k d ) = ∞ d , and if j ∈ { , . . . , d } is chosen so that k j < ∞ ,then ∂ Xk = 0 and ∂ k,X = 0 for all X ∈ u dk . Thus, our work so far does not provideus with any sort of differential calculus on fuzzy tori. It does however give us somesuggestions on how to proceed to define such a calculus, based upon a discreteversion of Expression (2.7), which we will explain in our next subsection. To thisend, we will use one more tool, which we now describe.Let ( k, σ ) ∈ Ξ d for some d ∈ N ∗ . There is another action of the C*-algebras C ∗ (cid:0) Z dk , σ (cid:1) on ℓ ( Z dk ) , which makes the latter Hilbert space into a bimodule over C ∗ (cid:0) Z dk , σ (cid:1) . Notation . For all d ∈ N ∗ and k ∈ N d ∗ , and for all ξ ∈ ℓ ( Z dk ) , we set J k ξ : m ∈ Z dk ξ ( − m ) .The operator J k is a conjugate linear isometric involution on ℓ ( Z dk ) .Let σ be chosen so that ( k, σ ) ∈ Ξ d . We then note that for all n ∈ Z dk : J k W nk,σ J k ξ : m ∈ Z dk W nk,σ J k ξ ( − m ) = σ ( − m, n ) J k ξ ( − m + n ) = σ ( − n, m ) ξ ( m − n ) .Thus, we easily verify, thanks to Expression (2.1), that ∀ n, m ∈ Z dk W nk,σ J k W mk,σ J k = J k W mk,σ J k W nk,σ .Therefore, we conclude that ∀ f, g ∈ ℓ ( Z dk ) π k,σ ( f ) J k π k,σ ( g ) J k = J k π k,σ ( g ) J k π k,σ ( f ) .Thus, the *-representation J k π k,σ ( · ) J k of ( ℓ ( Z dk ) , ∗ k,σ , · ∗ ) commutes with π k,σ . Bycontinuity, we conclude that J k aJ k commutes with b for all a, b ∈ C ∗ (cid:0) Z dk , σ (cid:1) . Notation . The right action of C ∗ (cid:0) Z dk , σ (cid:1) on ℓ ( Z dk ) will thus be defined by ∀ ξ ∈ ℓ ( Z dk ) ∀ a ∈ C ∗ (cid:0) Z dk , σ (cid:1) ξ · a = J k a ∗ J k ξ .In particular, if ξ ∈ ℓ ( Z dk ) and a ∈ C ∗ (cid:0) Z dk , σ (cid:1) , then we define: [ a, ξ ] = aξ − ξ · a . Remark . By the Young inequality, the twisted convolution product f ∗ k,σ g given by Expression (2.4) is a element of ℓ ( Z dk ) whenever either f ∈ ℓ ( Z dk ) and g ∈ ℓ ( Z dk ) — in which case π k,σ ( f ) g = f ∗ k,σ g — or when f ∈ ℓ ( Z dk ) and g ∈ ℓ ( Z dk ) — in which case, f · π k,σ ( g ) = f ∗ k,σ g . For our purpose, however, thepresentation in this section will prove more helpful. Remark . The *-representation π k,σ , seen as a *-representation of the C*-completion of ( ℓ ( Z dk ) , ∗ k,σ , ∗ ) , is the GNS representation for this completion andfor the tracial state f ∈ ℓ ( Z dk ) f (0) . In this case, the definition of J k is simplygiven as a natural extension of the adjoint operation. The operator J k could be usedto give what is known as a real structure [10] for the spectral triples on quantumand fuzzy tori which we will introduce in this paper, but this will not be our focushere. Spectral Triples on Fuzzy and Quantum Tori.
We now define the familyof spectral triples on the fuzzy and quantum tori, which will be the focus of ourpresent work.Our construction here aims at finding a natural family of spectral triples onfuzzy tori which approximate spectral triples on quantum tori, under the followingnatural conditions, which we will use throughout the remainder of this paper.
Hypothesis 2.31.
Fix d ∈ N ∗ . For each n ∈ N , let us be given ( k n , σ n ) ∈ Ξ d ,with k n = ( k n (1) , . . . , k n ( d )) , such that:(1) lim n →∞ σ n = σ ∞ ,(2) lim n →∞ k n = k ∞ = ∞ d .We write A n = C ∗ (cid:0) Z dk n , σ n (cid:1) for all n ∈ N .To simplify our notations, we assume that, for each j ∈ { , . . . , d } , we also have ( ∀ n ∈ N k n ( j ) ∈ N ) or ( ∀ n ∈ N k n ( j ) = ∞ ) . Remark . If ( x n ) n ∈ N is a sequence in N converges to ∞ , then either it is equalto ∞ after some N , or we can find a subsequence ( x y ( n ) ) n ∈ N such that, for all n ∈ N , we have x n ∈ N ⇐⇒ ∃ m ∈ N y ( m ) = n ; of course ( x y ( n ) ) n ∈ N convergesto ∞ . Therefore, our simplification in Hypothesis (2.31) can be done without lossof generality; we only include it to make our notation easier throughout this paper.As explained in our introduction, we generally need to embed our fuzzy tori inlarger fuzzy tori, in order to construct our spectral triples. What matters is theoverall consistency of this scheme along a sequence of fuzzy tori approximating somequantum torus, so we summarize the needed conditions in the following hypothesis,which we will again use throughout this paper. Hypothesis 2.33.
Assume Hypothesis (2.31). Let d ′ ∈ N ∗ , with d ′ > d . Let f be an order permutation of { , . . . , d ′ } with no fixed point. For each n ∈ N , let ( k ′ n , σ ′ n ) ∈ Ξ d ′ , such that(1) { d + 1 , . . . , d ′ } ⊆ f ( { , . . . , d } ) ,(2) we write k ′ n = ( k n (1) , . . . , k ′ n ( d ′ )) ∈ N d ′ , and the following holds: ∀ j ∈ { , . . . , d } k ′ n ( f ( j )) = k ′ n ( j ) = k n ( j ) ,(3) for all j, s ∈ { , . . . , d } , we have σ ′ n ( e j , e s ) = σ n ( e j , e s ) ,(4) for all j ∈ { , . . . , d } , then σ ′ n ( e j , e f ( j ) ) = ( exp (cid:16) iπk n ( j ) (cid:17) if k n ( j ) < ∞ , otherwise.(5) for all j ∈ { , . . . , d } , and if s ∈ { , . . . , d ′ } \ { j } , then σ ′ n ( e f ( j ) , e s ) = 1 ,(6) if j, s ∈ { d + 1 , . . . , d ′ } , and if ∞ ∈ { k ′ n ( j ) , k ′ n ( s ) } , then σ ′ n ( e j , e s ) = 1 ,(7) if j, s ∈ { d + 1 , . . . , d ′ } , then lim n →∞ σ ′ n ( e j , e s ) = 1 .Let B n = C ∗ (cid:16) Z d ′ k ′ n , σ ′ n (cid:17) . For each j ∈ { , . . . , d ′ } , the canonical unitary gener-ator W e j k ′ n ,σ ′ n of B n is simply denoted by U n,j . By construction, the C*-subalgebra C ∗ ( U n, , . . . , U n,d ) of B n is *-isomorphic to A n , with a canonical *-isomorphismsending W e j k n ,σ n to U n,j for all j ∈ { , . . . , d } . We henceforth identify A n with C ∗ ( U n, , . . . , U n,d ) in B n . We write G n = U d ′ k ′ n and c G n = Z d ′ k ′ n — of course, c G n is the Pontryagin dual of G n . The dual action α k ′ n ,σ ′ n of G n on B n , defined in Corollary (2.20), is simplydenoted by α n . For each z ∈ G n , we also simply denote by v zn the unitary v zk ′ n acting on ℓ ( c G n ) , as defined in Notation (2.17).Moreover, to further ease our notation, if z ∈ U dk n , then we identify z with theelement z, , . . . , | {z } d ′ − d times in c G n without further mention.For all j ∈ { , . . . , d ′ } , we also write ∂ n,j = ∂ k ′ n ,e j and ∂ jn = ∂ e j k ′ n .Last, let J n = J k ′ n be the conjugate linear isometric involution mapping ξ ∈ ℓ ( c G n ) to m ∈ c G n ξ ( − m ) .We make a few easy observations. First, note that since { d + 1 , . . . , d ′ } ⊆ f ( { , . . . , d } ) , and since f is a permutation, we must have d ′ d . Moreover,if j ∈ { d + 1 , . . . , d ′ } , then j = f ( s ) for s ∈ { , . . . , d } , so that f ( j ) = s ∈ { , . . . , d } .Therefore, if j = f ( s ) and j > d then s d — a fact we use in Condition (4) of Hy-pothesis (2.33). Moreover, we then see that { , . . . , d ′ } = { , . . . , d } ∪ f ( { , . . . , d } ) ,and thus, Condition (2) in Hypothesis (2.33) completely describes k ′ n .Second, we note that σ ′ n ( e j , e s ) , for j, s ∈ { d + 1 , . . . , d ′ } , can be any adequatevalue, if both k ′ n ( j ) and k ′ n ( s ) are finite (though it must be as soon as either k ′ n ( j ) or k ′ n ( s ) is infinite), though eventually, it must converge to as n goes to infinity.We place no further restriction on these values.Third, it is possible that the sequences ( σ n ) n ∈ N already meets all the givenrequirements to be chosen as the sequence ( σ ′ n ) n ∈ N , in which case d ′ = d and ( k ′ n ) n ∈ N ) = ( k n ) n ∈ N are acceptable choices. These will however be rather specialcircumstances. For instance, Condition (5) of Hypothesis (2.33) imposes that, forall j ∈ { , . . . , d } , the unitary U n,f ( j ) commutes with all other unitaries U n,l ( l ∈{ , . . . , d ′ } \ { d } ) except U n,j . Condition (4) also imposes a clear condition on σ n if we wish to choose σ ′ n = σ n for all n ∈ N . This said, such situations do occur —for instance, when working with the clock and shift matrices.Generically, we will prove that we can always can meet Hypothesis (2.33), if wechoose d ′ = 2 d . To understand this, it is helpful to briefly discuss the content ofHypothesis (2.33), using its notation. Fix n ∈ N . Notation . Let n ∈ N . Using the notation of Hypothesis (2.33), for all j ∈{ , . . . , d ′ } , we set ζ n,j = σ ′ n ( e j , e f ( j ) ) We then define: z n,j = ζ n,j e j .By construction, for all j, s ∈ { , . . . , d } , we have U n,j U n,s = σ n ( e j , e s ) U n,s U n,j so that A n is indeed *-isomorphic to C ∗ ( U n, , . . . , U n,d ) .Moreover, for all j, s ∈ { , . . . , d } , with j = s and f ( j ) > j , we also have U n,f ( j ) U n,j = ζ n,j U n,j U n,f ( j ) and U n,f ( j ) U n,s = U n,s U n,f ( j ) , so(2.8) ∀ j, s ∈ { , . . . , d } ( f ( j ) > j ) = ⇒ U n,f ( j ) U n,s U ∗ n,f ( j ) = α z n,j n ( U n,s ) .Therefore, if j ∈ { , . . . , d } and f ( j ) > j , for all a ∈ A n , we conclude that U n,f ( j ) aU ∗ n,f ( j ) = α z n,j n ( a ) .A similar computation shows that U n,f ( j ) aU ∗ n,f ( j ) = α z n,j n ( a ) if j ∈ { , . . . , d } and f ( j ) < d .We also note that, by construction, lim n →∞ ( k ′ n , σ ′ n ) = ( ∞ ( d ′ ) , σ ′∞ ) .In particular, if d ′ > d , then the canonical unitaries U ∞ ,d +1 , . . . , U ∞ ,d ′ are centralin B ∞ .We establish that we can always meet Hypothesis (2.33), given Hypothesis (2.31).Expression (2.8) suggests that we use the C*-crossed-product construction. Lemma 2.35.
If we assume Hypothesis (2.31), then there exist an order per-mutation of { , . . . , d } , and for all n ∈ N , an element ( k ′ n , σ ′ n ) ∈ Ξ d , such thatHypothesis (2.33) holds as well.Proof. We will work with d ′ = 2 d . Let f : { , . . . , d } → { , . . . , d } be definedas the order permutation such that f ( j ) = d + j for all j ∈ { , . . . , d } . Thepermutation f has no fixed point by construction.Let n ∈ N . For all j ∈ { , . . . , d } , we set k ′ n ( d + j ) = k ′ n ( j ) = k n ( j ) . Now, if z = ( z , . . . , z d ) ∈ Z dk n , we define z ′ = ( z ′ , . . . , z ′ d ) ∈ Z dk n by setting ∀ j ∈ { , . . . , d } z ′ j = ( z j if k n ( j ) < ∞ , otherwise.With this notation, we then define β zn = α z ′ n . It is easy to check that β n is a stronglycontinuous action of Z dk n on A n .Let B n = A n ⋊ β c G n . Thus, B n is the universal C*-algebra generated by d unitaries U n, ,. . . , U n, d , subject to the relations: ∀ j, s ∈ { , . . . , d } [ U n,f ( j ) , U n,f ( s ) ] = 0 ,while ∀ a ∈ A n U n,f ( j ) aU ∗ n,f ( j ) = β z n,j n ( a ) so that ∀ j = s ∈ { , . . . , d } U n,f ( j ) U n,j = ζ n,j U n,j U n,f ( j ) and U n,f ( j ) U n,s = U n,s U n,f ( j ) Moreover, ∀ j, s ∈ { , . . . , d } U n,j U n,s = σ n ( e j , e s ) U n,s U n,j .Universality of the C*-crossed-product and the C*-algebra A n ensures that B n isof the form C ∗ (cid:16) Z dk ′ n , σ ′ (cid:17) with σ ′ n = ς Ω n , where Ω n = (cid:18) Θ n r n − r n (cid:19) with ς Θ n = σ n , and r n = k n (1) k n ( d ) .We can easily check that we now have met all the requirements of (2.33) (notethat σ ′ n ( e j , e s ) = 1 for all n ∈ N and j, s ∈ { d + 1 , . . . , d } in this construction). (cid:3) Lemma (2.35) gives a general mean to implement Hypothesis (2.33), and indeed,it is the basic scheme we have in mind for our construction. However, we make twoquick observations. First, we technically could allow the unitaries U n,d +1 ,. . . , U n, d of finite order not to commute — though we do prescribe that the phase factorgiven by their multiplicative commutators must converge to as n → ∞ . Second,there are situations where we can work within the given fuzzy tori, without invokingthe scheme of Lemma (2.35). This is, in fact, a very commonly used example inphysics [27, 68, 3, 11]. Example 2.36.
We give an example where d = d ′ (= 2) . Let C n and S n bethe clock and shift matrices given by Expression (1.2). Note that C ∗ ( C n , S n ) = C ∗ (cid:16) Z n,n ) , ς r n (cid:17) with r n = (cid:18) n − n (cid:19) , as noted in Example (2.37). The C*-algebra C ∗ ( C n , S n ) is thus *-isomorphic to C ∗ (cid:18) Z Z n,n ) , σ n (cid:19) with σ n = ς r n − r ⊺ n , since ς r n and σ n are cohomologous. We thus have(2.9) U n, U n, = exp (cid:18) iπn (cid:19) U n, U n, .We let f : { , } 7→ { , } be given by f (1) = 2 and f (2) = 1 . Let n ∈ N . ByExpression (2.9), we have U n, U n, U ∗ n, = exp (cid:0) iπn (cid:1) U n, = α z n, n ( U n, ) .On the other hand, since C ( T ) = C ∗ ( Z , as C ∗ ( U ∞ , , U ∞ , ) with U ∞ , and U ∞ , two commuting unitaries, we see once again that, for all a ∈ C ( T ) , we have U ∞ , aU ∗∞ , = U ∞ , aU ∗∞ , = a .Thus Hypothesis (2.33) is met with f , and with d ′ = d = 2 , k ′ n = k n , σ ′ n = σ n for all n ∈ N .To illustrate our Hypothesis (2.33) further, note that there is another, obvious,approximation of the classical torus by fuzzy tori, and this time, we do need to usesome technique like the one given in Lemma (2.35) to meet Hypothesis (2.33). Example 2.37.
Let n ∈ N . The fuzzy torus C ( U n,n ) ) is Abelian, and thusall commutators are zero. In this situation, we employ auxiliary unitaries in or-der to meet Hypothesis (2.33). We embed C ( U n,n ) ) in the C*-crossed product C ( U n,n ) ) ⋊ α n Z n,n ) — which is *-isomorphic to the C*-algebra of n × n matricesover C . Note that α n , here, is just the action by translation.The C*-algebra C ( U n,n ) ) ⋊ α n Z n,n ) is the universal C*-algebra, generated byfour unitaries U n, ,. . . , U n, such that [ U n, , U n, ] = [ U n, , U n, ] = 0 , U n, U n, U ∗ n, = ζ n, U n, and U n, U n, U ∗ n, = ζ n, U n, .Thus, defining f : { , , , } → { , , , } by f (1) = 3 , f (2) = 4 , (and thus f (3) = 1 and f (4) = 2 ), we see that B n = C ( U n,n ) ) ⋊ α n Z n,n ) satisfy the needsof Hypothesis (2.33).We can generally mix and match the technique of Lemma (2.35) and Example(1.2), if the quantum torus A ∞ has a non-trivial center, giving us many possibleexamples of the convergence found in this work. When the limit quantum torusis simple, then Lemma (2.35) should be used, maybe with the slight modificationregarding the commutation between the auxiliary canonical unitaries of finite order.We now define the domains of the Dirac operators for our sequence of spectraltriples, based upon Hypothesis (2.33). Hypothesis 2.38.
