A survey of the Preservation of Symmetries by the Dual Gromov-Hausdorff Propinquity
aa r X i v : . [ m a t h . OA ] A ug A SURVEY OF THE PRESERVATION OF SYMMETRIES BYTHE DUAL GROMOV-HAUSDORFF PROPINQUITY
FRÉDÉRIC LATRÉMOLIÈRE
Abstract.
We survey the symmetry preserving properties for the dual propin-quity, under natural non-degeneracy and equicontinuity conditions. Theseproperties are best formulated using the notion of the covariant propinquitywhen the symmetries are encoded via the actions of proper monoids andgroups. We explore the issue of convergence of Cauchy sequences for thecovariant propinquity, which captures, via a compactness result, the fact thatproper monoid actions can pass to the limit for the dual propinquity.
The dual propinquity, a noncommutative analogue of the Gromov-Hausdorff dis-tance, enjoys some intrinsic symmetry-preserving properties, which is particularlyvaluable for the study of many types of noncommutative geometries arising fromgroup or semigroup actions, as well as the study of physical models where sym-metries play a central role. In order to capture these elusive properties, a notionof covariant propinquity can be defined, which enables a discussion of convergenceof actions of proper monoids, under the mild assumption that the actions are byLipschitz maps for an underlying quantum metric. The covariant propinquity isan interesting metric in its own right, and it contributes to our overall researchprogram, where we seek to encapsulate within certain hypertopologies on classes ofquantum structures, various basic geometric components of potential physical in-terest. This note surveys the covariance property of the dual propinquity observedin [28] and the covariant propinquity as introduced in [30, 29].Metric considerations in noncommutative metric geometry can be traced backto Connes’ introduction of spectral triples in [5], and its link with mathematicalphysics has been a matter of interest for a long time. Parallel to these observations,the metric theory of groups [10], and applications of metric geometric ideas toRiemannian manifolds centered around the Gromov-Hausdorff distance [3] haveproven to be very powerful tools. Moreover, the Gromov-Hausdorff distance forcompact metric spaces was in fact first introduced by Edwards [7], while consideringthe problem of defining a geometry for Wheeler’s superspace as an approach toquantum gravity [41]. In other words, the use of metric ideas, and specifically, oftopologizing appropriate classes of metric spaces using some version of the Gromov-Hausdorff distance, is a theme well worth exporting to noncommutative geometry.This endeavor was initiated by Rieffel [36, 37, 39], who first introduced theappropriate notion of a quantum compact metric space, inspired by Connes’ orig-inal work, and best understood as generalizing to noncommutative geometry the
Date : August 12, 2020.2000
Mathematics Subject Classification.
Primary: 46L89, 46L30, 58B34.
Key words and phrases.
Noncommutative metric geometry, Gromov-Hausdorff convergence,Monge-Kantorovich distance, quantum metric spaces, Lip-norms, proper monoids, Gromov-Hausdorff distance for proper monoids, C*-dynamical systems. metric introduced by Kantorovich [13, 14] in his study of Monge’s transportationproblem. Rieffel then defined the quantum Gromov-Hausdorff distance [39] on hisclass of quantum compact metric spaces. Some very nontrivial convergences ofnoncommutative spaces were established, including for instance finite dimensionalapproximations of quantum tori [16].However, defining a noncommutative analogue of the Gromov-Hausdorff dis-tance is not a trivial matter. For instance, Rieffel’s distance could be null betweennon-isomorphic C*-algebras, as long as these are endowed with isometric quantummetric in a natural sense. This issue grew into more than a curiosity when the studyof the behavior of structures associated to quantum compact metric spaces, such asmodules over such spaces, became a focus of the research in noncommutative metricgeometry. Indeed, it became apparent that Rieffel’s distance does not capture theentire C*-algebraic structure: in fact, the distance is defined on quantum compactmetric spaces built on top of order-unit spaces rather than algebras, and thus in-termediate estimates of Rieffel’s distance will often involve leaving the category ofC*-algebras. It should however be noted that Rieffel’s metric is very flexible andconvenient to use, and therein lies its power; it also has a clear intuitive meaningin term of distance between states, a notion at the core of C*-algebra theory andits application to physics.Several interesting alternative to Rieffel’s distance were introduced, often moti-vated by this coincidence property issue. A notable early example is due to Kerr[15], who replaced states with unital, completely positive maps from operator sys-tems to matrix algebras. Kerr then proved that distance zero between two quantumcompact metric spaces defined over C*-algebras indeed implied that the underlyingC*-algebras are *-isomorphic, in addition to the quantum metrics being isomet-ric in the sense of Rieffel. An interesting phenomenon emerged from Kerr’s work,where completeness for his metric relied on a relaxed notion of the Leibniz inequal-ity which plays no role (nor can be made sense of) in Rieffel’s notion of quantumcompact metric spaces. Yet, introducing this quasi-Leibniz condition breaks theproof of the triangle inequality as note in [15].This problem with the Leibniz, or even relaxed Leibniz, condition, again grewfrom a curious observation to a recurrent issue when working, for instance, withthe convergence of modules [40]. Eventually, it became clear that one wished fora noncommutative analogue of the Gromov-Hausdorff distance which would, atthe same time, be compatible with C*-algebras, be compatible with Leibniz orquasi-Leibniz quantum metrics, and keep the underlying interpretation of Rieffel’spioneering metric in term of states. Taken altogether, these requirements meantsolving the issues that plagued the field for its decade of existence.The resolution of this challenge came in the form of our dual Gromov-Hausdorffpropinquity [24, 21, 26, 25]. The dual propinquity addresses all the above needs byworking exclusively with quantum compact metric spaces defined over C*-algebrasand endowed with quasi-Leibniz quantum metrics, while being a complete metricup to full quantum isometry—meaning, in particular, that distance zero does im-ply *-isomorphism of the underlying C*-algebras. Since its introduction, the dualpropinquity seems to have become the metric of choice for this subject. A par-ticular version of the construction of our metric, called the quantum propinquity[24], which was in fact our first step toward this resolution, has proven particularlyvaluable.
YMMETRIES AND THE DUAL PROPINQUITY 3
Equipped with this new metric, we address some of the problems we hoped tostudy from a noncommutative metric geometric perspective, such as approximationof modules [32, 31, 27], or other structures of value when eyeing applications tomathematical physics. Such structures include symmetries and dynamics, encodedby actions of groups or semigroups on quantum compact metric spaces. It wouldappear very valuable to understand the interplay between convergence of quantumcompact metric spaces and convergence of some of their symmetries, for instance.Note that in some sense, such a study requires a Gromov-Hausdorff metric whichdoes indeed discriminate between *-automorphisms.This note surveys our progress regarding the problem of preservation of symme-tries of dynamics under convergence for the dual propinquity. We approach thisproblem by actually taking our construction of the dual propinquity further to in-clude entire proper monoid actions by Lipschitz maps, which requires just a smallamount of changes to the basic concepts—hiding the difficulties in the proofs of thebasic properties. This observation alone suggests that the dual propinquity is closeto remembering something of symmetries by itself. This is indeed vindicated by asort of Arzéla-Ascoli theorem where equicontinuity conditions ensure that conver-gence for the dual propinquity can be strengthened to convergence of entire monoidor group actions. The equicontinuity condition is expressed using natural conceptsassociated with quantum metrics and in particular, Lipschitz maps.In the first section of this survey, we introduce the category of quantum compactmetric spaces and their Lipschitz morphisms. When discussing monoid actions, wecertainly want to use the right notion of morphism, and it turns out that a naturalpicture emerges from the notion of quantum compact metric spaces. We introducethe very natural ingredient which will then be used to express our equicontinuityconditions on actions. The second section describes the dual propinquity, as thebasic object of our research. We then turn to the covariant propinquity in the lastsection. We construct this metric and then show a compactness-type result whichformalizes the notion of preservation of symmetries by the dual propinquity. Weconclude with a discussion of the completeness of the covariant propinquity and anexample of covariant metric convergence.1.