Assume Hypothesis (2.33). We also use Notation (2.34). We fixa faithful *-representation c of the Clifford algebra Cl ( C d + d ′ ) on a finite dimensional Hilbert space C .The Hilbert space ℓ ( c G n ) is denoted by H n . The *-representation π k ′ n ,σ ′ n on H n is simply denoted by π n .We write J n = H n ⊗ C .We identify J n with the Hilbert space ℓ ( c G n , C ) of functions from c G n to C suchthat X m ∈ c G n k ξ ( m ) k C < ∞ .Writing C for the identity on C , we then define for all bounded linear operator a on H n (including a ∈ B n ): a ◦ = a ⊗ C .For all n ∈ N , and for all m = ( m , . . . , m d ′ ) , m ′ = ( m ′ , . . . , m ′ d ′ ) ∈ c G n , we set h m, m ′ i n = h m, m ′ i k ′ n = X j ∈{ ,...,d ′ } k ′ n ( j )= ∞ m j m ′ j .We define the subspace dom (cid:0) /D n (cid:1) = ( ξ ∈ J n : (cid:18)q h m, m i n ξ ( m ) (cid:19) m ∈ c G n ∈ J n ) .Hypothesis (2.38) defines the domain on which the Dirac operators of our spectraltriples will be defined. We establish a few properties which will be helpful towardthis goal. Lemma 2.39.
Assume Hypothesis (2.38). If n ∈ N , then, for all linear map t : C → C , and if ξ ∈ dom (cid:0) /D n (cid:1) , then ( ∂ n,j ⊗ t ) ξ ∈ J n .Moreover, dom (cid:0) /D n (cid:1) contains all finitely supported functions from b G n to C . There-fore, dom (cid:0) /D n (cid:1) is dense in J n .Proof. The proof is an immediate computation. (cid:3)
Lemma 2.40.
Assume Hypothesis (2.38). For all n ∈ N , U ◦ n,j dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) , ( U ∗ n,j ) ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) .and ( J n ⊗ C )dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) .Proof. Let n ∈ N . Fix y ∈ c G n and write W y for the unitary W yk ′ n ,σ ′ n . Let ξ ∈ dom (cid:0) /D n (cid:1) . We compute X m ∈ b G (1 + h m, m i n ) (cid:13)(cid:13) ( W ◦ y ξ )( m ) (cid:13)(cid:13) C = X m ∈ b G (1 + h m, m i n ) k σ ′ n ( m, y ) ξ ( m + y ) k C = X m ∈ c G n (1 + h m, m i n )(1 + h m + y, m + y i n ) (1 + h m + y, m + y i n ) k ξ ( m + y ) k C = X m ∈ c G n (1 + h m + y, m + y i n )(1 + h m, m i n ) (1 + h m, m i n ) k ξ ( m ) k C .Since ξ ∈ dom (cid:0) /D n (cid:1) , we know that X m ∈ c G n (1 + h m, m i n ) k ξ ( m ) k C < ∞ .On the other hand, there exists N ∈ N such that, if h m, m i n > N , then (1+ h m + y,m + y i n )(1+ h m,m i n ) . As { m ∈ c G n : h m, m i n < N } is finite, we conclude that X m ∈ b G (1 + h m, m i n ) (cid:13)(cid:13) ( W ◦ y ξ )( m ) (cid:13)(cid:13) C < ∞ .Thus W ◦ y dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) , as claimed.Fix now some orthonormal basis f , . . . , f p of C (where p = dim C ). If x = P ps =1 x s f s , for x , . . . , x p ∈ C , then we write x = P ps =1 x s f s . An immediatecomputation shows that k x k C = k x k C .Now, we also note that if J ◦ n = J n ⊗ C , and if ξ ∈ dom (cid:0) /D n (cid:1) , then, X m ∈ c G n (1 + h m, m i n ) k ( J ◦ n ξ )( m ) k C = X m ∈ c G n (1 + h m, m i n ) (cid:13)(cid:13) ξ ( − m ) (cid:13)(cid:13) C = X m ∈ c G n (1 + h− m, − m i n ) k ξ ( − m ) k C < ∞ .This concludes our proof. (cid:3) We now are ready to define our spectral triples for fuzzy tori and quantumtori, by specifying the Dirac operators for these triples over the domains defined inHypothesis (2.38). Hypothesis 2.41.
Assume Hypothesis (2.38). We also will use Notation (2.34).For all n ∈ N , we then define the following objects.The identity of H n is denoted by n for the identity of H n .For all j ∈ { , . . . , d } , we now define the self-adjoint bounded operators X n,j = ℜ ( U n,f ( j ) ) = 12 (cid:16) U n,f ( j ) + U ∗ n,f ( j ) (cid:17) and Y n,j = fsgn( j ) ℑ ( U n,f ( j ) ) = fsgn( j )2 i (cid:16) U n,f ( j ) − U ∗ n,f ( j ) (cid:17) ,where fsgn( j ) = ( if f ( j ) > j , − if f ( j ) < j .We now define the following operators. Let j ∈ { , . . . , d } .(1) if j ∈ { , . . . , d } and k n ( j ) = ∞ , then Γ n,j = X n,j ∂ n,j and Γ n,d + j = Y n,j ∂ n,j ;(2) if j ∈ { , . . . , d } and k n ( j ) < ∞ , then Γ n,j = k n ( j )2 π [ Y n,j , · ] and Γ n,d + j = k n ( j )2 π [ X n,j , · ] ;(3) if j ∈ { d + 1 , . . . , d ′ } and k ′ n ( j ) = ∞ , then Γ n,d + j = ∂ n,j ;(4) if j ∈ { d + 1 , . . . , d ′ } and k ′ n ( j ) < ∞ , then Γ n,d + j = k ′ n ( j )2 π (cid:0) ℑ (cid:0) v z n,j n (cid:1) − n (cid:1) .We now define the following operator from dom (cid:0) /D n (cid:1) to J n : /D n = d + d ′ X j =1 Γ n,j ⊗ c ( γ j ) ,which is well-defined using Lemma (2.39).Thus, our candidate for an interesting, convergent sequence of metric spectraltriples on fuzzy and quantum tori is given by the sequence of triples ( A n , J n , /D n ) ,given by Hypothesis (2.41). Our first task is to prove that, indeed, these triples arespectral triples. The main concern is to prove that /D n is a self-adjoint operatorwith a compact resolvent, for all n ∈ N . While this is obvious when k n ∈ N d ∗ , thesituation is more involved when k n may have infinite component.2.3. Self-adjointness of the proposed Dirac operators.
In order to prove thatthe operator /D n is self-adjoint, we will use the following, standard, characterization[54, Sec. VIII.2]: a densely defined, symmetric operator T on a Hilbert space T isessentially self-adjoint if, and only if the ranges of both T + i and T − i are dense in T . Now, suppose that, in addition, the range of T + i is T . Let T be the closureof T . Let η ∈ dom (cid:0) T (cid:1) . Since T + i is surjective, there exists ξ ∈ dom ( T ) such that ( T + i ) ξ = ( T + i ) η . Of course, since T extends T , we also have ( T + i ) ξ = ( T + i ) ξ .Therefore, ( T + i ) ξ = ( T + i ) η , and since T is self-adjoint, T + i is injective, so ξ = η , i.e. η ∈ dom ( T ) . Thus dom (cid:0) T (cid:1) ⊆ dom ( T ) . Since dom ( T ) ⊆ dom (cid:0) T (cid:1) , weconclude dom ( T ) = dom (cid:0) T (cid:1) . Thus T = T , and thus T is self-adjoint.Since any nonzero, real multiple of a self-adjoint operator is again self-adjoint,we, in fact, will use the following equivalence. Lemma 2.42. for a densely-defined symmetric operator T on a Hilbert space T : • T is self-adjoint, • there exists z ∈ i R \ { } such that the image by T ± z of dom ( T ) is T . This strategy is implemented in the following lemmas. First, we prove that /D n is symmetric. Lemma 2.43.
Assume Hypothesis (2.41). For all n ∈ N , the operator /D n is adensely symmetric operator on dom (cid:0) /D n (cid:1) .Proof. By Lemma (2.39), the operator /D n is well-defined on dom (cid:0) /D n (cid:1) , which isindeed dense in J n .By Lemma (2.23), for all j ∈ { , . . . , d ′ } with k ′ n ( j ) = ∞ , we note that ∀ ξ, η ∈ dom ( ∂ n,j ) h η, ∂ n,j ξ i H n = h− ∂ n,j η, ξ i H n .On the other hand, if j ∈ { , . . . , d } , then X ∗ n,j = X n,j , and, if k n ( j ) = ∞ , theoperator X n,j commutes with ∂ n,j by Hypothesis (2.33), since f has no fixed point.Thus h η, X n,j ∂ n,j ξ i H n = h− ∂ n,j X n,j η, ξ i H n = h− X n,j ∂ n,j η, ξ i H n .The same result holds for Y n,j in place of X n,j , for all j ∈ { , . . . , d } with k n ( j ) = ∞ .Let now j ∈ { , . . . , d } with k n ( j ) < ∞ . We compute (cid:0) k ′ n ( j )2 π [ X n,j , · ] (cid:1) ∗ = − k n ( j )2 π ( π ( X n,j ) ∗ − ( Jπ ( X n,j ) J ) ∗ )= − k n ( j )2 π ( π n,j ( X n,j ) − Jπ ( X n,j ) J ) = − k n ( j )2 π [ X n,j , · ] .The same computation holds for Y n,j in place of X n,j .Similarly, (cid:16) k n ( j )2 π ( ℑ ( v z n,j j ) − n ) (cid:17) ∗ = − k n ( j )2 π ( ℑ ( v z n,j j ) − n ) for all j ∈ { d +1 , . . . , d ′ } with k ′ n ( j ) < ∞ .Therefore, for all j ∈ { , . . . , d + d ′ } , we have seen that Γ ∗ n,j = − Γ n,j on dom (cid:0) /D n (cid:1) , and thus, for all j ∈ { , . . . , d + d ′ } , since c ( γ j ) ∗ = − c ( γ j ) , we con-clude that ∀ η, ξ ∈ dom (cid:0) /D n (cid:1) h η, (Γ n,j ⊗ c ( γ j )) ξ i J n = h (Γ n,j ⊗ c ( γ j )) η, ξ i J n .Therefore, if ξ, η ∈ dom (cid:0) /D n (cid:1) , then we compute (cid:10) η, /D n ξ (cid:11) J n = d + d ′ X j =1 h η, Γ n,j ⊗ c ( γ j ) ξ i J n = d + d ′ X j =1 h (Γ n,j ⊗ c ( γ j )) η, ξ i J n = (cid:10) /D n η, ξ (cid:11) J n .Thus, indeed, /D n is a symmetric with the dense domain dom (cid:0) /D n (cid:1) . (cid:3) We now introduce a positive compact operator, which will be used to computethe inverse of M + /D n , for a constant M whose choice will justified in a few lemmas. Lemma 2.44.
We use the notation of Hypothesis (2.41). Let M = 5 d . Let n ∈ N .We define the multiplication operator K n on H n by setting, for all ξ ∈ H n K n ξ : m ∈ c G n M + h m, m i n ξ ( m ) .The operator K ◦ n is a positive, compact operator on J n , with ran (cid:0) √ K n (cid:1) H n ⊆ T j ∈{ ,...,d ′ } k n ( j )= ∞ dom ( ∂ n,j ) . Moreover (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)p K ◦ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n = 1 M .Moreover, for all linear map t : C → C , the following holds: p K ◦ n J n ⊆ dom (cid:0) /D n (cid:1) and ( ∂ n,j ⊗ t ) K ◦ n J n ⊆ dom (cid:0) /D n (cid:1) ,so that /D n K ◦ n J n ⊆ dom (cid:0) /D n (cid:1) .Therefore, K ◦ n J n ⊆ dom (cid:16) /D n (cid:17) , and K ◦ n J n ⊆ dom (cid:0) ( ∂ n,j (cid:1) ⊗ alg C ,for all j ∈ { , . . . , d ′ } with k ′ n ( j ) = ∞ , and for all linear map t : C → C .Furthermore, for all j ∈ { , . . . , d ′ } with k ′ n ( j ) = ∞ , and for all linear map t : C → C , the operator ( ∂ n,j ⊗ t ) p K ◦ n is a bounded operator on J n , with normat most ||| t ||| C .Proof. An obvious computation shows that K n is self-adjoint, and that ( δ m ) m ∈ Z dk isa Hilbert basis of eigenvectors of K n ; the spectrum of K n is n M + h m,m i n : m ∈ Z dk o .Thus K n is positive and compact. The norm of K n is, in particular, its spectralradius is M , and, similarly, the norm of √ K n is M .Since C is finite dimensional, K ◦ n is also positive, compact, √ K n ◦ = p K ◦ n , and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)p K ◦ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n = M .Fix n ∈ N . Let ξ ∈ J n . We compute: X m ∈ c G n (1 + h m, m i n ) (cid:13)(cid:13)(cid:13) ( p K ◦ n ξ )( m ) (cid:13)(cid:13)(cid:13) C = X m ∈ c G n (1 + h m, m i n ) 1( M + h m, m i n ) k ξ ( m ) k C X m ∈ c G n k ξ ( m ) k C = k ξ k J n < ∞ .Thus, p K ◦ n ξ ∈ dom (cid:0) /D n (cid:1) . Therefore, p K ◦ n J n ⊆ dom (cid:0) /D n (cid:1) .Now, let ξ ∈ J n . Let j ∈ { , . . . , d ′ } such that k ′ n ( j ) = ∞ , and let t : C → C be linear (hence bounded, since C is finite dimensional by Hypothesis (2.38)). Aneasy computation shows that if ξ ∈ dom (cid:0) /D n (cid:1) , then ( ∂ n,j ⊗ t ) p K ◦ n ξ = p K ◦ n ( ∂ n,j ⊗ t ) ξ . Since p K ◦ n J n ⊆ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) , we conclude that, for all ξ ∈ J n , thefollowing holds for all linear map t : C → C : ( ∂ n,j ⊗ t ) K ◦ n ξ = ( ∂ n,j ⊗ t ) p K ◦ n ( p K ◦ n ξ )= p K ◦ n ( ∂ n,j ⊗ t ) p K ◦ n ξ since p K ◦ n ξ ∈ dom (cid:0) /D n (cid:1) , ⊆ p K ◦ n J n ⊆ dom (cid:0) /D n (cid:1) ,so ( ∂ n,j ⊗ t ) K ◦ n J n ⊆ dom (cid:0) /D n (cid:1) . Of course, this could also be checked with asimple, direct computation.Since X ◦ n,j , Y ◦ n,j , [ X n,j , · ] ◦ and [ Y n,j , · ] ◦ all map dom (cid:0) /D n (cid:1) to itself as well byLemma (2.40), and since dom (cid:0) /D n (cid:1) is an algebraic subspace, we conclude that /D n K ◦ n J n ⊆ dom (cid:0) /D n (cid:1) .Moreover, we see that if t : C → C is linear, and if ξ ∈ J n , then (cid:13)(cid:13)(cid:13) ( ∂ n,j ⊗ t ) p K ◦ n ξ (cid:13)(cid:13)(cid:13) J n = X m =( m ,...,m d ′ ) ∈ c G n m j M + h m, m i n k tξ ( m ) k C ||| t ||| C X m ∈ c G n k ξ ( m ) k C ||| t ||| C k ξ k C and thus, ( ∂ n,j ⊗ t ) p K ◦ n is bounded on J n , with norm at most ||| t ||| C .Therefore, /D n p K ◦ n , as a linear combination of bounded operators on J n , isbounded on J n as well. (cid:3) Remark . An alternative proof that, for all n ∈ N and j ∈ { , . . . , d ′ } with k n ( j ) = ∞ , the operator ( ∂ n,j ⊗ t ) p K ◦ n is bounded would be to note that ∂ n,j isa closed operator (as a skew-adjoint operator), and thus ( ∂ n,j ⊗ t ) p K ◦ n is a closedoperator as well, and thus by the closed graph theorem, since ( ∂ n,j ⊗ t ) p K ◦ n isdefined on the entire Hilbert space J n , it is continuous. It is helpful, however, tohave an upper bound on the norm of ( ∂ n,j ⊗ t ) p K ◦ n .We now compute commutation relations. Lemma 2.46.
Assume Hypothesis (2.41). We use the notation of Lemma (2.44).If j, s ∈ { , . . . , d + d ′ } , and if ∞ ∈ { k ′ n ( j ) , k ′ n ( s ) } , then for all linear map t : C → C , the following estimate holds: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ([Γ n,j , Γ n,s ] ⊗ t ) p K ◦ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n ||| t ||| C if j = f ( s ) and j, s d , ||| t ||| C if j = f ( s ) and j > d , otherwise.Moreover, if j ∈ { , . . . , d } and s ∈ { d +1 , . . . , d + d ′ } , and if ∞ / ∈ { k ′ n ( j ) , k ′ n ( s ) } ,then [Γ n,j , Γ n,s ] = [Γ n,d + j , Γ n,s ] = 0 .Proof. For all j ∈ { , . . . , d } , we let Z n,j ∈ { X n,j , Y n,j } . For all j ∈ { d + 1 , . . . , d ′ } ,we let Z n,j = 1 .Let now j, s ∈ { , . . . , d ′ } . Assume first that ∞ = k ′ n ( j ) = k ′ n ( s ) . We compute,over the range of K n , Z n,j ∂ n,j Z n,s ∂ n,s = Z n,j ( ∂ n,j ( Z n,s ) ∂ n,s + Z n,s ∂ n,j ∂ n,s ) = ( Z n,j Z n,s ( ∂ n,s + ∂ n,j ∂ n,s ) if s = f ( j ) and s d , Z n,j Z n,s ∂ n,j ∂ n,s otherwise.Since ∂ n,j and ∂ n,s always commute, and since Z n,j and Z n,s commute by Hy-pothesis (2.33) — since k ′ n ( j ) = k ′ n ( s ) = ∞ — we conclude that [ Z n,j ∂ n,j , Z n,s ∂ n,s ] = Z n,j Z n,s ( ∂ n,j − ∂ n,s ) if j = f ( s ) and j, s d , Z n,j ∂ n,j if j = f ( s ) and j > d , otherwise.Therefore, by Lemma (2.44), we conclude that, for linear map t : C → C , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ([ Z n,j ∂ n,j , Z n,s ∂ n,s ] ⊗ t ) p K ◦ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n ||| t ||| C ||| Z n,j Z n,s ||| (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( ∂ n,j − ∂ n,s ) p K ◦ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) if j = f ( s ) and j, s d , ||| t ||| C ||| Z n,j ||| (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ n,j p K ◦ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) if j = f ( s ) and j > d , otherwise, ||| t ||| C if j = f ( s ) and j, s d , ||| t ||| C if j = f ( s ) and j > d , otherwise.On the other hand, a direct computation shows that, if k ′ n ( j ) = ∞ and k ′ n ( s ) < ∞ , and if s ∈ { , . . . , d } , then Z n,j ∂ n,j [ Z n,s , · ] n = [ Z n,s , Z n,j ∂ n,j · ] ,so [ Z n,j ∂ n,j , [ Z n,s , · ]] = 0 .The last statement of our lemma is easily shown since v z n,j n commutes with A n whenever j ∈ { d + 1 , . . . , d ′ } . This concludes the proof of our lemma. (cid:3) We are now ready for the core result of this section.
Lemma 2.47.
Assume Hypothesis (2.41). If n ∈ N , then /D n is a self-adjointoperator, defined on dom (cid:0) /D n (cid:1) , with compact resolvent.Proof. All the computations in this lemma are done over dom (cid:0) /D n (cid:1) . A directcomputation shows that /D n = d + d ′ X j =1 Γ n,j ⊗ C + X j ,Therefore, ||| Z n K ◦ n ||| J n d M = 12 d d < .We now use the notation of Lemma (2.44). A direct computation shows that ∀ ξ ∈ J n ( M + ∆ n ) K ◦ n ξ = ξ and ∀ ξ ∈ ran ( K ◦ n ) K ◦ n ( M + ∆ n ) ξ = ξ .Therefore, for all ξ ∈ J n , ( M + /D n ) K ◦ n ξ = ( M + ∆ n ) K ◦ n ξ + ( F n + Z n ) K ◦ n ξ = ξ + ( F n + Z n ) K ◦ n ξ = (1 J n + ( F n + Z n ) K ◦ n ) ξ .Now, by Lemma (2.46), we observe that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n p K ◦ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n X j ∈{ ,...,d } k n ( j )= ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [Γ n,j , Γ n,f ( j ) ] ⊗ c ( γ j γ f ( j ) ) p K ◦ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d .Since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)p K ◦ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n = M = d , we conclude ||| F n K ◦ n ||| J n d d < .Altogether, we just have shown that ||| ( F n + Z n ) K ◦ n ||| J n < .Consequently, J n + ( F n + Z n ) K ◦ n is invertible; let R n = (1 J n + ( F n + Z n ) K ◦ n ) − (note that R n is bounded).We then observe that, for all ξ ∈ J n , the vector K ◦ n R n ξ is in dom (cid:16) /D n (cid:17) byLemma (2.44). Moreover, we compute: ∀ ξ ∈ J n ( M + /D n ) K ◦ n R n ξ = (1 J n + ( F n + Z n ) K ◦ n ) R n ξ = ξ .Thus M + /D n is surjective. Since ( /D n + iM )( /D n − iM ) = ( /D n − iM )( /D n + iM ) = M + /D n on ran ( K ◦ n ) , we conclude that both /D n + iM and /D n − iM are surjective. Thus, /D n , as a densely defined symmetric operator (by Lemma (2.43)), such that /D n ± iM are surjective, is a self-adjoint operator from dom (cid:0) /D n (cid:1) , using Lemma(2.42).Moreover, since K ◦ n is compact and R n is bounded, the operator R n K ◦ n is com-pact. Now, since /D n is self-adjoint, the operator M + /D n is invertible, with inverse R n K ◦ n . Therefore, M + /D n has a compact inverse. Since, for any bounded oper-ator a , the operator aa ∗ is compact if, and only, if a is compact, we conclude that ( /D n + iM ) − is compact. Thus, /D n has a compact resolvent. (cid:3) We thus conclude this section by summarizing our work in the following theorem.
Theorem 2.48.
Assume Hypothesis (2.41). For all n ∈ N , the triple ( A n , J n , /D n ) is a spectral triple, where A n acts on J n via the *-representation a ∈ A n a ◦ ,such that, if a ∈ A n , and if a has finite support, then a ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) and [ /D n , a ◦ ] = (id ⊗ c ) ∇ n a where id is the identity function on B n , and (2.10) ∇ n a = X j ∈{ ,...,d } k n ( j )= ∞ (cid:0) X n,j ∂ jn ( a ) ⊗ γ j + Y n,j ∂ jn ( a ) ⊗ γ d + j (cid:1) + X j ∈{ ,...,d } k n ( j ) < ∞ k n ( j )2 iπ ([ X n,j , a ] ⊗ γ j + [ Y n,j , a ] ⊗ γ d + j ) .Proof. By Lemma (2.47), the operator /D n is self-adjoint, with a compact resolvent.Moreover, by Lemma (2.40), we have, for all j ∈ { , . . . , d } : U ◦ n,j dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) and ( U ∗ n,j ) ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) .Thus, for any finitely supported element a in A n , i.e., any linear combinations ofpowers of U n, ,. . . , U n,d and their adjoints, a ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) .In particular, since the finitely supported elements in A n are dense in A n , wehave shown that (cid:8) a ∈ A n : a ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) and [ /D n , a ◦ ] is bounded (cid:9) is dense in A n .Moreover, if a ∈ A n is finitely supported, then, noting that • [ ∂ n,j , a ] = ∂ jn a for all j ∈ { , . . . , d ′ } with k ′ n ( j ) = ∞ ; in particular ∂ jn a = 0 for j ∈ { d + 1 , . . . , d ′ } , • [ v z n,j n , a ] = 0 if j ∈ { d + 1 , . . . , d ′ } and k ′ n ( j ) < ∞ , • [[ X n,j , · ] , a ] = [ X n,j , a ] and [[ Y n,j , · ] , a ] = [ Y n,j , a ] for all j ∈ { , . . . , d } with k n ( j ) < ∞ , • [ X n,j , a ] = 0 and [ Y n,j , a ] = 0 for all j ∈ { , . . . , d } with k n ( j ) = ∞ ,we easily conclude that [ /D n , a ◦ ] is indeed given by the image by id ⊗ c of Expression(2.10). (cid:3) Metric Properties of the Spectral Triples
We have constructed families of spectral triples on fuzzy and quantum tori. Ourgoal is to prove that, under natural assumptions, these families are continuous forthe spectral propinquity. As a first, key step, we prove that our family of spectraltriples induce continuous families of quantum compact metric spaces.
Hypothesis 3.1.
Assume Hypothesis (2.41). For each n ∈ N , we define dom ( L n ) as the space n a ∈ sa ( A n ) : a ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ /D n , a ◦ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n < ∞ o ,and we define the seminorm L n on dom ( L n ) by ∀ a ∈ dom (cid:0) /D n (cid:1) L n ( a ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:2) /D n , a ◦ (cid:3)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n .We begin by recording properties which follow in general from constructing semi-norms from spectral triples, as seen in [42]. Lemma 3.2.
Assume Hypothesis (3.1). For all n ∈ N :(1) the domain dom ( L n ) of L n is dense in sa ( A n ) ,(2) the set { a ∈ dom ( L n ) : L n ( a ) } is closed in A n ,(3) for all a, b ∈ dom ( L n ) , ab + ba , ab − ba i ∈ dom ( L n ) ,and max (cid:26) L n (cid:18) ab + ba (cid:19) , L n (cid:18) ab − ba i (cid:19)(cid:27) k a k A n L n ( b ) + L n ( a ) k b k A n .Proof. By Theorem (2.48), the domain dom ( L n ) contains all finitely supportedelements in A n (i.e. elements of the form π n ( f ) for f finitely supported in ℓ ( c G n ) ),and thus dom ( L n ) is dense in sa ( A n ) .We refer to [61, 42] for the proof of that L n is lower semi-continuous on A n , since /D n is self-adjoint.Let a, b ∈ dom ( L n ) . First, note that b ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) , and thus ( ab ) ◦ dom (cid:0) /D n (cid:1) = a ◦ b ◦ dom (cid:0) /D n (cid:1) ⊆ a ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) .We then easily compute: (cid:2) /D n , ( ab ) ◦ (cid:3) = a ◦ (cid:2) /D n , b ◦ (cid:3) + (cid:2) /D n , a ◦ (cid:3) b ◦ ,and all the operators on the right hand side in the previous equation are boundedby assumption on dom ( L n ) . Thus (cid:2) /D n , ( ab ) ◦ (cid:3) is bounded, and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:2) /D n , ( ab ) ◦ (cid:3)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n = k a k A n L n ( b ) + L n ( a ) k b k A n .It is then an easy computation to see that (cid:0) ab + ba (cid:1) ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) and L n (cid:18) ab + ba (cid:19) = 12 (cid:16)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:2) /D n , ( ab ) ◦ (cid:3)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S n + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:2) /D n , ( ba ) ◦ (cid:3)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n (cid:17) k a k A n L n ( b ) + L n ( a ) k b k A n ,and similarly with the Lie product. (cid:3) The main tool which we need to both prove that the seminorms L n are L-seminorms on A n , and that the sequence ( A n , L n ) n ∈ N of quantum compact metricspaces converges, in the sense of the propinquity, to ( A ∞ , L ∞ ) , is given by a resultabout certain approximation of elements in dom ( L n ) by finitely supported elements,where the error in the approximation is controlled by the seminorms L n . This isthe matter of the next subsection.3.1. A Mean Value Theorem for our Spectral Triples.
We first observe thatthe dual actions of fuzzy and quantum tori are by Lipschitz automorphisms [37].We establish this in several steps.
Lemma 3.3.
Assume Hypothesis (2.41). Let n ∈ N . If z ∈ G n , then ( v zn ) ◦ dom (cid:0) /D n (cid:1) = dom (cid:0) /D n (cid:1) .Proof. Let z = ( z , . . . , z d ′ ) ∈ G n , then X m ∈ Z d ′ k ′ n (1 + h m, m i n ) k (( v zn ) ◦ ξ )( m ) k C = X m ∈ Z d ′ k ′ n (1 + h m, m i n ) k z m ξ ( m ) k C = X m ∈ Z d ′ k ′ n (1 + h m, m i n ) k ξ ( m ) k C < ∞ and thus ( v zn ) ◦ ξ ∈ dom (cid:0) /D n (cid:1) . Therefore, for all z ∈ U dk n , we have ( v zn ) ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) ; as ( v zn ) ◦ is a unitary, we then conclude ( v zn ) ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) = ( v zn ) ◦ ( v − zn ) ◦ dom (cid:0) /D n (cid:1) ⊆ ( v zn ) ◦ dom (cid:0) /D n (cid:1) ,i.e. ∀ z ∈ U dk n ( v zn ) ◦ dom (cid:0) /D n (cid:1) = dom (cid:0) /D zn (cid:1) .This establishes our lemma. (cid:3) We now rewrite our operator /D n in a convenient form. Lemma 3.4.
Assume Hypothesis (3.1). Let n ∈ N . Let E j = 1 n ⊗ c ( γ j ) for all j ∈ { , . . . , d + d ′ } . The following identity between operators on dom (cid:0) /D n (cid:1) holds: ∀ j ∈ { , . . . , d + d ′ } E j /D n + /D n E j = − n,j ⊗ C ,and, therefore, /D n = − d + d ′ X j =1 (cid:0) E j /D n + /D n E j (cid:1) E j .Proof. Owing to the standard relations in the Clifford algebra Cl ( C d ′ ) , we have, forall j, s ∈ { , . . . , d ′ } : E j E s + E s E j = ( − if j = s , otherwise.and, by construction, Γ n,s and E j commute. Of course, E j maps dom (cid:0) /D n (cid:1) toitself. Thus, on dom (cid:0) /D n (cid:1) , E j /D n + /D n E j = d + d ′ X s =1 Γ n,s ( E j E s + E s E j )= − n,j ⊗ C . Our lemma follows. (cid:3)
We deduce from Lemma (3.4) the following helpful estimates relating our L-seminorms candidates to the quantized calculus on fuzzy and quantum tori.
Lemma 3.5.
Assume Hypothesis (3.1). Let n ∈ N . If a ∈ dom ( L n ) , then, for all j ∈ { , . . . , d } , if k n ( j ) = ∞ , then, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:2) U n,f ( j ) ∂ n,j , a (cid:3)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H n L n ( a ) ,and k n ( j )2 π k [ U n,j , a ] k B n L n ( a ) .Proof. Let E j = 1 n ⊗ c ( γ j ) for j ∈ { , . . . , d + d ′ } . For the following computation,note that since E j dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) and a ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) , we alsohave a ◦ E j dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) . We also note that [ E j , a ◦ ] = 0 by construction.By Lemma (3.4), if n ∈ N , j ∈ { , . . . , d } and k n ( j ) = ∞ , then, we compute(over the space dom (cid:0) /D n (cid:1) ): [ X n,j ∂ n,j , a ] ◦ = − (cid:2) E j /D n + /D n E j , a ◦ (cid:3) = − E j (cid:2) /D n , a ◦ (cid:3) + /D n [ E j , a ◦ ] | {z } =0 +[ /D n , a ◦ ] E j + /D n [ E j , a ◦ ] | {z } =0 ,and therefore, since ||| E j ||| J n , we conclude ||| [ X n,j ∂ n,j , a ] ◦ ||| J n dom ( /D n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ /D n , a ◦ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n dom ( /D n ) = L n ( a ) .The same method would show that ||| [ Y n,j ∂ n,j , a ◦ ] ||| J n dom ( /D n ) L n ( a ) .Therefore, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:2) U n,f ( j ) ∂ n,j , a (cid:3) ◦ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n dom ( /D n )= (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) [ X n,j ∂ n,j , a ] ◦ + [ Y n,j ∂ n,j , a ] ◦ (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H n L n ( a ) .It thus immediately follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:2) U n,f ( j ) ∂ n,j , a (cid:3)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H n L n ( a ) .The other inequality is proven similarly. (cid:3) We are now able to compute the Lipschitz seminorms of the dual actions, withrespect to our L-seminorms.
Notation . If z = ( z , . . . , z d ′ ) ∈ G n , then we let dil ( z ) = X j ∈{ ,...,d } | − z f ( j ) | . Lemma 3.7.
Assume Hypothesis (3.1). Let n ∈ N . If a ∈ dom ( L n ) and z ∈ G n ,then α zn ( a ) ∈ dom ( L n ) , and | L n ( a ) − L n ( α zn ( a )) | dil ( z ) L n ( a ) . Proof.