The category of quantum compact metric spaces
A quantum metric space is a noncommutative analogue of the algebra of Lips-chitz functions over a metric space. The definition of such an analogue for compactmetric spaces has evolved from an observation of Connes [5] to the corner stoneof our approach to noncommutative metric geometry in a few key steps. Mostimportantly, Rieffel observed [36, 37] that the metric on state spaces in [5] wasin essence a special case of a noncommutative generalization of the constructionof Kantorovich of a distance on Radon probabilities measures by duality from theLipschitz seminorm [13, 14]. The core features of the Kantorovich metric includethe fact that it metrizes the weak* topology, and this is taken as a starting pointfor identifying those seminorms on C*-algebras which are candidates for noncom-mutative Lipschitz seminorms.Another feature of Lipschitz seminorms is that they satisfy the Leibniz inequality.The importance of this property was not fully evident at the beginning of thestudy of quantum compact metric spaces, though the difficulties it introduces inthe study of noncommutative analogues of the Gromov-Hausdorff distance were
FRÉDÉRIC LATRÉMOLIÈRE [40]. In fact, Rieffel’s distance [39] does not require that noncommutative Lipschitzseminorms, then named Lip-norms, possess any such Leibniz properties. We onlyrealized much later that some form of connection between the Lip-norms and theunderlying multiplicative structure is in fact a means to define a noncommutativeGromov-Hausdorff distance which is zero only when the underlying C*-algebras are*-isomorphic. More importantly, as seen in our work, such a connection enablesus to push forward our program by making it possible to study group actions onquantum compact metric spaces and appropriately defined modules over quantumcompact metric spaces.To express the connection between multiplication and noncommutative Lipschitzseminorms, we introduce, inspired by [15]:
Definition 1.1.
A function F : [0 , ∞ ) → [0 , ∞ ) is permissible when F is weaklyincreasing for the product order on [0 , ∞ ) and for all x, y, l x , l y > , we have F ( x, y, l x , l y ) > xl y + yl x .Our definition for quantum compact metric spaces is thus as follows. Notation . Throughout this paper, for any unital C*-algebra A , the norm of A is denoted by k·k A , the space of self-adjoint elements in A is denoted by sa ( A ) , theunit of A is denoted by A and the state space of A is denoted by S ( A ) . We alsoadopt the convention that if a seminorm L is defined on some dense subspace of sa ( A ) and a ∈ sa ( A ) is not in the domain of L , then L ( a ) = ∞ . Definition 1.3 ([5, 36, 37, 40, 24, 25]) . An F -quantum compact metric space ( A , L ) ,for a permissible function F , is an ordered pair consisting of a unital C*-algebra A and a seminorm L , called an L-seminorm , defined on a dense Jordan-Lie subalgebra dom ( L ) of sa ( A ) , such that:(1) { a ∈ sa ( A ) : L ( a ) = 0 } = R A ,(2) the Monge-Kantorovich metric mk L defined for any two states ϕ, ψ ∈ S ( A ) by: mk L ( ϕ, ψ ) = sup {| ϕ ( a ) − ψ ( a ) | : a ∈ dom ( L ) , L ( a ) } metrizes the weak* topology restricted to S ( A ) ,(3) L satisfies the F -quasi-Leibniz inequality, i.e. for all a, b ∈ dom ( L ) : max (cid:26) L (cid:18) ab + ba (cid:19) , L (cid:18) ab − ba i (cid:19)(cid:27) F ( k a k A , k b k A , L ( a ) , L ( b )) ,(4) L is lower semi-continuous with respect to k·k A .We say that ( A , L ) is Leibniz when F can be chosen to be F : x, y, l x , l y xl y + yl x .Definition (1.3) is modeled after the following classical picture. Example 1.4.
Let ( X, d ) be a compact metric space. For any f ∈ C ( X ) , where C ( X ) is the C*-algebra of continuous C -valued functions over X , we set L ( f ) = sup (cid:26) | f ( x ) − f ( y ) | d ( x, y ) : x, y ∈ X, x = y (cid:27) ,allowing for the value ∞ . Of course, L is the usual Lipschitz seminorm on C ( X ) .Now, by [13, 14], the Monge-Kantorovich metric mk L does metrize the weak* topol-ogy on the state space S ( C ( X )) , which by Radon-Riesz theorem is the space of allRadon probability measures. Moreover, L ( f g ) k f k C ( X ) L ( g ) + L ( f ) k g k C ( X ) forall f, g ∈ C ( X ) , so ( C ( X ) , L ) is a Leibniz quantum compact metric space. YMMETRIES AND THE DUAL PROPINQUITY 5
A truly noncommutative example of a Leibniz quantum compact metric spacewas obtained by Rieffel in [36].
Example 1.5 ([36]) . Let G be a compact group with identity element e and ℓ bea continuous length function over G . Let A be a unital C*-algebra such that thereexists a strongly continuous action α of G on A . For all a ∈ A , we define: L ( a ) = sup (cid:26) k α g ( a ) − a k A ℓ ( g ) : g ∈ G \ { e } (cid:27) ,allowing for the value ∞ . The condition that α is ergodic, i.e. { a ∈ A : ∀ g ∈ G α g ( a ) = a } = C A , which is clearly necessary for ( A , L ) to be a quantumcompact metric space, is proven by Rieffel in [36] to be sufficient as well.This family of examples include: • quantum tori , where A = C ∗ ( Z d , σ ) for some multiplier σ of Z d (with d > )and with G = T d the d -torus, using the dual action α , • fuzzy tori , where G is a finite subgroup of T d , and A = C ∗ ( b G, σ ) for somemultiplier σ of G , again using the dual action as α , • noncommutative solenoids [9], where G = S p with p ∈ N \ { , } : S p = (cid:8) ( z n ) n ∈ N ∈ T N : ∀ n ∈ N z pn +1 = z n (cid:9) is the solenoid group, and where A = C ∗ (cid:16) Z h p i × Z h p i , σ (cid:17) for somemultiplier σ of Z h p i × Z h p i , and α is again the dual action. In [33], wecomputed the multipliers of Z h p i × Z h p i which is naturally homeomorphicto the solenoid group S p , and we classified all noncommutative solenoids upto *-isomorphism in terms of their multipliers, computing their K-theory.In all our examples, any continuous length function would provide a quantum com-pact metric space. We will see later on that noncommutative solenoids and quan-tum tori can be seen as elements of the closure of the class of fuzzy tori, for ourGromov-Hausdorff propinquity, if we choose appropriately compatible continuouslength functions.A completely different set of examples is given by AF algebras. Example 1.6 ([2]) . Let A = cl (cid:0)S n ∈ N A n (cid:1) be a C*-algebra which is the closure ofan increasing union of a sequence of finite dimensional C*-subalgebras ( A n ) n ∈ N , i.e.an AF algebra. We assume that A carries a faithful tracial state µ . For all n ∈ N ,there exists a unique conditional expectation E n : A ։ A n such that µ ◦ E n = µ .We then define, for all a ∈ A : L ( a ) = sup (cid:26) k a − E n ( a ) k A β ( n ) : n ∈ N (cid:27) allowing the value ∞ , and with ( β ( n )) n ∈ N = ( A n ) n ∈ N , or any choice of a se-quence of positive numbers converging to .We note that for all a, b ∈ A , we have L ( ab ) k a k A L ( b )+ L ( a ) k b k A ) . So ( A , L ) is a quantum compact metric space.There are many other important examples of quantum compact metric spaces,such as hyperbolic group C*-algebras [35], nilpotent group C*-algebras [4], other FRÉDÉRIC LATRÉMOLIÈRE quantum metrics on quantum tori [38, 20], Podlès spheres [1], various deformationsof quantum metrics [22], and more.The notion of a quantum locally compact metric space is more delicate to define,since the behavior of the Monge-Kantorovich metric is more subtle in this case [6].We introduced such a notion in [17, 18], where we also study the noncommutativebounded-Lipschitz distance.The class of quantum compact metric spaces forms the class of objects of acategory, which is helpful, for instance, in the appropriate definition for actionsof groups and semigroups on quantum compact metric spaces. There are severalnatural definitions of what a Lipschitz morphism ought to be. Maybe the one whichis at first sight the least demanding is the following.
Definition 1.7 ([23]) . Let ( A , L A ) and ( B , L B ) be two quantum compact metricspaces. A Lipschitz morphism π : ( A , L A ) → ( B , L B ) is a unital *-morphism π : A → B such that π (dom ( L A )) ⊆ dom ( L B ) .We now observe that in fact, there are important consequences to being a Lips-chitz morphism, based on the following theorem: Theorem 1.8 ([23]) . Let ( A , L ) be a quantum compact metric space, with L lowersemi-continuous with domain dom ( L ) . Let S be a seminorm on dom ( L ) such that:(1) S is lower semi-continuous with respect to k · k A ,(2) S (1 A ) = 0 .Then there exists C > such that for all a ∈ dom ( L ) : S ( a ) C L ( a ) . We thus conclude, using Theorem (1.8) and [39], that at least three naturalnotions of Lipschitz morphisms are indeed equivalent.
Theorem 1.9 ([37, 39, 23]) . Let ( A , L A ) and ( B , L B ) be two quantum compactmetric spaces and let π : A → B be a unital *-morphism. The following assertionsare equivalent:(1) π : ( A , L A ) → ( B , L B ) is a Lipschitz morphism,(2) there exists k > such that L B ◦ π k L A ,(3) there exists k > such that ϕ ∈ S ( B ) ϕ ◦ π is a k -Lipschitz map from ( S ( B ) , mk L B ) to ( S ( A ) , mk L A ) .Moreover, the real number k in Assertion (2) and Assertion (3) can be chosen tobe the same.Furthermore, the composition of two Lipschitz morphisms is a Lipschitz mor-phism. We will find it useful also allow for the notion of a Lipschitz linear map, definedas follows:
Definition 1.10 ([30]) . Let ( A , L A ) and ( B , L B ) be two quantum compact metricspaces. A Lipschitz linear map µ : ( A , L A ) → ( B , L B ) is a positive unit-preservinglinear map µ : A → B for which there exists k > such that L B ◦ µ k L A .By Theorem (1.8), and with the notations of Definition (1.10), µ : A → B isLipschitz linear if it is linear, positive, unital, and µ (dom ( L A )) ⊆ dom ( L B ) .In general, there is a natural notion of dilation (or Lipschitz constant) for Lip-schitz morphisms and more generally, Lipschitz linear maps. This notion will be YMMETRIES AND THE DUAL PROPINQUITY 7 useful in formulating equicontinuity condition later on when working with actionsof groups and semigroups.
Notation . Let ( A , L A ) and ( B , L B ) be two quantum compact metricspaces. If π : A → B is a unital positive linear map, then dil ( π ) = inf { k > ∀ a ∈ sa ( A ) L ◦ π ( a ) k L ( a ) } .By definition, dil ( π ) < ∞ if and only if π is a Lipschitz linear map.In [23], we use the notion of dilation for Lipschitz morphisms to define the non-commutative version of the Lipschitz distance. The Lipschitz distance dominatesthe dual propinquity we will review in the next section, and in fact, closed ballsfor the Lipschitz distance are compact for the dual propinquity. Of course, as inthe classical picture, the Lipschitz distance is only interesting between *-isomorphicC*-algebras endowed with various metrics, so it is much too strong for most of ourpurpose.Another useful tool when working with actions via Lipschitz morphisms will be ametric on the space of *-morphisms, or more generally unital positive maps, inducedby L-seminorms. The following result generalizes slightly the last statement of [23]. Theorem 1.12 ([23]) . Let ( A , L A ) be a quantum compact metric space and let B be a unital C*-algebra. If for any two unital linear maps α , β from A to B , we set mkD L A ( α, β ) = sup {k α ( a ) − β ( a ) k B : a ∈ dom ( L ) , L ( a ) } ,then mkD L A is a distance on the space B ( A , B ) of unit preserving bounded linearmaps, which, on any norm-bounded subset, metrizes the initial topology induced bythe family of seminorms { α ∈ B