We note that, since a ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) , since ( v zn ) ◦ dom (cid:0) /D n (cid:1) =dom (cid:0) /D n (cid:1) , and since α zn ( a )) = v zn av − zn , we conclude that α zn ( a ) ◦ dom (cid:0) /D n (cid:1) ⊆ dom (cid:0) /D n (cid:1) .Let z = ( z , . . . , z d ′ ) ∈ G n . By construction, since v − zn U n,f ( j ) v zn = z f ( j ) U n,f ( j ) ,if j ∈ { , . . . , d } , and k n ( j ) = ∞ , then (performing the following computations overthe dense subspace F n of H n consisting of finitely supported elements): [ U n,f ( j ) ∂ n,j , α zn ( a )] = [ U n,f ( j ) ∂ n,j , v zn av − zn ]= U n,f ( j ) ∂ n,j v zn av − zn − v zn av − zn U n,f ( j ) ∂ n,j = U n,f ( j ) v zn ∂ n,j av − zn − v zn az f ( j ) U n,f ( j ) v − zn,j ∂ n,j = z f ( j ) v zn (cid:0) U n,f ( j ) ∂ n,j a − aU n,j ∂ n,j (cid:1) v − zn = v zn (cid:0) z f ( j ) (cid:2) U n,f ( j ) ∂ n,j , a (cid:3)(cid:1) v − zn .In particular, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ U n,f ( j ) ∂ n,j , α zn ( a )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H n F n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ U n,f ( j ) ∂ n,j , a ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H n F n < ∞ .Thus, using the density of F n , we conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ U n,f ( j ) ∂ n,j , α zn ( a )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H n < ∞ .Since a ∈ dom ( L n ) , we know that a = a ∗ , we easily conclude as well that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ U ∗ n,f ( j ) ∂ n,j , α zn ( a )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ U ∗ n,f ( j ) ∂ n,j , a ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H n < ∞ .Therefore, for all j ∈ { , . . . , d } , if k n ( j ) = ∞ , then ||| [Γ n,j , α zn ( a )] ||| H n < ∞ and ||| [Γ n,d + j , α zn ( a )] ||| H n < ∞ .By construction, v zn commutes with ∂ n,j and v wn , for all j ∈ { d + 1 , . . . , d ′ } and w ∈ G n . Thus, if j ∈ { d + 1 , . . . , d ′ } , then by construction, v zn Γ n,j v − zn = Γ n,j , so [Γ n,j , α zn ( a )] = v zn [Γ n,j , a ] v − zn . Thus, for all j ∈ { d + 1 , . . . , d + d ′ } : ||| [Γ n,j , α zn ( a )] ||| H n = ||| [Γ n,j , a ] ||| H n .We thus conclude that L n ( α zn ( a )) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ /D n , a ◦ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H n < ∞ ,and therefore, α zn ( a ) ∈ dom ( L n ) , as claimed.To conclude our estimation of the Lipschitz constant for the dual action, wemake a few quick observations. If j ∈ { , . . . , d } and k n ( j ) < ∞ , then [ U n,f ( j ) , α zn ( a )] = v zn (cid:0) z f ( j ) [ U n,f ( j ) , a ] (cid:1) v − zn .Thus, a direct computation shows that, for all j ∈ { , . . . , d } , and k ′ n ( j ) < ∞ ,then again, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v zn [Γ n,j , a ] v − zn − [Γ n,j , α zn ( a )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H n | − z f ( j ) | L n ( a ) .Since v zn is unitary, we have L n ( a ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( v zn ) ◦ [ /D n , a ◦ ]( v − zn ) ◦ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n .Therefore, we conclude: | L n ( a ) − L n ( α zn ( a )) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( v zn )[ /D n , a ◦ ]( v − zn ) ◦ − [ /D n , α zn ( a ) ◦ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J n d + d ′ X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v zn [Γ n,j , a ] v − zn − [Γ n,j , α zn ( a )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈{ ,...,d } | − z f ( j ) | L n ( a ) dil ( z ) L n ( a ) .This concludes our proof. (cid:3) We can approximate elements of a quantum or fuzzy torus to elements by finitelysupported elements, in a manner which gives us control over the support of theapproximations. For this purpose, we use integral operators on A n defined usingkernels, which ultimately we will choose to be Fejer kernels. Notation . If f ∈ C ( T d ) is a C -valued continuous function over the d -torus T d ,if n ∈ N , and if µ n is the Haar probability measure on U dk n , then we then define,for all a ∈ A n , the following element of B n : α fn ( a ) = Z U dkn f ( z ) α zn ( a ) dµ n ( z ) . Lemma 3.9.
Assume Hypothesis (3.1), and let n ∈ N . If f ∈ C ( U dk n ) , and if f ( U dk n ) ⊆ [0 , ∞ ) , then L n ( α fn ( a )) Z U dkn f ( z ) | − dil ( z ) | dµ n ( z ) ! L n ( a ) .Proof. First, we note that α fn ( a ) ∈ sa ( A n ) whenever a ∈ sa ( A n ) (which is werewe use the observation that σ ′ n is normalized). In this proof, we recall that L ( a ) iswell-defined, if possibly infinite, for all a ∈ A n .Since L n is a lower semi-continuous seminorm, we conclude L n ( α fn ( a )) Z U dkn | f ( z ) | L n ( α zn ( a )) dµ n ( z )= Z U dkn f ( z ) | − dil ( z ) | L n ( a ) dµ n ( z ) by Corollary (3.7) and f > , = Z U dkn f ( z ) | − dil ( z ) | dµ n ( z ) ! L n ( a ) .This completes our proof. (cid:3) Lemma 3.10.
Assume Hypothesis (3.1). Let n ∈ N . If ( f m ) m ∈ N is a sequence ofpositive elements in C ( U dk n ) such that, for all g ∈ C ( T d ) , lim m →∞ Z U dkn f m ( z ) g ( z ) dµ n ( z ) = g (1 , . . . , ,then ∀ a ∈ B n lim m →∞ (cid:13)(cid:13) a − α f m n ( a ) (cid:13)(cid:13) A n = 0 . Proof.
Let n ∈ N and let a ∈ A n . For any m ∈ N , we simply compute: (cid:13)(cid:13) a − α f m n ( a ) (cid:13)(cid:13) A n Z U dkn f m ( z ) k a − α zn ( a ) k A n dµ n ( z ) m →∞ −−−−→ k a − a k A n = 0 .Thus, our lemma is proven. (cid:3) We thus obtain a first approximation result for Lipschitz elements. To thisend, we use the following well-known lemma as a source of kernels — in the givenreference, these kernels are Féjer kernels.
Lemma 3.11 ([31]) . Let q ∈ N ∗ . Let k ∈ N q ∗ , and let µ k be the Haar probabilitymeasure on U dk . There exists a sequence ( f m ) m ∈ N in C ( U qk ) such that:(1) ∀ m ∈ N f ( U qk ) ⊆ [0 , ∞ ) ,(2) ∀ m ∈ N f m (0) = 0 ,(3) ∀ m ∈ N ∀ n ∈ Z qk f m ( − n ) = f m ( n ) ,(4) ∀ m ∈ N R U k f m ( z ) dµ m ( z ) = 1 ,(5) ∀ g ∈ C ( U dk n ) lim m →∞ R U dk f ( z ) g ( z ) dµ k ( z ) = g (1 , . . . , ,(6) for all m ∈ N , the function f m is a linear combination of characters of U dk .Proof. We refer to [31, Lemma 3.1, Lemma 3.2, Lemma 3.6] for this lemma. (cid:3)
Putting these observations together, we conclude:
Corollary 3.12.
Assume Hypothesis (3.1). Let n ∈ N . For all ε > , and for all a ∈ dom ( L n ) , there exists b ∈ dom ( L n ) , with b finitely supported , such that L n ( b ) (1 + ε ) L n ( a ) and k a − b k A n < ε .Proof. Let n ∈ N and let ( f m ) m ∈ N be a sequence in C ( U dk n ) given by Lemma (3.11)(for q = d and k = k n ).Let a ∈ dom ( L n ) . Let ε > . By Lemma (3.10), there exists M ∈ N suchthat, if m > M , then (cid:13)(cid:13) a − α f m n ( a ) (cid:13)(cid:13) A n < ε . There exists M ′ ∈ N such that, if m > M ′ , then (cid:12)(cid:12)(cid:12) − R U dkn f ( z ) | − dil ( z ) | dµ n ( z ) (cid:12)(cid:12)(cid:12) < ε — since | − dil (1) | = 1 . Let f = f max { M,M ′ } ,and let b = α fn ( a ) .By Lemma (3.9), L n ( b ) (1 + ε ) L n ( a ) . By [31], if S is the support of the Fouriertransform of f , the range of α fn is the image by π n of (cid:8) f ∈ ℓ ( Z dk n ) : f (cid:0) Z dk n \ S (cid:1) = { } (cid:9) .Since S is finite by assumption on f , the range of α fn consists of finitely supportedelements in A n . This concludes our lemma. (cid:3) Theorem (2.48) gives us, for all n ∈ N , an explicit formula for L n ( a ) whenever a ∈ dom ( L n ) is finitely supported. We now use this expression to obtain a naturalbound on L n ( a ) in terms of the action of the Lie algebra of G n on A n . Lemma 3.13.
Assume Hypothesis (3.1). Let n ∈ N . If a ∈ dom ( L n ) is finitelysupported , then, max n(cid:13)(cid:13) ∂ jn ( a ) (cid:13)(cid:13) A n : j ∈ { , . . . , d } , k n ( j ) = ∞ o L n ( a ) and max ((cid:13)(cid:13)(cid:13)(cid:13) k n ( j )2 π ( α z n,j n ( a ) − a ) (cid:13)(cid:13)(cid:13)(cid:13) A n : j ∈ { , . . . , d } , k n ( j ) < ∞ ) L n ( a ) .Proof. If a ∈ sa ( A n ) is finitely supported, then a direct computation shows that [ ∂ n,j , a ] = ∂ jn ( a ) . We then conclude the following, for all j ∈ { , . . . , d } . If k n ( j ) = ∞ , then (cid:13)(cid:13) ∂ jn ( a ) (cid:13)(cid:13) A n = ||| [ ∂ n,j , a ] ||| H n = (cid:13)(cid:13) U n,f ( j ) ∂ n,j ( a ) (cid:13)(cid:13) B n as U n,f ( j ) is unitary, L n ( a ) using Lemma (3.5).Similarly, if k n ( j ) < ∞ , then, precisely thanks to Hypothesis (2.33), (cid:13)(cid:13)(cid:13)(cid:13) k n ( j )2 π ( α z n,j n ( a ) − a ) (cid:13)(cid:13)(cid:13)(cid:13) A n (cid:13)(cid:13)(cid:13)(cid:13) k n ( j )2 π ( U n,f ( j ) aU ∗ n,f ( j ) − a ) (cid:13)(cid:13)(cid:13)(cid:13) B n = (cid:13)(cid:13)(cid:13)(cid:13) k n ( j )2 π [ U n,f ( j ) , a ] U ∗ n,f ( j ) (cid:13)(cid:13)(cid:13)(cid:13) B n = (cid:13)(cid:13)(cid:13)(cid:13) k n ( j )2 π [ U n,f ( j ) , a ] (cid:13)(cid:13)(cid:13)(cid:13) B n L n ( a ) using Lemma (3.5).This completes our proof. (cid:3) We now prove some basic estimates, relating the geometry of the torus and thequantum torus. To this end, we define a translation invariant Riemannian metricon T d by endowing the Lie algebra u d ∞ d = R d of T d with the usual inner product h· , ·i R d of R d (whose associated norm we denote by k·k R d ). The distance from theunit of T d to any point for the path metric induced by this Riemannian metric isa continuous length function on T d , given by the expression: ∀ z ∈ T d len ( z ) = min {k X k R d : z = exp T d ( X ) } .While U dk has a trivial Lie group structure when k ∈ N d ∗ , it does inherit from themetric structure on T d a length function by restricting len ( · ) to it. Remark . Assume Hypothesis (2.41). For all n ∈ N , and for all j ∈ { , . . . , d } ,the following equality holds: len ( z n,j ) = ( πk n ( j ) if k n ( j ) < ∞ , if k n ( j ) = ∞ .We now compute an estimate for (cid:13)(cid:13)(cid:13) a − α zk,σ ( a ) (cid:13)(cid:13)(cid:13) A n , valid for any ( k, σ ) ∈ Ξ d , forall a ∈ C ∗ (cid:0) Z dk , σ (cid:1) , and for all z ∈ U dk . We obtain this estimate in three steps. Lemma 3.15.
Let d ∈ N ∗ . Let ( k, σ ) ∈ Ξ d with k = ( k (1) , . . . , k ( d )) . If j ∈{ , . . . , d } such that k ( j ) < ∞ , if z k,j = exp (cid:16) πk ( j ) (cid:17) e j , then for all a ∈ C ∗ (cid:0) Z dk , σ (cid:1) ,for all n ∈ Z , and setting z = z nk,j , we compute: (cid:13)(cid:13) a − α zk,σ ( a ) (cid:13)(cid:13) C ∗ ( Z dk ,σ ) len ( z ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α z k k,σ ( a ) − a πk ( j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C ∗ ( Z dk ,σ ) Proof.
Let ( k, σ ) ∈ Ξ d with k = ∞ d . Let j ∈ { , . . . , d } such that k ( j ) < ∞ , andlet a ∈ C ∗ (cid:0) Z dk , σ (cid:1) .Let z = z nk for n ∈ N . Our lemma’s conclusion is trivial when n = 0 , so weassume that z = 1 .Let β = α z k,j k,σ and note that β is an isometry of C ∗ (cid:0) Z dk , σ (cid:1) . Now: (cid:13)(cid:13) a − α zk,σ ( a ) (cid:13)(cid:13) C ∗ ( Z dk ,σ ) n − X w =0 (cid:13)(cid:13) β w ( a ) − β w +1 ( a ) (cid:13)(cid:13) C ∗ ( Z dk ,σ )= n − X w =0 k β w ( a − β ( a )) k C ∗ ( Z dk ,σ )= n − X w =0 k a − β ( a ) k C ∗ ( Z dk ,σ )= n len ( z k,j ) (cid:13)(cid:13)(cid:13)(cid:13) β ( a ) − a len ( z k,j ) (cid:13)(cid:13)(cid:13)(cid:13) C ∗ ( Z dk ,σ ) .If n ∈ Z , and z = z nk,j , then since (cid:13)(cid:13) a − α zk,σ ( a ) (cid:13)(cid:13) C ∗ ( Z dk ,σ ) = (cid:13)(cid:13) α zk,σ ( a ) − a (cid:13)(cid:13) C ∗ ( Z dk ,σ ) ( − n ) len ( z k,j ) (cid:13)(cid:13)(cid:13)(cid:13) β ( a ) − a len ( z k,j ) (cid:13)(cid:13)(cid:13)(cid:13) C ∗ ( Z dk ,σ ) .Therefore, (cid:13)(cid:13) a − α zk,σ ( a ) (cid:13)(cid:13) C ∗ ( Z dk ,σ ) | n | len ( z k,j ) (cid:13)(cid:13)(cid:13)(cid:13) β ( a ) − a len ( z k,j ) (cid:13)(cid:13)(cid:13)(cid:13) C ∗ ( Z dk ,σ ) ,and therefore, since len ( z ) = min n | n | k ( j ) : z nk,j = z o , we conclude that our lemmaholds. (cid:3) We follow [58, Theorem 3.1] for our next lemma.
Lemma 3.16.
Let ( k, σ ) ∈ Ξ d . If j ∈ { , . . . , d } , if k j = ∞ , and if z = exp U dk ( te j ) for some t ∈ R , then for all finitely supported element a ∈ C ∗ (cid:0) Z dk , σ (cid:1) : (cid:13)(cid:13) a − α zk,σ ( a ) (cid:13)(cid:13) C ∗ ( Z dk ,σ ) len ( z ) (cid:13)(cid:13) ∂ e j k ( a ) (cid:13)(cid:13) C ∗ ( Z dk ,σ ) .Proof. Let a be finitely supported. Let z = exp U dk ( te j ) , for some t ∈ R . Now, let f : s ∈ R α exp U dk ( se j ) k,σ ( a ) . Note that, for all s, r ∈ R , with r = 0 , we have: r (cid:16) α exp(( s + r ) e j ) k,σ ( a ) − α exp( se j ) k,σ ( a ) (cid:17) = α exp U dk ( se j ) k,σ (cid:18) r (cid:18) α exp U dk ( re j ) k,σ ( a ) − a (cid:19)(cid:19) r → −−−→ α exp U dk ( se j ) k,σ ( ∂ e j k ( a )) .Therefore, (cid:13)(cid:13) a − α zk,σ ( a ) (cid:13)(cid:13) C ∗ ( Z dk ,σ ) = (cid:13)(cid:13)(cid:13)(cid:13)Z t α exp U dk ( se j ) k,σ (cid:16) ∂ e j k,σ ( a ) (cid:17) ds (cid:13)(cid:13)(cid:13)(cid:13) C ∗ ( Z dk ,σ ) Z t (cid:13)(cid:13)(cid:13)(cid:13) α exp U dk ( se j ) k,σ (cid:16) ∂ e j k,σ ( a ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) C ∗ ( Z dk ,σ ) ds | t | (cid:13)(cid:13) ∂ e j k ( a ) (cid:13)(cid:13) C ∗ ( Z dk ,σ ) .Now, min {| t | : exp U dk ( te j ) = z } = len ( z ) and thus our lemma is proven. (cid:3) While Lemma (3.16) could be improved to provide a similar estimate for any z in the connected component of U dk , the estimate in Lemma (3.15) does not workquite as nicely. Instead, we obtain the following lemma, which will suffice. Lemma 3.17.