7→ k α ( a ) k B : a ∈ A } .Proof. Let ( α n ) n ∈ N be a sequence of unit preserving linear maps converging to someunital linear map α ∞ for mkD L A ( , ) , and for which there exists some B > suchthat for all n ∈ N ∪ {∞} , we have ||| α n ||| AB B , where |||·||| AB is the operator normfor linear maps from A to B . Let a ∈ sa ( A ) and ε > . Since dom ( L A ) is dense in sa ( A ) , there exists a ′ ∈ dom ( L A ) such that k a − a ′ k A < ε B . By definition, thereexists N ∈ N such that for all n > N , we have mkD L A ( α n , α ∞ ) < ε L A ( a ′ )+1) .Thus k α ∞ ( a ) − α n ( a ) k B k α n ( a − a ′ ) k B + k α n ( a ′ ) − α ∞ ( a ′ ) k B + k α ∞ ( a − a ′ ) k B < ε .Thus for all a ∈ sa ( A ) , the sequence ( α n ( a )) n ∈ N converges to α ∞ ( a ) . By linearity,we then conclude ( α n ( a )) n ∈ N converge to α ∞ ( a ) for k·k B .Conversely, assume that for all a ∈ A , the sequence ( α n ( a )) n ∈ N converges to α ∞ ( a ) in B , and again assume that there exists B > such that for all n ∈ N ∪{∞} ,we have ||| α n ||| AB B . Let ε > and fix µ ∈ S ( A ) . As L A is a L-seminorm, L = { a ∈ sa ( A ) : L A ( a ) , µ ( a ) = 0 } is totally bounded. Thus, there existsa finite ε B -dense set F ⊆ L of L . As F is finite, by assumption, there exists N ∈ N such that for all n > N and all a ∈ F , we have k α n ( a ) − α ∞ ( a ) k B < ε .If n > N and a ∈ sa ( A ) such that L A ( a ) then there exists a ′ ∈ F such that k a − µ ( a )1 A − a ′ k A < ε B , and thus k α n ( a ) − α ∞ ( a ) k B k α n ( a − µ ( a )1 A ) − α ∞ ( a − µ ( a )1 A ) k B k α n ( a − µ ( a )1 A − a ′ ) k B FRÉDÉRIC LATRÉMOLIÈRE + k α n ( a ′ ) − α ∞ ( a ′ ) k B + k α ∞ ( a − µ ( a )1 A ) − a ′ k B B ε B + ε B ε B < ε .Thus for n > N , we have mkD L A ( α n , α ∞ ) < ε . (cid:3) There are of course at least four common notions of morphisms over the cate-gory of metric spaces: continuous functions, uniformly continuous functions, Lips-chitz functions, and isometries. Continuous functions correspond to *-morphism ofcourse. As we work with compact metric space, uniform continuity and continuityare equivalent. We just define the notion of Lipschitz morphism. For our work, wealso will need to understand what a quantum isometry should be.Rieffel observed that McShane’s theorem [34] on extension of real-valued
Lips-chitz functions can be used to characterize isometries. The emphasis on real-valuedLipschitz functions, rather than complex valued, means for our purpose that the iso-metric property will only involve self-adjoint elements. The definition of a quantumisometry is thus given by:
Definition 1.13 ([37, 39]) . Let ( A , L A ) and ( B , L B ) be quantum compact metricspaces. • A quantum isometry π : ( A , L A ) → ( B , L B ) is a *-epimorphism from A onto B such that for all b ∈ dom ( L A ) L B ( b ) = inf { L A ( a ) : π ( a ) = b } . • A full quantum isometry π : ( A , L A ) → ( B , L B ) is a *-isomorphism from A onto B such that L B ◦ π = L A .We observe that in [39], Rieffel proved that quantum isometries can be chosenas morphisms for a category over quantum compact metric spaces; this is a subcat-egory of the category of quantum compact metric spaces with Lipschitz morphisms(as quantum isometries are obviously -Lipschitz morphisms).The notion of full quantum isometry is essential to our work: it is the notionwhich we take as the basic equivalence between quantum compact metric spaces,i.e. two quantum compact metric spaces are, for our purpose, the same when theyare fully quantum isometric.2. The Gromov-Hausdorff Propinquity
The Gromov-Hausdorff distance [11, 10] is a complete metric up to full isometryon the class of proper metric spaces, initially described by Edwards for compactmetric spaces [7]. This metric is an intrinsic version of the Hausdorff distance [12].It is constructed by taking the infimum of the Hausdorff distance between any twoisometric copies of given compact metric spaces. By duality, as we have a notion ofquantum isometry, we obtain a notion of something we shall call a tunnel betweentwo quantum compact metric spaces.
Definition 2.1 ([21]) . Let ( A , L ) and ( A , L ) be two F -quantum compact metricspaces for a permissible function F . An F -tunnel τ = ( D , L , π , π ) from ( A , L ) to ( A , L ) is a F –quasi-Leibniz quantum compact metric space ( D , L ) and for each j ∈ { , } , a quantum isometry π j : ( D , L ) ։ ( A j , L j ) . The space ( A , L ) is the domain dom ( τ ) of τ , while the space ( A , L ) is the codomain codom ( τ ) of τ . YMMETRIES AND THE DUAL PROPINQUITY 9 ( Z, d Z )( X, d X ) +(cid:11) ι X sssssssss ( Y, d Y ) ι Y e e ❏❏❏❏❏❏❏❏❏ Figure 1.
Isometric Embeddings ( D , L D ) π A y y y y ttttttttt π B % % % % ❑❑❑❑❑❑❑❑❑❑ ( A , L A ) ( B , L B ) Figure 2.
A tunnel ( S ( D ) , mk L D )( S ( A ) , mk L A ) ) (cid:9) π ∗ A ♠♠♠♠♠♠♠♠♠♠♠♠♠ ( S ( B ) , mk L B ) π ∗ B h h ◗◗◗◗◗◗◗◗◗◗◗◗◗ Figure 3.
Isometric Embeddings of state spaces induced by tunnelsWe need to associate a quantity to any given tunnel which estimates how farapart its domain and codomain are. The following quantity is the choice whichgives us the best construction of our propinquity [26], though a certain alternativemethod can be found in [21] where the dual propinquity was originally developed.
Definition 2.2 ([26]) . The extent χ ( τ ) of a tunnel τ = ( D , L , π , π ) is given asthe real number max j ∈{ , } Haus mk L ( S ( D ) , { ϕ ◦ π j : ϕ ∈ S ( A j ) } ) .Tunnels between two quantum compact metric spaces involve a third one, andwe have to choose what properties, if any, this third quantum compact metric spaceshould possess. The most important restriction we must impose for our theory towork is a particular choice of permissible function, i.e. of a quasi-Leibniz inequalitysatisfied by all three L-seminorms. The key is that the choice must be uniformthroughout our construction of the propinquity, so that the propinquity is indeed ametric up to full quantum isometry. The quasi-Leibniz inequality is used to obtainthe multiplicative property.There may be situations which require more properties for L-seminorms. Anexample is the strong Leibniz property introduced by Rieffel in [40]. Thus, it ishelpful to keep some level of generality in our construction by allowing flexibility inrestricting the class of tunnels used, beyond the quasi-Leibniz restriction. In orderfor our construction to lead to a metric, we ask that a chosen class of tunnel meetthe following definition. Definition 2.3 ([26]) . Let F be a permissible function. A class T of F -tunnels is appropriate for a nonempty class C of F -quantum compact metric spaces when(1) for every τ ∈ T , we have dom ( τ ) , codom ( τ ) ∈ C ,(2) for every A , B ∈ C , there exists τ ∈ T with domain A and codomain B ,(3) if τ = ( D , L , π, ρ ) ∈ T then τ − = ( D , L , ρ, π ) ∈ T ,(4) if ε > , and if τ , τ ∈ T with codom ( τ ) = dom ( τ ) , then there exists τ ∈ T with dom ( τ ) = dom ( τ ) , codom ( τ ) = codom ( τ ) , and χ ( τ ) χ ( τ ) + χ ( τ ) + ε ,(5) if ( A , L A ) , ( B , L B ) are in C and if there exists a full quantum isometry π : ( A , L A ) → ( B , L B ) , then ( A , L A , π, id) ∈ T , with id the identity *-automorphism of A .We proved in [26] that the class of all F -tunnels is appropriate for the class ofall F -quantum compact metric spaces, and this is, for our own work, the sort ofclass we work with (note that we must impose the restriction to work with tunnelswhich all share the same quasi-Leibniz inequality, as parameterized by F ). Notation . Let T be a class of tunnels appropriate for a nonempty class of F -quantum compact metric spaces C . The set of all F -tunnels in T from ( A , L A ) to ( B , L B ) , both chosen in C , is denoted by Tunnels h ( A , L A ) T −→ ( B , L B ) i .When T is simply the class of all F -tunnels, we then write the class of all F -tunnelsfrom ( A , L A ) to ( B , L B ) , for any two F -quantum compact metric spaces A , B , by Tunnels h ( A , L A ) F −→ ( B , L B ) i .The dual Gromov-Hausdorff propinquity is now constructed using the same tech-nique as Edwards’ and Gromov’s. It enjoys many good properties, among which isbeing a metric up to full quantum isometry. Theorem-Definition 2.5 ([24, 21, 26, 25]) . Let T be a class of tunnels appropriatefor a nonempty class of F -quantum compact metric spaces for some permissiblefunction F . We define the dual T -propinquity between any two quantum compactmetric spaces ( A , L ) and ( B , L ) in C as the real number Λ ∗T (( A , L A ) , ( B , L B )) = inf n χ ( τ ) : τ ∈ Tunnels h ( A , L A ) T −→ ( B , L B ) io .The dual propinquity is a metric up to quantum full isometry on C , i.e. it is apseudo-metric on C such that any A , B ∈ C are fully quantum isometry if and onlyif Λ ∗T ( A , B ) = 0 .Moreover, if ( X, d X ) and ( Y, d Y ) are two compact metric spaces and if L X and L Y are the Lipschitz seminorms induced respectively on C ( X ) by d X and C ( Y ) by d Y , and if GH is the usual Gromov-Hausdorff distance, then GH((
X, d X ) , ( Y, d Y )) Λ ∗T (( C ( X ) , L X ) , ( C ( Y ) , L Y )) ,as long as ( C ( X ) , L X ) , ( C ( Y ) , L Y ) ∈ C , and the topology induced on the class ofclassical metric spaces in C via Λ ∗ T is the same as the topology induced by theGromov-Hausdorff distance.Last, let us denote by Λ ∗ F the dual propinquity induced on all F -quantum compactmetric spaces using all possible F -tunnels. Then Λ ∗ F is complete if F is continuous. YMMETRIES AND THE DUAL PROPINQUITY 11
Thus in particular, Λ ∗ F is a complete metric up to full quantum isometry whichinduces the same topology as the Gromov-Hausdorff distance on classical compactmetric spaces. This is the main tool for our research.We note that completeness is a desirable property for obvious reasons, includingthe study of compactness. We derive an analogue of Gromov’s compactness theoremin [25].Now, in order to actually prove convergence results for the dual propinquity,it is of course desirable to have a prolific source of tunnels for any two quantumcompact metric spaces. We actually first discovered this source in [24] before weintroduced tunnels [21]. The idea behind a bridge could be said to be a far-reachinggeneralization of the idea of an intertwiner between two *-representations of a C*-algebra, though we now work with representations of two different C*-algebras,restricting ourselves to faithful unital representations on the same space, and wewill learn to measure how good of an “approximate intertwiner” a particular bridgeis. This informal idea begins with the following definition. Definition 2.6 ([24]) . Let A and A be two unital C*-algebras. A bridge γ =( D , x, π , π ) is a given by(1) a unital C*-algebra D ,(2) for each j ∈ { , } , a unital *-monomorphism π j : A j ֒ → D ,(3) an element x ∈ D , called the pivot of γ , for which there exists a state ϕ ∈ S ( D ) such that ϕ ((1 − x ) ∗ (1 − x )) = ϕ ((1 − x )(1 − x ) ∗ ) = 0 .The domain dom ( γ ) of γ is A while the codomain codom ( γ ) of γ is A .To measure how far apart the domain and codomain of a bridge are, the followingtwo objects which arise immediately from our definition will be helpful. Notation . Let γ = ( D , x, π A , π B ) be a bridge from A to B , where A and B are unital C*-algebras. The -level set of x in D is the set of states S ( D | x ) = { ϕ ∈ S ( D ) | ϕ ((1 − x ) ∗ (1 − x )) = ϕ ((1 − x )(1 − x ) ∗ ) = 0 } .Moreover, we define a seminorm on A ⊕ B by setting for all a ∈ A , b ∈ B : bn γ ( a, b ) = k π A ( a ) x − xπ B ( b ) k D .We now associate a number to our bridge. In the following definition, it may behelpful to think of the height as measuring how far the pivot is from the identity ina manner employing the quantum metrics. On the other hand, the reach measureshow far apart the domain and the codomain are using our almost intertwiner, thepivot, and the seminorm it defines. Importantly, all the quantities used to quantifya bridge involve the quantum metrics, which are not used in defining the bridgeitself. Definition 2.8 ([24]) . Let ( A , L ) and ( A , L ) be two quantum compact metricspace. If γ = ( D , x, π , π ) is a bridge from A to A then:(1) the height ς ( γ | L A , L B ) of γ is the real number max j ∈{ , } Haus mk L j ( S ( A j ) , { ϕ ◦ π j : ϕ ∈ S ( D | x ) } ) ,(2) the reach ̺ ( γ | L , L ) of γ is the real number Haus bn γ ( { ( a,
0) : a ∈ sa ( A ) , L ( a ) } , { (0 , b ) : b ∈ sa ( A ) , L ( b ) } ) ,(3) the length λ ( γ | L , L ) of γ is max { ς ( γ | L , L ) , ̺ ( γ | L , L ) } . We now see that bridges provide a mean to build tunnels. This is how most non-trivial tunnels are constructed in our work so far. Maybe a main reason for thisfact is that constructing L-seminorms is usually delicate, but bridges provide suchL-seminorms in a systematic manner and moreover, their length gives an estimateon the extent of the resulting tunnel.
Theorem 2.9 ([24, 21]) . Let ( A , L A ) and ( B , L B ) be two F -quantum compact met-ric spaces for some permissible function F . If γ = ( D , x, π A , π B ) is a bridge from A to B , if ε > is chosen so that λ = λ ( γ | L A , L B ) + ε > , and if, for all ( a, b ) ∈ sa ( A ) ⊕ sa ( B ) , we set L ( a, b ) = max (cid:26) L A ( a ) , L B ( b ) , λ bn γ ( a, b ) (cid:27) ,then ( A ⊕ B , L , ρ A , ρ B ) , where ρ A : ( a, b ) ∈ A ⊕ B a and ρ B : ( a, b ) ∈ A ⊕ B b ,is an F -tunnel from ( A , L A ) to ( B , L B ) , of extent at most λ .In particular Λ ∗ F (( A , L A ) , ( B , L B )) λ ( γ | L A , L B ) . We used the construction of appropriate bridges to prove the following examplesof convergence for the dual propinquity.
Example 2.10 ([21]) . Let ℓ be a continuous length function on T d . For any G ⊆ T d a closed subgroup and σ a multiplier of the Pontryagin dual b G of G , forany a ∈ C ∗ ( b G, σ ) , we set as in Example (1.5): L G,σ ( a ) = sup ( k α g ( a ) − a k C ∗ ( b G,σ ) ℓ ( g ) : g ∈ G \ { } ) where α is the dual action of G on C ∗ ( b G, σ ) .If ( G n ) n ∈ N is a sequence of closed subgroups of T d converging to T d for the Haus-dorff distance Haus ℓ , and if ( σ n ) n ∈ N is a sequence of multipliers of Z d convergingpointwise to some σ , with σ n ( g ) = 1 if g is the coset of for c G n , then: lim n →∞ Λ ∗ (( C ∗ ( c G n , σ n ) , L c G n ,σ n ) , ( C ∗ ( Z d , σ ) , L Z d ,σ )) = 0 .In particular, the function which maps a multiplier to a quantum torus is con-tinuous for the dual propinquity, and quantum tori are limits of fuzzy tori. Example 2.11 ([9]) . Noncommutative solenoids are limits of quantum tori, andconsequently, limits of fuzzy tori, for the dual propinquity, for the appropriatechoice of a metric on the solenoid groups.
Example 2.12 ([2]) . We use the same notation as in Example (1.6). We thenhave: • ( A , L ) = Λ ∗ − lim n →∞ ( A n , L ) , • the natural map from the Baire space to UHF algebras is Lipschitz.Another example is given by the Effros-Shen algebras [8]. For θ ∈ (0 , \ Q , let θ = lim n →∞ p θn q θn with p θn q θn = 1 a + 1 a + 1 .. . + a n for a , . . . ∈ N . YMMETRIES AND THE DUAL PROPINQUITY 13
Set AF θ = lim −→ n →∞ (cid:0) M q n ⊕ M q n − , ψ n,θ (cid:1) where ψ n,θ is defined by: ( a, b ) ∈ M q n ⊕ M q n − a . . . a b , a .Let L θ the L-seminorm for this data as in Example (1.6). For all θ ∈ (0 , \ Q ,we have: lim ϑ → θϑ Q Λ ∗ (( AF ϑ , L ϑ ) , ( AF θ , L θ )) = 0 .A final remark in this section concerns the following question: could we choose,in the construction of the dual propinquity, only those tunnels which emerge frombridges as in Theorem (2.9)? We do not know, nor believe, that tunnels obtainedfrom bridges form an appropriate class with the class of all F -quantum compactmetric spaces for any particular permissible function F . The problem can besummed up by saying that given two bridges γ and γ with codom ( γ ) = dom ( γ ) ,we do not know how to build a single bridge whose length is approximately the sumof the lengths of γ and γ , going from dom ( γ ) to codom ( γ ) . This, in turn, meansthat our construction of the propinquity would fail to satisfy the triangle inequality.This issue was nontrivial and is the tip of the iceberg regarding the difficulties ofworking with Leibniz or quasi-Leibniz seminorms.However, [26, Theorem 3.1] does essentially show us how to take the “closure” ofthe class of all tunnels-from-bridges to obtain an appropriate class for all quantumcompact metric spaces. It turns out that in the case of tunnels constructed frombridges, [26, Theorem 3.1] can be seen as introducing between γ and γ , as above,a very short bridge from codom ( γ ) to dom ( γ ) . Thus, [26, Theorem 3.1] dictatesthat we really want to work, not just with bridges, but with tunnels built by afinite collection of bridges, each ending where the next starts. The very short in-between bridges can in fact be taken so short as to have length zero in this case,and disappear—leaving us with the idea of treks which we used, in our first workon the propinquity [24], to define a first form of the propinquity called the quantumpropinquity.We note that the length of a bridge dominates the extent of its canonicallyassociated tunnel by Theorem (2.9), so even reconciling the idea of treks with theidea of almost composition of tunnels as in [26, Theorem 3.1] does not mean thequantum propinquity, which uses the length of bridges directly rather than theextent of the associated tunnels, is equal to the dual propinquity—the former stilldominates the latter as far as we can tell. But the two pictures are now closertogether.As is seen for instance with quantum tori, group actions can be used as a means oftransport of structure to define a noncommutative geometry. Semigroups, or rathersemigroups of completely positive unital maps, can be interpreted as a form of non-commutative heat semigroups, and give rise to a differential calculus where the gen-erator of the semigroup is a noncommutative Laplacian. Both are very interestingand important models for noncommutative geometry. More generally, symmetrieshave been a central concept in the development of mathematical physics modelsin particle physics, so, with our motivation for this research program in mind, wewant to understand: what is the interplay between symmetries as encoded in group actions, and dynamics encoded as group or semigroup actions, and the dual propin-quity, and our approach to noncommutative metric geometry? There are actuallysome rather pleasant facts regarding these matters, and we begin by describing howto capture group and semigroup actions in our metric framework—only to see latera nice compactness-type result which shows that the dual propinquity is keen toremember some symmetries of spaces, under natural non-degeneracy conditions.3. The Covariant Propinquity
We begin by defining the objects on which we are going to define an extension ofthe dual propinquity. The idea is to bring together a metrized group or semigroup,a quantum compact metric space, and an action of the former on the latter byLipschitz morphisms, or at least Lipschitz linear maps.We thus begin with:
Definition 3.1. A proper monoid ( G, δ ) is a monoid (i.e. a set G endowed withan associative binary operation and with an identity element) and a left-invariantdistance δ on G whose closed balls are all compact.For any two proper monoids ( G, δ G ) and ( H, δ H ) , a proper monoid morphism π : ( G, δ G ) → ( H, δ H ) is a map from G to H such that: • π maps the identity element of G to the identity element of H , • ∀ g, h ∈ G π ( gh ) = π ( g ) π ( h ) , • π is continuous. Definition 3.2 ([30]) . Let F be a permissible function. A Lipschitz dynamical F -system ( A , L , G, δ, α ) is given by:(1) an F -quantum compact metric space ( A , L ) ,(2) a proper monoid ( G, δ ) ,(3) a strongly continuous action α of G on A : for all a ∈ A and g ∈ G , we have: lim h → g (cid:13)(cid:13) α h ( a ) − α g ( a ) (cid:13)(cid:13) A = 0 ,(4) g ∈ G dil ( α g ) is locally bounded: for all ε > and g ∈ G there exist D > and a neighborhood U of g in G such that if h ∈ U then dil (cid:0) α h (cid:1) D .A Lipschitz C ∗ -dynamical F -system ( A , L , G, δ, α ) is a Lipschitz dynamical sys-tem where G is a proper group and α g is a Lipschitz unital *-automorphism for all g ∈ G .We now wish to endow classes of Lipschitz dynamical systems with a sort ofdual propinquity. To this end, we first must understand how to define a Gromov-Hausdorff distance for proper monoids. We propose the following construction,which encompasses natural ideas, but in a manner which defines a nice metric. Notation . For a metric space ( X, δ ) , if x ∈ X and r > , then the closed ball in ( X, δ ) centered at x , of radius r , is denoted X δ [ x, r ] , or simply X [ x, r ] . If ( G, δ ) isa metric monoid with identity element e ∈ G , and if r > , then G [ e, r ] is denotedby G [ r ] .We define our distance between two proper metric monoids ( G , δ ) and ( G , δ ) by measuring how far a given pair of maps ς : G → G and ς : G → G arefrom being an isometric isomorphism and its inverse. YMMETRIES AND THE DUAL PROPINQUITY 15