We use the notations of Lemmas (3.15) and (3.16). Let d ∈ N ∗ .Let ( k, σ ) ∈ Ξ d , where k = ( k (1) , . . . , k ( d )) . Let Υ = { j ∈ { , . . . , d } : k ( j ) = ∞} .For all finitely supported a ∈ C ∗ (cid:0) Z dk , σ (cid:1) , and for all z ∈ U dk , we estimate: (cid:13)(cid:13) a − α zk,σ ( a ) (cid:13)(cid:13) C ∗ ( Z dk ,σ ) √ d len ( z ) max max j ∈ Υ (cid:13)(cid:13) ∂ e j k ( a ) (cid:13)(cid:13) C ∗ ( Z dk ,σ ) , max j ∈{ ,...,d } j / ∈ Υ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α z k,j k,σ ( a ) − a πk ( j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C ∗ ( Z dk ,σ ) .Proof. Let z = ( z , . . . , z d ) ∈ U dk . By construction, we observe with an easy com-putation that for all ( z , . . . , z d ) ∈ U dk : len (( z , . . . , z d )) = d X j =1 len ( z j e j ) where { e , . . . , e d } is the canonical basis of C d .Thus P dj =1 len ( z j e j ) √ d len ( z ) by the Cauchy Shwarz inequality.Let K = max ( max j ∈ Υ (cid:13)(cid:13) ∂ e j k ( a ) (cid:13)(cid:13) C ∗ ( Z dk ,σ ) , max j ∈{ ,...,d } j / ∈ Υ (cid:13)(cid:13)(cid:13)(cid:13) α zk,jk,σ ( a ) − a len ( z k,j ) (cid:13)(cid:13)(cid:13)(cid:13) C ∗ ( Z dk ,σ ) ) .We conclude, using both Lemma (3.15) and (3.16): (cid:13)(cid:13) a − α zk,σ ( a ) (cid:13)(cid:13) C ∗ ( Z dk ,σ ) d X j =1 len ( z j e j ) K √ d len ( z ) K ,as desired. (cid:3) We thus arrive at our conclusion for this section.
Corollary 3.18.
Assume Hypothesis (3.1). For all n ∈ N , if a ∈ dom ( L n ) , then (3.1) ∀ z ∈ U dk n k a − α zn ( a ) k A n √ d len ( z ) L n ( a ) .Proof. Expression (3.1) holds if a is finitely supported by Lemmas (3.13) and (3.17).Now, let a ∈ dom ( L n ) . Let ε > . By Corollary (3.12), there exists a finitelysupported element b ∈ dom ( L n ) such that L n ( b ) (cid:18) ε √ d L n ( a )) (cid:19) L n ( a ) and k a − b k A n < ε √ d L n ( a )) .Therefore, for all z ∈ U dk n : k a − α zn ( a ) k A n k a − b k A n + k b − α zn ( b ) k A n + k α zn ( b − a ) k A n < ε √ d L n ( a )) + √ d len ( z ) L n ( b ) + ε √ d L n ( a )) ε √ d L n ( a ) (cid:16) √ d L n ( a ) (cid:17) + √ d len ( z ) L n ( a ) ε + √ d len ( z ) L n ( a ) .As ε > is arbitrary, we have proven our lemma. (cid:3) As a consequence of Corollary (3.18), we obtain one more necessary propertytoward the proof that our spectral triples are, indeed, metric.
Corollary 3.19.
Assume Hypothesis (3.1). For all n ∈ N , and for all a ∈ dom ( L n ) , L n ( a ) = 0 ⇐⇒ a ∈ R A n .Proof. Of course, L n ( t A n ) = 0 for all t ∈ R .Now, let a ∈ dom ( L n ) such that L n ( a ) = 0 . By Corollary (3.18), ∀ z ∈ U dk n k a − α zn ( a ) k A n √ d len ( z ) L n ( a ) = 0 and thus α zn ( a ) = a for all z ∈ U dk n . As α n is ergodic on A n , we conclude that a ∈ R A n , as desired. (cid:3) We now bring together the tools developed in this section, to obtain our principalapproximation theorem.
Theorem 3.20 ([31],[36, Theorem 3.3.2]) . Assume Hypothesis (3.1). For all ε > , there exists N ∈ N , and a function f ∈ C ( T d ) , whose Fourier transform issupported on a finite subset S of Z d with − S = S and ∈ S , and such that for all n ∈ N , if n > N , and for all a ∈ dom ( L n ) , we have: (cid:13)(cid:13) a − α fn ( a ) (cid:13)(cid:13) A n ε L n ( a ) and L n ( α fn ( a )) (1 + ε ) L n ( a ) .In particular, α fn ( a ) is in the linear span of { δ m : m ∈ q ( S ) } , where q : Z d ։ Z dk n is the canonical surjection.Proof. Let ε > .By Lemma (3.11), there exists linear combination f : T d → [0 , ∞ ) of charactersof T d such that R T d f dµ ∞ = 1 and Z T d f ( z ) len ( z ) dµ ∞ ( z ) ε √ d and Z T d f ( z ) | − dil ( z ) | dµ ∞ ( z ) ε .The Fourier transform b f of f is an element of ℓ ( Z d ) with finite support, denotedby S . Moreover, the range of α fn is { f ∈ ℓ ( Z d ) : ∀ z / ∈ S f ( z ) = 0 } . In particular, α fn has finite rank.By [31, Lemma 3.6], since both f and f len ( · ) is continuous, we conclude that: lim n →∞ Z U dkn f dµ n = Z T d f dµ ∞ = 1 ,and lim n →∞ Z U dkn f len ( · ) dµ n = Z T d f len ( · ) dµ ∞ . Therefore, there exists N ∈ N such that for all n ∈ N , if n > N then: Z U dkn f ( z )dil ( z ) dµ n ( z ) ε and Z U dkn f ( z ) len ( z ) dµ n ε √ d .By Lemma (3.9), we conclude that L n (cid:0) α fn ( a ) (cid:1) (1 + ε ) L n ( a ) .From Corollary (3.18), for all a ∈ dom ( L n ) , we compute: (cid:13)(cid:13) a − α fn ( a ) (cid:13)(cid:13) A n Z U dkn f ( z ) k a − α zn ( a ) k A n dµ n ( z ) Z U dkn f ( z ) √ d L n ( a ) len ( z ) dµ n ( z )= √ d L n ( a ) Z U dkn f ( z ) len ( z ) dµ n ( z ) ε L n ( a ) .as desired. (cid:3) We can finish the proof that ( A n , J n , /D n ) is a metric spectral triple, for all n ∈ N . Theorem 3.21.
Assume Hypothesis (3.1). For all n ∈ N , the ordered pair ( A n , L n ) is a quantum compact metric space.Proof. Let τ : a ∈ A n Z U dkn α zn ( a ) dµ n ( z ) .Since µ n is the probability Haar measure of U dk n , it is easy, and well-known, is that τ is a state of A n , invariant for the action α n .We now prove that the set T = { a ∈ dom ( L n ) : L n ( a ) and τ ( a ) = 0 } is totally bounded in A n (note that we already know, by Lemma (3.2), that T isclosed, and thus complete, since A n is complete; thus T is totally bounded if, andonly if it is compact).First, by Corollary (3.18), we then note that if a ∈ dom ( L n ) and L n ( a ) , then(3.2) k a − τ ( a ) k A n Z U dkn k a − α zn ( a ) k A n dµ n ( z ) √ d diam (cid:0) T d , len ( · ) (cid:1) = 2 π √ d .Therefore, T is bounded in A n .Let ε > . By Theorem (3.20), there exists f ∈ C ( T d ) such that α fn has finitedimensional range, and ∀ a ∈ T (cid:13)(cid:13) a − α fn ( a ) (cid:13)(cid:13) A n ε .Now, α fn ( T ) is a bounded subset, as it is the image, by a continuous linear map,of a bounded set (by Expression (3.2)). Therefore, as a bounded subset of the finitedimensional space α fn ( A n ) , the set α fn ( T ) is actually totally bounded. Thus, there exists a ε -dense finite subset F ⊆ α fn ( T ) of α fn ( T ) . Consequently, for all a ∈ T ,there exists b ∈ F such that k a − b k A n (cid:13)(cid:13) a − α fn ( a ) (cid:13)(cid:13) A n + (cid:13)(cid:13) α fn ( a ) − b (cid:13)(cid:13) A n < ε .Therefore, since ε > is arbitrary, we conclude that T is totally bounded.Using Lemma (3.2) and Corollary (3.18), we thus have completed our proof. (cid:3) Convergence of the Quantum Metrics.
Quantum and fuzzy tori are thefibers of a continuous field of C*-algebras, with the C*-algebra of continuous sectionsgiven by a twisted groupoid C*-algebra, where the groupoid is a bundle over thebase space Ξ d , with fiber Z dk over each point ( k, σ ) ∈ Ξ d , as explained in [31]. Forour purpose, we record the following consequence, which will suffice. Notation . Assume Hypothesis (2.41). Let n ∈ N and j ∈ { , . . . , d ′ } . For all x ∈ R , we denote max { n ∈ Z : n x } as ⌊ x ⌋ . If k ′ n ( j ) < ∞ then we let: C jn = (cid:26)(cid:22) − k ′ n ( j )2 (cid:23) , (cid:22) − k ′ n ( j )2 (cid:23) + 1 , . . . , (cid:22) k ′ n ( j ) − (cid:23)(cid:27) while, if k ′ n ( j ) = ∞ , then we let C jn = Z . We then define the sets: C n = d ′ Y j =1 C jn ⊆ Z ( d ′ ) .The canonical surjection q n : Z d ′ ։ c G n restricts to an bijection from C n onto c G n , whose inverse, by abuse of notation, we denote by q − n : c G n ֒ → C n . Note that q − ∞ is the identity map.To ease our notations further, we also adopt the following convention. Notation . If k ∈ N d , if F ⊆ Z ( d ′ ) k , and p ∈ [1 , ∞ ) , then we define ℓ p ( Z ( d ′ ) k | F ) = n ξ ∈ ℓ p ( Z ( d ′ ) k ) : ∀ m ∈ Z ( d ′ ) k m / ∈ F = ⇒ ξ ( m ) = 0 o . Remark . If F ⊆ C n for some n ∈ N ∗ , then the map f ∈ ℓ p ( Z ( d ′ ) | F ) f ◦ q − n ∈ ℓ p ( c G n ) is a well-defined linear isometry from ℓ p ( Z ( d ′ ) | F ) onto ℓ p ( c G n | F ) . Notation . If n ∈ N , and if ξ ∈ H n , then we define ρ n ξ : m ∈ Z d ′ ( ξ ( q ( m )) if m ∈ C n , otherwise.and note that ρ n is an isometry from H n into H ∞ , such that ∀ ξ ∈ H ∞ ρ ∗ n ξ : m ∈ c G n ξ ( q − n ( m )) .The following lemma establishes the asymptotic behavior of the operators Γ n,j ,for all j ∈ { , . . . , d + d ′ } , as n → ∞ ; this result is the basis of the idea that oursequence of spectral triples in Hypothesis (2.41) converges to the expected limit. Lemma 3.26.
Assume Hypothesis (2.41). Let F ⊆ Z ( d ′ ) be finite. For all j ∈{ , . . . , d + d ′ } , the following limit holds: lim n →∞ ||| ρ n Γ n,j ρ ∗ n − Γ ∞ ,j ||| H ∞ ℓ ( Z ( d ′ ) | F ) = 0 . Proof.
We record, of course, that for all m ∈ Z , and for all j ∈ { , . . . , d ′ } suchthat k ′ n ( j ) < ∞ for all n ∈ N , ζ mn,j − len ( ζ n,j ) = exp (cid:16) πmk ′ n ( j ) (cid:17) − πk ′ n ( j ) n →∞ −−−−→ im .Let ε > . Since F is finite, and therefore, the set S = n m ∈ Z ( d ′ ) : m ∈ F or ∃ j ∈ { , . . . , d ′ } m + e j ∈ F or m − e j ∈ F o is finite, there exists K ∈ N such that if n > K then(1) the set S ⊆ C n ,(2) the following holds, max j ∈{ ,...,d ′ } k ′ n ( j ) < ∞ max m =( m ,...,m d ′ ) ∈ S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ m j n,j − len ( ζ n,j ) − im j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε .Let now n > K . First, let ξ ∈ H ∞ . Note that since S ⊆ C n , we conclude that q − n ◦ q n , restricted to S , is the identity function. Moreover, we also note that, forall m ∈ b G n , ρ n v z n,j n ρ ∗ n ξ ( m ) = ( ( z ( · ) n,j ξ ( q − n ( · )))( q n ( m )) if m ∈ C n , otherwise; = ( z mn,j ξ ( m ) if m ∈ C n , otherwise; = v z n,j ∞ ρ n ρ ∗ n ξ ( m ) ,where we used that z q n ( m ) n,j = z mn,j by definition of this notation. Therefore, ρ n v z n,j n ρ ∗ n ξ = v z n,j ∞ ρ n ρ ∗ n ξ .Thus, if ξ ∈ ℓ ( Z d ′ | F ) , then ρ n v z n,j n ρ ∗ n ξ = v z n,j ∞ ξ .Now, fix ξ ∈ ℓ ( Z ( d ′ ) | F ) with k ξ k H ∞ . We compute that, for all j ∈{ , . . . , d ′ } with k ′ n ( j ) < ∞ , (cid:13)(cid:13)(cid:13)(cid:13) k ′ n ( j )2 π ρ n (cid:0) v z n,j n − n (cid:1) ρ ∗ n ξ − ∂ ∞ ,j ξ (cid:13)(cid:13)(cid:13)(cid:13) H ∞ = (cid:13)(cid:13)(cid:13)(cid:13) k ′ n ( j )2 π ( v z n,j ∞ − − ∂ ∞ ,j ) ξ (cid:13)(cid:13)(cid:13)(cid:13) H ∞ = X m =( m ,...,m d ′ ) ∈ F | ξ ( m ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ m j n,j − len ( ζ n,j ) − im j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) max m =( m ,...,m d ′ ) ∈ S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ m j n,j − len ( ζ n,j ) − im j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p X m ∈ c G n | ξ ( m ) | max j ∈{ ,...,d ′ } k ′ n ( j )= ∞ max m =( m ,...,m d ′ ) ∈ F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ m j n,j − len ( ζ n,j ) − im j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ξ k H ∞ (cid:16) ε (cid:17) .Thus, in particular, if j ∈ { d + 1 , . . . , d ′ } and k ′ n ( j ) < ∞ , and if ξ ∈ ℓ ( Z d ′ | F ) ,then lim n →∞ k ρ n Γ n,d + j ρ ∗ n ξ − Γ ∞ ,d + j ξ k H ∞ = 0 .Now, fix j ∈ { , . . . , d } with k n ( j ) < ∞ . We now recall that if ξ ∈ H n and j ∈ { , . . . , d ′ } , then(3.3) U n,j ξ : m ∈ c G n σ ′ n ( m, e j ) ξ ( m + e j ) .Note that, if ξ ∈ ℓ ( Z ( d ′ ) | F ) , then J n U ∞ ,j J n ξ ( m ) = 0 whenever m ∈ Z d ′ \ S , since m S = ⇒ m − e j / ∈ F , and by Expression (3.3).By Hypothesis (2.33), and as recalled in Expression (3.3), if ξ ∈ H ∞ , then U ∞ ,f ( j ) ξ = J ∞ U ∗∞ ,f ( j ) J ∞ ξ . We will write ξ · a for J n a ∗ J n ξ whenever a is a boundedlinear operator on H n , and ξ ∈ H n . Thus, in particular, [ U ∞ ,j , ξ ] = U ∞ ,j ξ − ξ · U ∞ ,j = 0 . Note that the right action · of B n on H n obviously depends on n , butthe context will make it clear which right action we are using at all time.Let ξ ∈ ℓ (cid:16) Z d ′ | F (cid:17) with k ξ k H ∞ . A simple computation then shows that, if ξ ∈ ℓ ( c G n | F ) , and if k ξ k H ∞ , and if we let ξ j : m ∈ c G n ξ ( m − e n,j ) ,then(3.4) (cid:13)(cid:13) ρ n ( ρ ∗ n ξ · U n,f ( j ) ) − ξ · U ∞ ,j (cid:13)(cid:13) H ∞ = X m ∈ C n (cid:12)(cid:12) ( ξ ◦ q − n · U n,f ( j ) ) ◦ q n ( m ) − ξ · U ∞ ,f ( j ) ( m ) (cid:12)(cid:12) = X m ∈ C n (cid:12)(cid:12) ζ mn,j ξ f ( j ) ( m ) − ξ f ( j ) ( m ) (cid:12)(cid:12) = X m ∈ F (cid:12)(cid:12)(cid:12) ζ m + e j n,j ξ ( m ) − ξ ( m ) (cid:12)(cid:12)(cid:12) (cid:16) ε (cid:17) .With this in mind, for all n > K , and for all j ∈ { , . . . , d } , if k n ( j ) < ∞ and f ( j ) > j , and if ξ ∈ ℓ ( Z d ′ | F ) with k ξ k H ∞ , then we conclude that, since ρ ∗ n ρ n = 1 , and, crucially, since Hypothesis (2.33) implies: ∀ η ∈ H n U n,f ( j ) η · U ∗ n,f ( j ) − η = ( v z n,j n − η ,the following computation holds: (cid:13)(cid:13)(cid:13)(cid:13) k n ( j )2 π ρ n [ U n,f ( j ) , ρ ∗ n ξ ] − U ∞ ,f ( j ) ∂ ∞ ,j ξ (cid:13)(cid:13)(cid:13)(cid:13) H ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) k n ( j )2 π ρ n (cid:16) U n,f ( j ) ρ ∗ n ξ · U ∗ n,f ( j ) − ρ ∗ n ξ (cid:17) · U n,j (cid:19) − ( ∂ ∞ ,j ξ ) · U ∞ ,f ( j ) (cid:13)(cid:13)(cid:13)(cid:13) H ∞ = (cid:13)(cid:13)(cid:13)(cid:13) k n ( j )2 π ρ n (cid:16) ( v z n,j n − ρ ∗ n ξ · U n,f ( j ) (cid:17) − ( ∂ ∞ ,j ξ ) · U ∞ ,f ( j ) (cid:13)(cid:13)(cid:13)(cid:13) H ∞ = (cid:13)(cid:13)(cid:13)(cid:13) k n ( j )2 π ρ n (cid:16) ρ ∗ n ρ n ( v z n,j n − ρ ∗ n ξ · U n,f ( j ) (cid:17) − ( ∂ ∞ ,j ξ ) · U ∞ ,f ( j ) (cid:13)(cid:13)(cid:13)(cid:13) H ∞ (cid:13)(cid:13)(cid:13)(cid:13) ρ n (cid:16) k n ( j )2 π ( v z n,j n − ρ ∗ n ξ (cid:17) · U ∞ ,f ( j ) − ( ∂ ∞ ,j ξ ) · U ∞ ,f ( j ) (cid:13)(cid:13)(cid:13)(cid:13) H ∞ + ε (cid:13)(cid:13)(cid:13)(cid:13) ρ n (cid:16) k n ( j )2 π ( v z n,j n − ρ ∗ n ξ (cid:17) − ( ∂ ∞ ,j ξ ) (cid:13)(cid:13)(cid:13)(cid:13) H ∞ + ε ε .Similarly, we can prove with the same method as for the proof of Expression(3.4) that for all n ∈ N , j ∈ { , . . . , d } such that k n ( j ) < ∞ , and for all ξ ∈ H ∞ , (cid:13)(cid:13)(cid:13) ρ n ( U ∗ n,f ( j ) ρ ∗ n ξ ) − U ∗∞ ,j ξ (cid:13)(cid:13)(cid:13) H ∞ ε ;and therefore, if f ( j ) > j , then (cid:13)(cid:13)(cid:13)(cid:13) k n ( j )2 π ρ n [ U ∗ n,j , ρ ∗ n ξ ] + U ∗∞ ,j ∂ ∞ ,j ξ (cid:13)(cid:13)(cid:13)(cid:13) H ∞ ε ,using the observation that h U ∗ n,f ( j ) , ρ ∗ n ξ i n = U ∗ n,f ( j ) (cid:0) (1 n − v z n,j n ) ρ ∗ n ξ (cid:1) .Therefore, for all n > K , if j ∈ { , . . . , d } , if k n ( j ) < ∞ , and if f ( j ) > j , thenwe conclude that k ( ρ n Γ n,j ρ ∗ n − Γ ∞ ,j ) ξ k H ∞ = (cid:13)(cid:13)(cid:13)(cid:13) k n ( j )2 π ρ n [ Y n,j , ρ ∗ n ξ ] n − X ∞ ,j ∂ ∞ ,j ξ (cid:13)(cid:13)(cid:13)(cid:13) H ∞ (cid:18) (cid:13)(cid:13)(cid:13)(cid:13) k n ( j )2 π ρ n [ U n,j , ρ ∗ n ξ ] n − U ∞ ,j ∂ ∞ ,j ξ (cid:13)(cid:13)(cid:13)(cid:13) H ∞ + (cid:13)(cid:13)(cid:13)(cid:13) k n ( j )2 π ρ n (cid:2) U ∗ n,j , ρ ∗ n ξ (cid:3) n + U ∗∞ ,j ∂ ∞ ,j ξ (cid:13)(cid:13)(cid:13)(cid:13) H ∞ (cid:19) ε ,and similarly, k ( ρ n Γ n,d + j ρ ∗ n − Γ ∞ ,d + j ) ξ k H ∞ = (cid:13)(cid:13)(cid:13)(cid:13) k n ( j )2 π ρ n [ X n,j , ρ ∗ n ξ ] − Y ∞ ,j ∂ ∞ ,j ξ (cid:13)(cid:13)(cid:13)(cid:13) H ∞ ε .The only difference when working with n ∈ N , j ∈ { , . . . , d } , and f ( j ) < j , ifthat the roles of U n,j and U ∗ n,j are reversed in the above computation (i.e. X n,j = ℜ ( U ∗ n,j ) , Y n,j = ℑ ( U ∗ n,j ) , and U ∗ n,f ( j ) ξ · U n,f ( j ) = v z n,j n ξ , for all ξ ∈ ℓ ( c G n ) ).Thus, we conclude that for all j ∈ { , . . . , d + d ′ } with k n ( j ) < ∞ : k ( ρ n Γ n,j ρ ∗ n − Γ ∞ ,j ) ξ k H ∞ ε .Last, if j ∈ { , . . . , d } and k ′ n ( j ) = ∞ , then an easy computation shows, usingExpression (3.4) once more, that k ( ρ n Γ n,j ρ ∗ n − Γ ∞ ,j ) ξ k H ∞ ε .This concludes our proof. (cid:3) A first major consequence of Lemma (3.26) is a form of continuity for the L-seminorms from Hypothesis (3.1). Theorem 3.27 ([31]) . Assume Hypothesis (3.1). If F ⊆ Z d be a nonempty finitesubset of Z d , and if a ∈ ℓ ( Z d | F ) , then lim n →∞ L n ( a ◦ q − n ) = L ∞ ( a ) .Proof. In this proof, we use the notation of Example (2.12), so that Cl ( C d + d ′ ) = C ∗ (cid:16) Z d + d ′ (2 ,..., , ς r (cid:17) , for the matrix r defined in that example. As Cl ( C d + d ′ ) is finitedimensional, we note that, as vector spaces , Cl ( C d + d ′ ) = ℓ ( Z d + d ′ ,..., ) — the normsare different, of course. In any case, we also recall that γ j = ie j (see Notation(2.11)).We also identify B n with the C ∗ completion of ℓ ( c G n ) via the *-isomorphism π n , without further mention, to keep notations simpler.Let G = ` n ∈ N (cid:16) c G n × Z d + d ′ (2 ,..., (cid:17) be the disjoint union of the spaces c G n × Z d + d ′ (2 ,..., — i.e. x ∈ G if and only if x = ( n, m, l ) for n ∈ N , m ∈ c G n , and l ∈ Z d + d ′ (2 ,..., .The function ( n, m, l ) ∈ G → ( n, q − ( m ) , l ) ∈ N × Z d ′ × Z d + d ′ (2 ,..., is trivially an injection, with range { ( n, m, l ) : n ∈ N , m ∈ C n , l ∈ Z d + d ′ (2 ,..., } ; we en-dow G with the unique topology such that this injective map is an homeomorphismonto its image, where N × Z d ′ × Z d + d ′ (2 ,..., is endowed with the product topology. Thetopology on G is thus metrizable, and a sequence ( n p , m p , l p ) p ∈ N in G convergesif, and only if, it is eventually constant, or lim p →∞ n p = ∞ and ( m p , l p ) p ∈ N iseventually constant.The set G is trivially a groupoid, where the source and target maps are bothgiven by s : ( n, m, l ) ∈ G n ∈ N (and thus, the space of G (0) is N ), the partial multiplication is defined on the space G (2) = { (( n, m, l ) , ( n ′ , m ′ , l ′ )) ∈ G : n = n ′ } by setting: ∀ (( n, m, l ) , ( n, m ′ , l ′ )) ∈ b G (2) ( n, m, l )( n, m ′ , l ′ ) = ( n, m + m ′ , l + l ′ ) ,and the inverse of ( n, m, l ) ∈ b G is ( n, − m, − l ) . If, moreover, for all n ∈ N , we let λ n be the counting measure on the discrete group b G n = s − ( { n } ) , then G is easilychecked to be a locally compact groupoid with (left) Haar measure ( λ n ) n ∈ N .For any ( n, m, l ) , ( n, m ′ , l ′ ) ∈ G (2) , we then set: β (( n, m, l ) , ( n, m ′ , l ′ )) = σ ′ n ( m, m ′ ) ς r ( l, l ′ ) .Thus defined, β is a continuous -cocycle of G ((see, e.g., [31, Sec. 2.1.] and ofcourse, [55])).For each n ∈ N , we record that s − ( { n } ) = { n } × c G n × Z d + d ′ (2 ,..., , and that β ,restricted to s − ( { n } ) , is β n : ( n, m, l ) , ( n, m, ′ l ′ ) ∈ c G n × Z d + d ′ (2 ,..., = σ ′ n ( m, m ′ ) ς r ( l, l ′ ) ,so that B n ⊗ Cl ( C d + d ′ ) is *-isomorphic to C ∗ ( c G n × Z d + d ′ ,..., , β n ) , where our chosen *-isomorphism sends any element of the form a ⊗ b ∈ ℓ ( c G n ) ⊗ Cl ( C d + d ′ ) to the function ( m, l ) ∈ c G n × Z d + d ′ (2 ,..., a ( m ) b ( l ) (the fact that thismap extends to a *-isomorphism follows form a standard argument based on theuniversality of the C*-algebras involved, which are all fuzzy tori).We therefore conclude by [31, Theorem 2.6] that C ∗ ( G , β ) is the C*-algebra ofcontinuous sections for the family ( B n ⊗ Cl ( C d + d ′ )) n ∈ N of C*-algebras.In particular, if g : G → C is continuous and compactly supported on G , and if,for all n ∈ N , we define g n by g n : ( m, l ) ∈ c G n × Z d + d ′ (2 ,..., g ( n, m, l ) ,then lim n →∞ k g n k B n ⊗ Cl ( C d + d ′ ) = k g ∞ k B ∞ ⊗ Cl ( C d + d ′ ) .Let now a ∈ ℓ ( Z ( d ′ ) | F ) , and let N ∈ N be chosen such that, for all n ∈ N with n > N , we have F ⊆ C n . We define the function: g : ( n, m, l ) ∈ G ( ∇ n (cid:0) a ◦ q − n (cid:1) ( m, l ) if n > N , otherwise,using the notation defined in Expression (2.10).It is immediate to check that g is compactly supported on G . Let now ( n p , m p , l ) p ∈ N be a sequence in b G such that lim p →∞ n p = ∞ , and q n p ( m p ) = m for all p ∈ N . Weaim at showing that ( g ( n p , m p , l )) p ∈ N converges to g ( ∞ , m, l ) .We make the simple observation that, for all n ∈ N with n > N , since a ◦ q − n isfinitely supported, it is an element of H n , and with this observation, ∇ n a = d X j =1 Γ n,j ( a ◦ q − n ) ⊗ γ j .Using Expression (2.10), we then compute, for all p ∈ N : (cid:12)(cid:12) ∇ n p a ◦ q − n p ( m p , l ) − ∇ ∞ a ( m, l ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∇ n p (cid:16) a ◦ q − n p (cid:17) ( q n p ( m ) , l ) − ∇ ∞ a ( m, l ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X j =1 (cid:16) Γ n p ,j ( a ◦ q − n p )( q n p ( m )) − Γ ∞ ,j ( a )( m ) (cid:17) γ j ( l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X j =1 (cid:12)(cid:12)(cid:12)(cid:16) Γ n p ,j ( a ◦ q − n p )( q n p ( m )) − Γ ∞ ,j ( a )( m ) (cid:17) γ j ( l ) (cid:12)(cid:12)(cid:12) d X j =1 (cid:13)(cid:13)(cid:13) ρ n p Γ n p ,j ρ ∗ n p a − Γ ∞ a (cid:13)(cid:13)(cid:13) H ∞ p →∞ −−−→ .Therefore, the function g is continuous on G .Consequently, using Theorem (2.48), L ∞ ( a ) = k∇ ∞ a k B ∞ ⊗ Cl ( C d + d ′ ) = lim n →∞ (cid:13)(cid:13) ∇ n (cid:0) a ◦ q − n (cid:1)(cid:13)(cid:13) B n ⊗ Cl ( C d + d ′ ) = lim n →∞ L n ( a ◦ q − n ) . This concludes our proof. (cid:3)
Now, Theorem (3.20), Theorem (3.21), and Theorem (3.27) are all we need toapply the methods of [36] to conclude:
Theorem 3.28.
If we assume Hypothesis (3.1), then lim n →∞ Λ ∗ (( A n , L n ) , ( A ∞ , L ∞ )) = 0 . The proof of Theorem (3.28) is essentially the same as the proof of [38, Theorem5.2.5], once we replace some intermediate results in [34] with results proven here.
Proof of Theorem 3.28.
Let S ⊆ Z d be finite, and let V = ℓ ( Z d | S ) ⊆ A n . Set F = S ×{ (0 , . . . , } ⊆ Z ( d ′ ) , and let N ∈ N be chosen so that n > N = ⇒ F ⊆ C n .We write N N = { n ∈ N : n > N } .If f ∈ V , then n ∈ N L n ( f ◦ q − n ) is continuous, by Theorem (3.27), andsince N is compact, we conclude that sup n ∈ N L n ( f ◦ q − n ) < ∞ . We thus define theseminorm m : f ∈ V sup n ∈ N L n ( f ◦ q − n ) . Therefore, by [36, Lemma 4.2.5], andby Theorem (3.27), the map ( n, f ) ∈ N N × V L n ( f ◦ q − n ) is continuous (since V is finite dimensional, it is not important to specify the normon V , though of course, it is natural to choose m ). This can now be used in place ofthe conclusion of [31, Theorem 4.2.7], together with Theorem (3.20), in [36, Section5], to conclude our proof. We record the following main tools used in [36, Theorem5.2.5].Let ε > . There exist a finite subset S ⊆ Z d , a natural number N ∈ N and afinite rank operator T N on ℓ ( Z d ) with ||| T N ||| ℓ ( Z d ) = 1 , such that, for all n > N ,there exists a *-representation ϑ n of A n on ℓ ( Z d ) , with the following property: ifwe set, for all ( a, b ) ∈ A n ⊕ A ∞ , • T n ( a, b ) = max n L n ( a ) , L ∞ ( b ) , ε ||| ϑ n ( a ) T N − T N ϑ ∞ ( b ) ||| ℓ ( Z d ) o , • y n ( a, b ) = a and y ∞ ( a, b ) = b ,then: τ n = ( A n ⊕ A ∞ , T n , y n , y ∞ ) is a tunnel from ( A n , L n ) to ( A ∞ , L ∞ ) whose extent is at most ε . This concludesour proof, but we record two more details about the structure of our tunnels.First, if a ∈ ℓ ( Z d | S ) , then: ||| [ T n , ϑ n ( a )] ||| ℓ ( Z d ) ε L n ( a ) .This completes the summary of the properties of the tunnels τ n constructed in[36, Theorem 5.2.5]. (cid:3) We now adjust the presentation of the tunnels constructed in the proof of The-orem (3.28), in preparation for our work with the Dirac operators. To this end, weuse the same notation as in the proof of Theorem (3.28).
Notation . The Hilbert space J ∞ = ℓ ( Z d ′ , C ) is isometrically isomorphic tothe Hilbert space ℓ ( Z d ) ⊗ ℓ ( Z d ′ − d , C )) , via the usual unitary, which extends thefunction which sends ξ ⊗ η ∈ ℓ ( Z d ) ⊗ ℓ ( Z d ′ − d , C ) to ( m , . . . , m d ′ ) ∈ Z ( d ′ ) ξ ( m , . . . , m d ) η ( m d +1 , . . . , m d ′ ) , with the convention that a function of variables (if d ′ = d above) is a constant.With this identification, let θ n ( a ) = ϑ n ( a ) ⊗ ℓ ( Z d ′− d , C ) , where ℓ ( Z d ′− d , C ) isthe identity of ℓ ( Z d ′ − d , C ) .In particular, θ ∞ ( a ) = a ◦ for all a ∈ A ∞ . Again, using this identification, wethen note that if we set R N = T N ⊗ ℓ ( Z d ′− d , C ) , then, for all a ∈ A n and b ∈ A ∞ :(3.5) ||| θ n ( a ) R N − R N θ ∞ ( b ) ||| J ∞ = ||| ϑ n ( a ) T N − T N ϑ ∞ ( b ) ||| ℓ ( Z d ) .Therefore, the L-seminorm T n of tunnels τ n of the proof of Theorem (3.28) canbe rewritten, for all a ∈ A n and b ∈ A ∞ , as T n ( a, b ) = max (cid:26) L n ( a ) , L ∞ ( b ) , ε ||| θ n ( a ) R N − R N θ ∞ ( b ) ||| J ∞ (cid:27) .We now record two more properties proven in [34, Theorem 5.2], adjusted toour current notation. First, we also note that, for all n > N , if a ∈ A n is finitelysupported, then(3.6) ||| [ a ◦ , R N ] ||| J n = ||| [ a, T N ] ||| ℓ ( Z d ) ε L n ( a ) .Second, we also note that, for any finite subset S ⊆ Z ( d ′ ) , we can always choose R N such that R N restricted to J S ∞ is the identity.The advantage of this new formulation is that, now, our tunnels τ n are builtusing the Hilbert space J n , on which our spectral triples are also constructed.We are now ready to conclude that our sequence of metric spectral triples fromHypothesis (2.41) converge.3.3. Convergence of the Spectral Triples.
We now conclude with our mainresult about the convergence of the spectral triples of Hypothesis (2.41), where weuse the spectral propinquity, as explained in the introduction.