Definition 3.4 ([30]) . Let ( G , δ ) and ( G , δ ) be two metric monoids with iden-tity elements e and e . An r -local ε -almost isometric isomorphism ( ς , ς ) , for ε > and r > , is an ordered pair of maps ς : G [ r ] → G and ς : G [ r ] → G such that for all { j, k } = { , } : ∀ g, g ′ ∈ G j [ r ] ∀ h ∈ G k [ r ] | δ k ( ς j ( g ) ς j ( g ′ ) , h ) − δ j ( gg ′ , ς k ( h )) | ε ,and ς j ( e j ) = e k .The set of all r -local ε -almost isometric isomorphism is denoted by: UIso ε (( G , δ ) → ( G , δ ) | r ) .Our covariant Gromov-Hausdorff distance over the class of proper metric monoidsis then defined along the lines of Gromov’s distance. The bound √ is just hereto ensure that our metric satisfies the triangle inequality. Our construction followsGromov’s insight on how to define an intrinsic Hausdorff distance between pointed,proper spaces—rather than the Edwards definition we used for quantum compactmetric spaces—where we chose as base point the identity elements. Definition 3.5 ([30]) . The
Gromov-Hausdorff monoid distance
Υ(( G , δ ) , ( G , δ )) between two proper metric monoids ( G , δ ) and ( G , δ ) is given by: Υ(( G , δ ) , ( G , δ )) =min ( √ , inf (cid:26) ε > (cid:12)(cid:12)(cid:12)(cid:12) UIso ε (cid:18) ( G , δ ) → ( G , δ ) (cid:12)(cid:12)(cid:12)(cid:12) ε (cid:19) = ∅ (cid:27)) .We then record: Theorem 3.6 ([30]) . For any proper metric monoids ( G , δ ) , ( G , δ ) and ( G , δ ) :(1) Υ(( G , δ ) , ( G , δ )) √ ,(2) Υ(( G , δ ) , ( G , δ )) = Υ(( G , δ ) , ( G , δ )) ,(3) Υ(( G , δ ) , ( G , δ )) Υ(( G , δ ) , ( G , δ )) + Υ(( G , δ ) , ( G , δ )) ,(4) If Υ(( G , δ ) , ( G , δ )) = 0 if and only if there exists a monoid isometricisomorphism from ( G , δ ) to ( G , δ ) .In particular, Υ is a metric up to metric group isometric isomorphism on the classof proper metric groups.Moreover, if ( G, δ G ) and ( H, δ H ) are two proper metric monoids with units e G and e H then: GH((
G, δ G , e G ) , ( H, δ H , e H )) Υ((
G, δ G ) , ( H, δ H )) ,where GH is the Gromov-Hausdorff distance for pointed, proper metric spaces [10] . We now have a metric on the class of proper monoids and a metric on the classof quantum compact metric spaces—the dual propinquity discussed in the previoussection. We want to bring them together. We propose to merge the notion of tunneland the notion of almost isometry as follows.
Definition 3.7 ([30]) . Let ε > and F be a permissible function. Let ( A , L , G ,δ , α ) and ( A , L , G , δ , α ) be two Lipschitz dynamical F -systems. A ε -covariant F -tunnel τ = ( υ, ς , ς ) from ( A , L , G , δ , α ) to ( A , L , G , δ , α ) is given by an F -tunnel υ from ( A , L ) to ( A , L ) and a pair ( ς , ς ) ∈ UIso ε (cid:18) ( G , δ ) → ( G , δ ) (cid:12)(cid:12)(cid:12)(cid:12) ε (cid:19) .We make two remarks. First, a covariant tunnel does not involve any action onthe underlying tunnel: the actions of the domain and codomain are not involvedin the definition of the covariant tunnels themselves. We will include these actionsin our quantification of a covariant tunnel later on. Second, we do work with aquantified almost isometry, rather than just a pair of unit-preserving maps. Thisconstruct is the path we use to define the covariant propinquity, as it seems to makeit easiest to prove such properties as the triangle inequality.We now quantify covariant tunnels. Of course, covariant tunnels come with anumber which is related to the metric Υ by definition, and we also have the extentof the underlying tunnel available to us. What is left is to involve the actual actionsin some measurement. The following concept is a generalization of the reach of atunnel as defined in [21]. Definition 3.8 ([30]) . Let ε > . Let A = ( A , L , G , δ , α ) and A = ( A , L ,G , δ , α ) be two Lipschitz dynamical systems. The ε -reach ρ ( τ | ε ) of a ε -covarianttunnel τ = ( D , L D , π , π , ς , ς ) from A to A is given as: max { j,k } = { , } sup ϕ ∈ S ( A j ) inf ψ ∈ S ( A k ) sup g ∈ G j [ ε ] mk L D ( ϕ ◦ α gj ◦ π j , ψ ◦ α ς j ( g ) k ◦ π k ) We now bring all the data we have so far on covariant tunnels into one quantity.
Definition 3.9 ([30]) . The ε -magnitude µ ( τ | ε ) of a ε -covariant tunnel τ is themaximum of its ε -reach and its extent: µ ( τ | ε ) = max { ρ ( τ | ε ) , χ ( τ ) } .As with the dual propinquity, we have a natural notion of an appropriate classof covariant tunnels. Definition 3.10.
Let F be a permissible function. Let C be a nonempty class ofLipschitz dynamical F -systems. A class T of covariant F -tunnels is appropriate for C when:(1) for all A , B ∈ C , there exists a ε -covariant tunnel from A to B for some ε > ,(2) if τ ∈ T , then there exist A , B ∈ C such that τ is a covariant tunnel from A to B ,(3) if A = ( A , L A , G, δ G , α ) , B = ( B , L B , H, δ H , β ) are elements of C , and ifthere exists an equivariant full quantum isometry ( π, ς ) from A to B , then: (cid:0) A , L A , id A , π, ς, ς − (cid:1) , (cid:0) B , L B , π − , id B , ς − , ς (cid:1) ∈ T , where id A , id A are the identity *-automorphisms of A and B ,(4) if τ = ( D , L , π, ρ, ς, κ ) ∈ T then τ − = ( D , L , ρ, π, κ , ς ) ∈ T ,(5) if ε > and if τ , τ ∈ T are √ -tunnels, then there exists δ ∈ (0 , ε ] suchthat τ ◦ δ τ ∈ T . YMMETRIES AND THE DUAL PROPINQUITY 17