Notation . Assume Hypothesis (3.1). For all n ∈ N and for all ξ ∈ J n , we set: DN n ( ξ ) = k ξ k J n + (cid:13)(cid:13) /D n ξ (cid:13)(cid:13) J n .We thus have: qvb (cid:0) A n , J n , /D n (cid:1) = ( J n , DN n , C , , A n , L n ) where we regard J n as a module over A n via the *-representation a ∈ A n a ◦ .By [42, ], qvb (cid:0) A n , J n , /D n (cid:1) is a metrical C ∗ -correspondence.We construct a modular tunnel between the metrized quantum vector bundlesgiven by our spectral triples, namely the quadruples of the form ( J n , DN n , C , .We note that ( C , is a trivial quantum compact metric space (the space with onepoint), but we will actually need to work with Hilbert modules over C ⊕ C whenbuilding our tunnel — so we do need the generality of Definition (1.13). Notation . For any set F ⊆ Z ( d ′ ) , we write J Fn for the subspace of J n consisting of vectors ξ : b G n → C of J n such that ξ ( n ) = 0 whenever n ∈ b G n \ F .A corollary of Lemma (3.26) gives us a first form of continuity for our D-norms. Lemma 3.32.
Assume Hypothesis (2.41). If F ⊆ Z ( d ′ ) be a finite subset of Z ( d ′ ) ,and if, for all n ∈ N , we set ̺ n = ρ n ⊗ C , then we conclude: lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ̺ n ◦ /D n ̺ ∗ n − /D ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J F ∞ = 0 ,and thus in particular: lim n →∞ sup ξ ∈ J F ∞ k ξ k J n | DN n ( ̺ ∗ n ξ ) − DN ∞ ( ξ ) | = 0 .Proof. We compute, for all n ∈ N , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ̺ n ◦ /D n ◦ ̺ ∗ n − /D ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J F ∞ d + d ′ X j =1 ||| ( ρ n ◦ Γ n,j ◦ ρ ∗ n − Γ ∞ ,j ) ⊗ c ( γ j ) ||| J F ∞ d + d ′ X j =1 ||| ρ n Γ n,j ρ ∗ n − Γ ∞ ,j ||| ℓ ( Z d ′ | F ) ||| c ( γ j ) ||| C n →∞ −−−−→ , by Lemma (3.26).In particular, for all n ∈ N , since ̺ ∗ n is, by construction, an isometry from J F ∞ ,we compute: sup ξ ∈ J F ∞ k ξ k J ∞ | DN n ( ̺ ∗ n ξ ) − DN ∞ ( ξ ) | sup ξ ∈ J F ∞ k ξ k J ∞ (cid:16) k ̺ ∗ n ξ k J n − k ξ k J ∞ + (cid:13)(cid:13) /D n ̺ ∗ n ξ (cid:13)(cid:13) J n − (cid:13)(cid:13) /D ∞ ξ (cid:13)(cid:13) J ∞ (cid:17) = sup ξ ∈ J F ∞ k ξ k J ∞ (cid:16)(cid:13)(cid:13) ̺ n ◦ /D n ◦ ̺ ∗ n ξ (cid:13)(cid:13) J ∞ − (cid:13)(cid:13) /D ∞ ξ (cid:13)(cid:13) J ∞ (cid:17) sup ξ ∈ J F ∞ k ξ k J ∞ (cid:13)(cid:13) ̺ n ◦ /D n ◦ ̺ ∗ n ξ − /D ∞ ξ (cid:13)(cid:13) J ∞ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ̺ n ◦ /D n ◦ ̺ − n − /D ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J F ∞ n →∞ −−−−→ .This concludes our proof. (cid:3) The same argument as given in Lemmas (3.15) and Lemma (3.16) applies to givethe following result.
Lemma 3.33.
Assume Hypothesis (2.41). For all n ∈ N , and for all ξ ∈ J n , thefollowing holds ∀ z ∈ G n (cid:13)(cid:13) ξ − ( v zn ) ◦ ( ξ ) (cid:13)(cid:13) J n √ d len ( z ) DN ( ξ ) .Proof. For all ξ ∈ dom (cid:0) /D n (cid:1) and j ∈ { , . . . , d + d ′ } , k (Γ n,j ⊗ c ( γ j )) ξ k J n (cid:13)(cid:13)(cid:13)(cid:13) (cid:0) (1 n ⊗ c ( γ j )) /D n + /D n (1 n ⊗ c ( γ j )) (cid:1) ξ (cid:13)(cid:13)(cid:13)(cid:13) J n DN n ( ξ ) . Therefore, k (Γ n,j ⊗ C ) ξ k J n = k (Γ n,j ⊗ c ( γ j ))(1 n ⊗ c ( γ j )) ξ k J n k (Γ n,j ⊗ c ( γ j )) ξ k J n DN n ( ξ ) .Now, the same proof as in Lemmas (3.15) and (3.16) shows that, for all z ∈ G n , k ( v zn − ◦ ξ k J n √ d len ( z ) K where K = max max j ∈{ ...,d ′ } k ′ n ( j )= ∞ k ( ∂ n,j ⊗ C ) ξ k J n , max j ∈{ ,...,d ′ } k ′ n ( j ) < ∞ (cid:13)(cid:13)(cid:13)(cid:13) len ( z n,j ) ( v z n,j n − J n ) ◦ ξ (cid:13)(cid:13)(cid:13)(cid:13) J n .Therefore, k ( v zn − ◦ ξ k √ d len ( z ) DN n ( ξ ) ,as desired. (cid:3) From this, and the fact that our Dirac operators are closed, since they are self-adjoint, we now conclude the following lemma.
Notation . We use the notation of Hypothesis (2.41). Let n ∈ N . For all f ∈ C ( G n ) , and for all ξ ∈ J n , we set V fn ξ = Z G n f ( z ) ( v zn ) ◦ ξ dµ n ( z ) where µ n is the Haar probability measure on G n . Lemma 3.35.
For all ε > , there exists a positive function f ∈ C ( T ( d ′ ) ) , whoseFourier transform is supported on a finite set S ⊆ Z d ′ , and N ∈ N , such that, forall n > N , and for all ξ ∈ dom (cid:0) /D n (cid:1) , (cid:13)(cid:13) ξ − V fn ξ (cid:13)(cid:13) J n ε DN n ( ξ ) ,and DN n ( V fn ( ξ )) (1 + ε ) DN n ( ξ ) .Note that V fn ξ is finitely supported, with support included in S .Proof. Let ε > . Note that, if j ∈ { , . . . , d } and k n ( j ) = ∞ , then for all z =( z , . . . , z d ′ ) ∈ G n , over the space dom (cid:0) /D n (cid:1) , (cid:2) Γ n,j ⊗ c ( γ j ) , ( v zn ) ◦ (cid:3) = ( v zn ) ◦ (cid:16) (( z f ( j ) − U n,f ( j ) + ( z f ( j ) − U ∗ n,f ( j ) ) ∂ n,j (cid:17) ⊗ c ( γ j ) .Therefore, if ξ ∈ dom (cid:0) /D n (cid:1) , then (cid:13)(cid:13)(cid:2) Γ n,j ⊗ c ( γ j ) , ( v zn ) ◦ (cid:3) ξ (cid:13)(cid:13) J n dil ( z ) (cid:18)(cid:13)(cid:13) ( U n,f ( j ) ∂ n,j ) ⊗ c ( γ j ) ξ (cid:13)(cid:13) J n + (cid:13)(cid:13)(cid:13) ( U ∗ n,f ( j ) ∂ n,j ) ⊗ c ( γ j ) ξ (cid:13)(cid:13)(cid:13) J n (cid:19) dil ( z ) DN n ( ξ ) .A similar computations show that, if j ∈ { , . . . , d } and k n ( j ) = ∞ , then k [Γ n,d + j ⊗ c ( γ j ) , ( v zn ) ◦ ] ξ k J n dil ( z ) DN n ( ξ ) . If j ∈ { , . . . , d } and k n ( j ) < ∞ , then n,j , v zn ] = U n,f ( j ) v zn ξ − v zn U n,f ( j ) ξ + U ∗ n,f ( j ) v zn ξ − v zn U ∗ n,f ( j ) ξ = (cid:16) ( z f ( j ) − n ) U n,f ( j ) + ( z f ( j ) − U ∗ n,f ( j ) (cid:17) ξ + v zn n,j ξ and thus k [Γ n,j , v zn ] ξ k H n dil ( z ) k ξ k H n + k Γ n,j ⊗ c ( γ j ) ξ k H n dil ( z ) DN n ( ξ ) .Similarly, k [Γ n,d + j ⊗ c ( γ j ) , v zn ] ξ k H n dil ( z ) DN n ( ξ ) .Last, [Γ n,j , v zn ] = 0 if j ∈ { d + 1 , . . . , d + d ′ } .Using Lemma (3.11), there exists f ∈ C ( T ( d ′ ) ) , with f > , and, for all n > N , Z G n f ( z )dil ( z ) dµ n ( z ) < εd + d ′ and Z G n f ( z ) len ( z ) √ d dµ n ( z ) < ε .Note that, for all ξ ∈ J n , the vector V fn ξ is finitely supported, so in particular, V fn ξ ∈ dom (cid:0) /D n (cid:1) . As /D n is closed, we then easily deduce that /D n V fn ξ = Z G n f ( z ) /D n V zn ξ dµ n ( z ) .Therefore, if ξ ∈ dom (cid:0) /D n (cid:1) , then (cid:13)(cid:13) V fn /D n ξ − /D n V fn ξ (cid:13)(cid:13) J n d + d ′ X j =1 Z G n f ( z ) k [Γ n,j , V zn ] ξ k J n dµ n ( z ) d + d ′ X j =1 Z G n f ( z )dil ( z ) DN n ( ξ ) dµ n ( z ) d + d ′ X j =1 εd + d ′ DN n ( ξ ) = ε DN n ( ξ ) .Therefore, for all n > N and ξ ∈ dom (cid:0) /D n (cid:1) , DN n ( V fn ξ ) = (cid:13)(cid:13) V fn ξ (cid:13)(cid:13) J n + (cid:13)(cid:13) /D n V fn ξ (cid:13)(cid:13) J n k ξ k J n + (cid:13)(cid:13) V fn /D n ξ (cid:13)(cid:13) J n + (cid:13)(cid:13) [ /D n , V fn ] ξ (cid:13)(cid:13) J n k ξ k J n + (cid:13)(cid:13) /D n ξ (cid:13)(cid:13) J n + ε DN n ( ξ )= (1 + ε ) DN n ( ξ ) .Moreover, as in Corollary (3.18), using Lemma (3.33), (cid:13)(cid:13) ξ − V fn ξ (cid:13)(cid:13) J n Z G n f ( z ) len ( z ) √ d dµ n ( z ) DN n ( ξ ) ε DN n ( ξ ) .This concludes our proof. (cid:3) We now have the tools needed to conclude our proof of convergence for spectraltriples. We begin with our modular tunnels.
Theorem 3.36.
For all ε > , there exists N ∈ N such that, if n > N , and if weset (1) for all ξ ∈ dom (cid:0) /D n (cid:1) and η ∈ dom (cid:0) /D ∞ (cid:1) , TN n ( ξ, η ) = max (cid:26) DN n ( ξ ) , DN ∞ ( η ) , ε k ̺ n ( ξ ) − η k J ∞ (cid:27) ,(2) Q ( z, w ) = ε | z − w | for all z, w ∈ C ,(3) Y n : ( ξ, η ) ∈ J n ⊕ J ∞ ξ and Y ∞ : ( ξ, η ) ∈ J n ⊕ J ∞ η ,(4) x n : ( z, w ) ∈ C ⊕ C z and x ∞ : ( z, w ) w ,then (3.7) τ mod n = ( J n ⊕ J ∞ , TN n , C ⊕ C , Q , ( Y n , x n ) , ( Y ∞ , x ∞ )) is a modular tunnel from ( J n , DN n , C , to ( C ∞ , DN ∞ , C , , with extent at most ε .Proof. Let ε ∈ (0 , . Let f ∈ C ( T ( d ′ ) ) , S ⊆ ℓ ( Z ( d ′ ) ) and N ∈ N be given byLemma (3.35), for ε in place of ε .By Lemma (3.32), there exists N ′ ∈ N such that if n > N ′ , and if ξ ∈ J S ∞ , then | DN k ( ̺ ∗ n ξ ) − DN ∞ ( ξ ) | ε k ξ k J ∞ .Let n > max { N, N ′ } . Let η ∈ dom (cid:0) /D n (cid:1) with DN k ( η ) . Let χ = ̺ n ( V fn ( η )) ∈ dom (cid:0) /D ∞ (cid:1) .By construction, using Lemma (3.35), DN ∞ ( χ ) (cid:16) ε (cid:17) DN n ( V fn η ) (cid:16) ε (cid:17) and k ̺ n ( η ) − χ k J ∞ = (cid:13)(cid:13) ̺ n ( η − V fn ( η )) (cid:13)(cid:13) J ∞ = (cid:13)(cid:13) η − V fn ( η ) (cid:13)(cid:13) J n ε .So (cid:13)(cid:13)(cid:13)(cid:13) ̺ n ( η ) − ε ) χ (cid:13)(cid:13)(cid:13)(cid:13) J ∞ k ̺ n ( η ) − χ k J ∞ + (cid:18) ε ε (cid:19) k χ k J ∞ ε .Therefore, TN n (cid:18) ( ε ) χ, η (cid:19) . Since TN n ( ξ, η ) > DN n ( η ) for all ξ ∈ J ∞ by construction, we conclude that DN n is the quotient of TN n by the canonicalsurjection Y n from J n ⊕ J ∞ onto J n .A similar reasoning applies to show that DN ∞ is the quotient of TN n by thecanonical surjection Y ∞ from J n ⊕ J ∞ onto J ∞ .Now, by construction, the closed unit ball of TN n is a closed subset of the productof the closed unit ball of DN n and DN ∞ , both of which are compact, and thus, theclosed unit ball of TN n is compact as well. We now consider J n ⊕ J ∞ as a Hilbert module over C ⊕ C , with inner product: h ( ξ, η ) , ( ξ ′ , η ′ ) i C ⊕ C = (cid:16) h ξ, ξ ′ i J n , h η, η ′ i J ∞ (cid:17) and ( ξ, η )( z, w ) = ( zξ, wη ) for all ( z, w ) ∈ C ⊕ C , ξ, ξ ′ ∈ J n and η, η ′ ∈ J ∞ .For all ( ξ, η ) ∈ J n ⊕ J ∞ , we compute: TN n ( ξ, η ) > max { DN n ( ξ ) , DN ∞ ( η ) } > max {k ξ k J n , k η k J ∞ } = k ( ξ, η ) k J n ⊕ J ∞ .It is also easy to check, using Cauchy-Schwarz, that: Q ( h ( ξ, η ) , ( ξ ′ , η ′ ) i C ⊕ C ) = 1 ε (cid:12)(cid:12)(cid:12) h ξ, ξ ′ i J n − h η, η ′ i J ∞ (cid:12)(cid:12)(cid:12) = 1 ε (cid:16) h ̺ n ξ, ̺ n ξ ′ i J ∞ − h η, η ′ i J ∞ (cid:17) ε (cid:16) k ̺ n ξ − η k J ∞ d k ξ ′ k J n + k η k C ∞ k ̺ n ξ ′ − η ′ k J ∞ (cid:17) TN n ( ξ, η ) TN n ( ξ ′ , η ′ ) .It is then trivial to check that the canonical surjections x n and x ∞ are quantumisometries from ( C ⊕ C , Q ) onto ( C , . Thus we have established that Expression(3.7) does define a modular tunnel, as claimed. The extent of our tunnel is no morethan ε (it is no more than the distance between the two points in the spectrum of C ⊕ C for the metric induced by the Lipschitz seminorm Q ). (cid:3) Remark . Theorem (3.36) implies that: lim n →∞ Λ ∗ mod (( J n , DN n , C , , ( J ∞ , DN ∞ , C , where Λ ∗ mod is the modular propinquity [45, 44].We now discuss how close the metrical C*-correspondence in Notation (3.30) are.All which is needed is to check that the C ⊕ C modules in the modular tunnels ofTheorem (3.36) are indeed modules over the appropriate fuzzy or quantum torus. Theorem 3.38.
Assume Hypothesis (3.1). We conclude: lim n →∞ Λ ∗ met (cid:0) qvb (cid:0) A n , J n , /D n (cid:1) , qvb (cid:0) A ∞ , J ∞ , /D ∞ (cid:1)(cid:1) = 0 .Proof. Let ε ∈ (0 , . Let N ∈ N , f ∈ C ( T ( d ′ ) ) and S ⊆ Z d ′ be given by Lemma(3.35), Let N ∈ N be given by Theorem (3.36). Let N = max { N , N } . For each n > N , let τ n be the tunnel given by Theorem (3.36), with R N adjusted so that R N restricted to J Sn is the identity.Let a ∈ dom ( L n ) and b ∈ dom ( L ∞ ) . By Expression (3.6), we have ||| [ b ◦ , R N ] ||| J ∞ ε L ∞ ( b ) .For all n ∈ N , we set ̺ n = ρ n ⊗ C .Let ξ ∈ J Sn and η ∈ J S ∞ . Since R N J ∞ ⊆ J S ′ ∞ , we then check that: k θ n ( a ) ̺ n ξ − θ ∞ ( b ) η k J ∞ = k θ n ( a ) R N ̺ n ξ − b ◦ R N η k J ∞ k θ n ( a ) R N ̺ n ξ − R N b ◦ η k J ∞ + ε L ∞ ( b ) k θ n ( a ) R N − R N b ◦ k J ∞ k η k J ∞ + k a k A ∞ k ̺ n ξ − η k J ∞ + ε L ∞ ( b ) (cid:0) (1 + ε ) T n ( a, b ) + k a, b k A n ⊕ A ∞ (cid:1) DN n ( ξ, η ) .Now, let ξ ∈ dom (cid:0) /D n (cid:1) and η ∈ dom (cid:0) /D ∞ (cid:1) . We compute k θ n ( a ) ̺ n ξ − θ ∞ ( b ) η k J ∞ k a k A n (cid:13)(cid:13) ξ − V fn ξ (cid:13)(cid:13) J n + k b k A ∞ (cid:13)(cid:13) η − V f ∞ η (cid:13)(cid:13) J ∞ + (cid:13)(cid:13) θ n ( a ) ̺ n V fn ξ − θ ∞ ( b ) V fn η (cid:13)(cid:13) J ∞ ε (cid:0) k a k A n + k b k A ∞ (cid:1) TN n ( ξ, η )+ (cid:13)(cid:13) θ n ( a ) ρ n V fn ξ − θ ∞ ( b ) V fn η (cid:13)(cid:13) J ∞ ε k ( a, b ) k A n ⊕ A ∞ TN n ( ξ, η )+ (cid:0) (1 + ε ) T n ( a, b ) + k a, b k A n ⊕ A ∞ (cid:1) (1 + ε ) TN n ( ξ, η ) (cid:0) k ( a, b ) k A n ⊕ A ∞ + T n ( a, b ) (cid:1) TN n ( ξ, η ) .Thus the extent of the metrical tunnel τ met n = ( τ mod n , τ n ) is the maximum ofthe modular tunnel τ mod n and the tunnel τ n , which is no more than ε > . Thiscompletes our proof. (cid:3) We now establish our main theorem, proving that the sequence of spectral triplesconstructed in Hypothesis (3.1) is convergent, and giving its limit.
Theorem 3.39.
If Hypothesis (3.1) is assumed, then lim n →∞ Λ spec (cid:0) ( A n , J n , /D n ) , ( A ∞ , J ∞ , /D ∞ ) (cid:1) = 0 .Proof. For all n ∈ N and t ∈ R , we set S tn = exp(2 iπ /D n ) .Let ̺ n = ρ n ⊗ C for all n ∈ N .Now, let ε ∈ (0 , . Using Lemma (3.11), there exists a function f ∈ C ( T ( d ′ ) ) ,whose Fourier transform is an element in ℓ ( Z d ′ ) supported on some finite set S ,and there exists N ∈ N such that, if n > N , then S ⊆ C n , and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ̺ n /D n ̺ − n − /D ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ ( S ) ⊗ C ε ,while, for all ξ ∈ dom (cid:0) /D n (cid:1) , (cid:13)(cid:13) ξ − V fn ξ (cid:13)(cid:13) J n ε DN n ( ξ ) while DN n ( V fn ξ ) (1 + ε ) DN n ( ξ ) .Let N ∈ N be given, for our chosen ε > , as in the proof of Theorem (3.38),adjusting T N as needed so that T N restricted to ℓ ( Z d | S ) is the identity. We usethe notation employed in this proof, as well.Let n > N = max { N , N } . Let ξ ∈ dom (cid:0) /D ∞ (cid:1) . Let η = ρ n ( V fn ( ξ )) . Wecompute: sup ζ =( ζ ,ζ ) ∈ J n ⊕ J ∞ TN n ( ζ ) (cid:12)(cid:12)(cid:10) S tn ξ, Y n ( ζ ) (cid:11) − (cid:10) S t ∞ η, Y ∞ ( ζ ) (cid:11)(cid:12)(cid:12) = sup ζ =( ζ ,ζ ) ∈ J n ⊕ J ∞ TN n ( ζ ) (cid:12)(cid:12)(cid:10) S tn ξ, ζ (cid:11) − (cid:10) S t ∞ η, ζ (cid:11)(cid:12)(cid:12) = sup ζ =( ζ ,ζ ) ∈ J n ⊕ J ∞ TN n ( ζ ) (cid:12)(cid:12)(cid:10) S tn ξ − S tn V fn ( ξ ) , ζ (cid:11) + (cid:10) S tn V fn ( ξ ) , ζ (cid:11) − (cid:10) S t ∞ η, ζ (cid:11)(cid:12)(cid:12) (cid:13)(cid:13) ξ − V fn ( ξ ) (cid:13)(cid:13) + sup ζ =( ζ ,ζ ) ∈ J n ⊕ J ∞ TN n ( ζ ) (cid:12)(cid:12)(cid:10) ρ n S tn V fn ( ξ ) , ρ n ζ (cid:11) − (cid:10) S t ∞ d η, ζ (cid:11)(cid:12)(cid:12) = ε + sup ζ =( ζ ,ζ ) ∈ J n ⊕ J ∞ TN n ( ζ ) (cid:12)(cid:12)(cid:10) ρ n S tn ρ ∗ n η − S t ∞ η, ρ n ζ (cid:11) + (cid:10) S t ∞ η, ζ − ρ n ζ (cid:11)(cid:12)(cid:12) ε + (cid:13)(cid:13)(cid:0) exp( itρ n /D n ρ ∗ n ) − exp( it /D ∞ ) (cid:1) η (cid:13)(cid:13) J ∞ .Note that, by construction, S t ∞ J S ∞ = J S ∞ , for all t ∈ R . We then compute, forall t ε , using [26, Ch. 9, eq. (2.3), p. 497] (cid:13)(cid:13)(cid:0) exp( it̺ n /D n ̺ ∗ n ) − exp( it /D ∞ ) (cid:1) η (cid:13)(cid:13) J S ∞ Z t (cid:13)(cid:13) ̺ n S t − sn ̺ ∗ n · (cid:0) ̺ n /D n ̺ ∗ n − /D ∞ (cid:1) S s ∞ η (cid:13)(cid:13) J S ∞ ds Z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ̺ n /D n ̺ ∗ n − /D ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J S ∞ ds t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ̺ n /D n ̺ ∗ n − /D ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J S ∞ tε ε ε = ε .Therefore, the magnitude of τ n is at most ε , so, for all n > N , we have Λ spec (( A n , J n , /D n ) , ( A ∞ , J ∞ , /D ∞ )) ε .This concludes our proof. (cid:3) References
1. K. Aguilar and J. Kaad,
The podleś sphere as a spectral metric space. , J. Geom. Phys. (2018), 260–278.2. K. Aguilar and F. Latrémolière,
Quantum ultrametrics on AF algebras and the Gro-mov–Hausdorff propinquity , Studia Mathematica (2015), no. 2, 149–194, ArXiv:1511.07114.3. J. Barrett,
Matrix geometries and fuzzy spaces as finite spectral triples , J. Math. Phys. (2015), no. 8, 082301, 25pp.4. J. Bhowmick, C. Voigt, and J. Zacharias, Compact quantum metric spaces from quantumgroups of rapid decay , Submitted (2014), 19 Pages, arXiv:1406.0771.5. O. Bratteli,
Inductive limits of finite dimensional C ∗ -algebras , Trans. Amer. Math. Soc. (1972), 195–234.6. M. Christ and M. A. Rieffel, Nilpotent group C ∗ -algebras-algebras as compact quantum metricspaces , Canadian Mathematical Bulletin (2017), no. 1, 77–94, ArXiv: 1508.00980.7. E. Christensen, C. Ivan, and M. Lapidus, Dirac operators and spectral triples for some fractalsets built on curves , Adv. Math. (2008), no. 1, 42–78.8. A. Connes,
C*–algèbres et géométrie differentielle , C. R. de l’Academie des Sciences de Paris(1980), no. series A-B, 290.9. A. Connes,
Compact metric spaces, Fredholm modules and hyperfiniteness , Ergodic Theoryand Dynamical Systems (1989), no. 2, 207–220.10. , Noncommutative geometry , Academic Press, San Diego, 1994.11. A. Connes, M. Douglas, and A. Schwarz,
Noncommutative geometry and matrix theory: Com-pactification on tori , JHEP (1998), hep-th/9711162.12. L. Dąbrowski and A. Sitarz,
Curved noncommutative torus and Gauß-Bonnet , Journal ofMathematical Physics (2013).
13. ,
Asymmetric noncommutative torus , Submitted (2014), 10 pages, ArXiv: 1406.4645.14. K. R. Davidson,
C*–algebras by example , Fields Institute Monographs, American Mathemat-ical Society, 1996.15. R. L. Dobrushin,
Prescribing a system of random variables by conditional probabilities , Theoryof probability and its applications (1970), no. 3, 459–486.16. D. Edwards, The structure of superspace , Studies in Topology (1975), 121–133.17. F. Latrémolière and J. Packer,
Noncommutative solenoids and the Gromov-Hausdorff propin-quity , Proceedings of the American Mathematical Society (2017), no. 5, 1179–1195,ArXiv: 1601.02707.18. O. Gabriel and M. Grensing,
Ergodic actions and spectral triples , J. Oper. Theory (2016),no. 2, 307–334.19. J. Glimm, On a certain class of operator algebras , Trans. Amer. Math. Soc. (1960),318–340.20. M. Gromov, Groups of polynomial growth and expanding maps , Publications mathématiquesde l’ I. H. E. S. (1981), 53–78.21. , Metric structures for Riemannian and non-Riemannian spaces , Progress in Mathe-matics, Birkhäuser, 1999.22. F. Hausdorff,
Grundzüge der Mengenlehre , Verlag Von Veit und Comp., 1914.23. A. Hawkins, A. Skalski, S White, and J. Zacharias,
On spectral triples on crossed productsarising from equicontinuous actions , Math. Scand. (2013), 262–291, arXiv:1103.6199.24. L. V. Kantorovich,
On one effective method of solving certain classes of extremal problems ,Dokl. Akad. Nauk. USSR (1940), 212–215.25. L. V. Kantorovich and G. Sh. Rubinstein, On the space of completely additive functions ,Vestnik Leningrad Univ., Ser. Mat. Mekh. i Astron. (1958), no. 7, 52–59, In Russian.26. T. Kato, Perturbation theory for linear operators , Springer, 1995.27. Y. Kimura,
Noncommutative gauge theories on fuzzy sphere and fuzzy torus from matrixmodel , Nuclear Phys. B (2001), no. 1–2, 121–147.28. A. Kleppner,
Multipliers on Abelian groups , Mathematishen Annalen (1965), 11–34.29. A. Kleppner,
Continuity and measurability of multiplier and projective representations , J.Funct. Anal. (1974), 214–226.30. T. Landry, M. Lapidus, and F. Latrémolière, Metric approximations of the spectral triple onthe sierpinky gasket and other fractals , Submitted, 30 pages.31. F. Latrémolière,
Approximation of the quantum tori by finite quantum tori for the quan-tum Gromov-Hausdorff distance , Journal of Functional Analysis (2005), 365–395,math.OA/0310214.32. ,
Bounded-lipschitz distances on the state space of a C*-algebra , Tawainese Journal ofMathematics (2007), no. 2, 447–469, math.OA/0510340.33. , Quantum locally compact metric spaces , Journal of Functional Analysis (2013),no. 1, 362–402, ArXiv: 1208.2398.34. ,
Convergence of fuzzy tori and quantum tori for the quantum Gromov–Hausdorffpropinquity: an explicit approach. , Münster Journal of Mathematics (2015), no. 1, 57–98,ArXiv: math/1312.0069.35. , Curved noncommutative tori as Leibniz compact quantum metric spaces , Journal ofMathematical Physics (2015), no. 12, 123503, 16 pages, ArXiv: 1507.08771.36. , The dual Gromov–Hausdorff propinquity , Journal de Mathématiques Pures et Ap-pliquées (2015), no. 2, 303–351, ArXiv: 1311.0104.37. ,
Equivalence of quantum metrics with a common domain , Journal of MathematicalAnalysis and Applications (2016), 1179–1195, ArXiv: 1604.00755.38. ,
The quantum Gromov-Hausdorff propinquity , Trans. Amer. Math. Soc. (2016),no. 1, 365–411.39. ,
A compactness theorem for the dual Gromov-Hausdorff propinquity , Indiana Univer-sity Journal of Mathematics (2017), no. 5, 1707–1753, ArXiv: 1501.06121.40. , The triangle inequality and the dual Gromov-Hausdorff propinquity , Indiana Univer-sity Journal of Mathematics (2017), no. 1, 297–313, ArXiv: 1404.6633.41. , The dual-modular Gromov-Hausdorff propinquity and completeness , Accepted in J.noncomm. geometry (2018).42. ,
The Gromov-Hausdorff propinquity for metric spectral triples , Submitted, ArXiv:1811.10843 (2018).
43. ,
Actions of categories by lipschitz morphisms on limits for the gromov-hausdorffpropinquity , J. Geom. Phys. (2019), 103481, 31 pp., ArXiv: 1708.01973.44. ,
Convergence of Cauchy sequences for the covariant Gromov-Hausdorff propinquity ,Journal of Mathematical Analysis and Applications (2019), no. 1, 378–404, ArXiv:1806.04721.45. ,
The modular Gromov–Hausdorff propinquity , Dissertationes Mathematicae (2019), 1–70, ArXiv: 1608.04881.46. ,
Convergence of Heisenberg modules for the modular Gromov-Hausdorff propinquity ,J. Oper. Theory (2020), no. 1, 211–237.47. , The covariant Gromov-Hausdorff propinquity , Studia Math. (2020), no. 2,135–169, ArXiv: 1805.11229.48. ,
Heisenberg modules over quantum -tori are metrized quantum vector bundles ,Canad. J. Math (2020), no. 4, 1044–1081, ArXiv: 1703.07073.49. H. Li, Compact quantum metric sspace and ergodic actions of compact quantum groups , J.Funct. Anal. (2009), no. 10, 3368–3408, ArXiv: 0411178.50. H. Li,
Metrics aspects of noncommutative homogenous space , J. Funct. Anal. (2009),no. 7, 2325–2350, ArXiv: 0810.4694.51. G. W. Mackey,
Unitary representations of group extensions, i , Acta Math. (1958), 265–311.52. Nathan Seiberg and Edward Witten, String theory and noncommutative geometry , JHEP (1999), no. 32, ArXiv: hep-th/9908142.53. N. Ozawa and M. A. Rieffel,
Hyperbolic group C ∗ -algebras and free product C ∗ -algebras ascompact quantum metric spaces , Canadian Journal of Mathematics (2005), 1056–1079,ArXiv: math/0302310.54. M. Reed and B. Simon, Functional analysis , Methods of modern Mathematical Physics, Aca-demic Press, San Diego, 1980.55. J. Renault,
A groupoid approach to C*–algebras , Lecture Notes in Mathematics, vol. 793,Springer-Verlag, 1980.56. M. A. Rieffel,
C*-algebras associated with irrational rotations , Pacific Journal of Mathematics (1981), 415–429.57. , Non-commutative tori — a case study of non-commutative differentiable manifolds ,Contemporary Math (1990), 191–211.58. ,
Metrics on states from actions of compact groups , Documenta Mathematica (1998),215–229, math.OA/9807084.59. , Metrics on state spaces , Documenta Math. (1999), 559–600, math.OA/9906151.60. , Group C ∗ -algebras as compact quantum metric spaces , Documenta Mathematica (2002), 605–651, ArXiv: math/0205195.61. , Gromov-Hausdorff distance for quantum metric spaces , Memoirs of the AmericanMathematical Society (2004), no. 796, 1–65, math.OA/0011063.62. ,
Matrix algebras converge to the sphere for quantum Gromov–Hausdorff distance ,Mem. Amer. Math. Soc. (2004), no. 796, 67–91, math.OA/0108005.63. ,
Distances between matrix alegbras that converge to coadjoint orbits , Proc. Sympos.Pure Math. (2010), 173–180, ArXiv: 0910.1968.64. , Leibniz seminorms for “Matrix algebras converge to te sphere" , Clay Math. Proc. (2010), 543–578.65. , Standard deviation is a strongly Leibniz seminorm , Submitted (2012), 24 pages,ArXiv: 1208.4072.66. ,
Matricial bridges for "matrix algebras converge to the sphere" , Submitted (2015), 31pages, ArXiv: 1502.00329.67. T.S. Santhanam and K. B. Sinha,
Quantum mechanics in finite dimensions , Aust. J. Phys. , 233–238.68. P. Schreivogl and H. Steinacker, Generalized fuzzy torus and its modular properties , SIGMA (2013), no. 060, 23 pages.69. G. T’Hooft, Determinism beneath quantum mechanics , Presentation at "Quo Vadis QuantumMechanics?", Temple University, Philadelphia (2002), quant-ph/0212095.70. A. Vourdas,
Quantum systems with finite hilbert space , Rep. Progr. Phys. (2004), no. 4,267–320.71. L. N. Wasserstein, Markov processes on a countable product space, describing large systemsof automata , Problemy Peredachi Infomatsii (1969), no. 3, 64–73, In Russian.
72. H. Weyl,
The theory of groups and quantum mechanics , Dover Publication, N.Y., 1950, Trans-lated from the second (revised) German edition by H. P. Robertson.73. G. Zeller-Meier,
Produits croisés d’une C*-algèbre par un groupe d’ Automorphismes , J. Math.pures et appl. (1968), no. 2, 101–239. Email address : [email protected] URL :