Notation . Let T be a class of covariant tunnels appropriate for a nonemptyclass of Lipschitz F -dynamical systems C . The set of all ε -covariant F -tunnels in T , for any ε > , from A to B , both chosen in C , will be denoted by: Tunnels h A T −→ B (cid:12)(cid:12)(cid:12) ε i .When T is simply the class of all Lipschitz F -dynamical systems, we shall thenwrite the class of all F -tunnels from A to B as: Tunnels h A F −→ B (cid:12)(cid:12)(cid:12) ε i .We now can define the covariant propinquity between Lipschitz dynamical sys-tems. Definition 3.12 ([30]) . Let C be a nonempty class of Lipschitz F -dynamical sys-tems for a permissible function F and let T be a class of covariant tunnels ap-propriate for C . For A , B ∈ C , the covariant T -propinquity Λ cov T ( A , B ) is definedas: min ( √ , inf n ε > (cid:12)(cid:12)(cid:12) ∃ τ ∈ Tunnels h A T −→ B (cid:12)(cid:12)(cid:12) ε i µ ( τ | ε ) ε o) .Definition (3.12) indeed defines a metric up to the equivariant full quantumisometries: Theorem 3.13 ([30]) . Let C be a nonempty class of Lipschitz F -dynamical systemsfor a permissible function F and let T be a class of covariant tunnels appropriatefor C . If ( A , L A , G, δ G , α ) and ( B , L B , H, δ H , β ) in C then: Λ cov T (( A , L A , G, δ G , α ) , ( B , L B , H, δ H , β )) = 0 if and only if there exists a full quantum isometry π : ( A , L A ) → ( B , L B ) and anisometric isomorphism of monoids ς : G → H such that: ∀ g ∈ G ϕ ◦ α g = β ς ( g ) ◦ ϕ ,i.e. ( A , L A , G, δ G , α ) and ( B , L B , H, δ H , β ) are isomorphic as Lipschitz dynamicalsystems. We now wish to explore the issue of completeness of the covariant propinquity.The important observation in the following theorem is if we start from a convergentsequence of F -quantum compact metric spaces for the dual propinquity, and someconverging sequence of proper monoids for Υ , and if the monoids act on the quantumcompact metric spaces entry-wise, then a simple equicontinuity condition, expressedusing the notion of dilation for Lipschitz morphisms, is all that is required to geta subsequence of the sequence of Lipschitz dynamical systems thus constructed toconverge for the covariant propinquity. In other words, the dual propinquity wantsto remember symmetries, as long as they do not degenerate. This idea is capturedin its full power in our work in [28], which establishes a very general result regardingsemigroupoid actions. The following theorem is a consequence of [28] and capturesthis idea formally. Theorem 3.14 ([29]) . Let ( A , L ) be an F -quantum compact metric space and let ( G, δ ) be a proper monoid. Let ( A n , L n , G n , δ n , α n ) n ∈ N be a sequence of Lipschitzdynamical systems and let D : [0 , ∞ ) → [0 , ∞ ) be a locally bounded function suchthat: (1) for all n ∈ N and g ∈ G n , we have dil ( α gn ) D ( δ n ( e n , g )) ,(2) lim n →∞ Υ(( G n , δ n ) , ( G, δ )) = 0 ,(3) lim n →∞ Λ cov (( A n , L n ) , ( A , L )) = 0 ,(4) for all ε > , there exists ω > and N ∈ N such that if n > N and if g, h ∈ G n with δ n ( g, h ) < ω , then mkD L n (cid:0) α gn , α hn (cid:1) < ε .Then there exists a strictly increasing function j : N → N and a Lipschitz dynamicalsystem ( A , L , G, δ, α ) such that: Λ cov (( A j ( n ) , L j ( n ) , G j ( n ) , δ j ( n ) , α j ( n ) ) , ( A , L , G, δ, α )) n →∞ −−−−→ .Moreover: • if for all n ∈ N , the map α n is a *-endomorphism, then α is also a *-endomorphism, • if for all n ∈ N , the monoid G n is a proper group, the map α n is a fullquantum isometry, then α is also a full quantum isometry, • if for all n ∈ N , the monoid G n is a compact group, and if the action α n is ergodic, then α is ergodic as well. Before we apply Theorem (3.14) to finding sufficient conditions on Cauchy se-quences of Lipschitz dynamical systems, we observe some of its more immediateimportant consequences.It is usually very difficult to determine the closure, for the dual propinquity, ofa given set of quantum compact metric spaces. For instance, note that quantumtori are all members of the closure of all Leibniz quantum compact metric spacesover finite dimensional C*-algebras by [19]. In that same closure, one also findsall classical compact metric spaces, and noncommutative solenoids. Relaxing theLeibniz inequality to work within some also quasi-Leibniz class of quantum com-pact metric spaces (specifically, the so-called (2 , -quasi-Leibniz quantum compactmetric spaces of [25]), we then find that the closure of finite dimensional (2 , –quasi-Leibniz quantum compact metric spaces for the dual propinquity contains allAF algebras [2], and all Leibniz quantum compact metric spaces whose underlyingC*-algebras are nuclear quasi-diagonal by [25]. One technique to compute closuresis to invoke a compactness result. For instance, various classes of AF algebras areshown to be compact for the dual propinquity in [2]. Theorem (3.14) offers anothertechnique.As an example [28], let M be the class of all finite dimensional Leibniz quantumcompact metric spaces carrying an ergodic action of SU (2) by quantum isometries.Then by Theorem (3.14), any limit of any convergent sequence in M must alsocarry an ergodic action of SU (2) , so it must be of type I—in fact, it must be abundle of matrix algebras over a homogeneous space for SU (2) . This is a verynontrivial observation showing the power of Theorem (3.14).To study the completeness of the covariant propinquity, it is helpful to first have ageneral idea of what conditions on Cauchy sequences for Υ make them convergent.The condition we exhibit is a form of equicontinuity for right translations. Toformulate this condition, we write: Y n ∈ N G n = { ( g n ) n ∈ N : ∃ M > ∀ n ∈ N g n ∈ G n [ M ] } .Our equicontinuity condition will be expressed using the following notion ofregularity. YMMETRIES AND THE DUAL PROPINQUITY 19
Definition 3.15 ([29]) . Let ( G n , δ n ) n ∈ N be a sequence of proper monoids. Theset of regular sequences R (( G n , δ n ) n ∈ N ) is: ( g n ) n ∈ N ∈ Y n ∈ N G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∀ ε > ∃ ω > ∃ N ∈ N ∀ n > N ∀ h, k ∈ G n δ n ( h, k ) < ω = ⇒ δ n ( hg n , kg n ) < ε . .We can now phrase a sufficient condition for a Cauchy sequence of propermonoids to converge for Υ . Theorem 3.16 ([29]) . Let ( G n , δ n ) n ∈ N be a sequence such that for all n ∈ N , thereexist ε n > and ( ς n , κ n ) ∈ UIso ε n (cid:18) ( G n , δ n ) → ( G n +1 , δ n +1 ) (cid:12)(cid:12)(cid:12)(cid:12) ε n (cid:19) such that:(1) P ∞ n =0 ε n < ∞ ,(2) for all N ∈ N and g ∈ G N h P ∞ n = N ε n i : ̟ N ( g ) = g n = e n if n < N , g n = g if n = N , g n = ς n − ( g n − ) if n > N n ∈ N ∈ R (( G n , δ n ) n ∈ N ) .Then there exists a proper monoid ( G, δ ) such that lim n →∞ Υ(( G n , δ n ) , ( G, δ )) = 0 . We note that if we work with proper monoids endowed with bi-invariant metrics,then our regularity condition is automatic since all sequences are regular in the senseof Definition (3.15). More generally, any reasonable form of uniform control of theLipschitz constant of right translations can be used to prove that sequences areregular. In [29], we show how to exploit this idea to construct a complete version of Υ on a class of proper monoids including proper monoids with bi-invariant metricsas a proper subclass.Putting together our compactness theorem and our sufficient condition for con-vergence of Υ -Cauchy sequences of proper monoids, we get the following sufficientcondition for convergence of Cauchy sequences in the covariant propinquity. Corollary 3.17.
Let F be permissible and continuous and let D : [0 , ∞ ) → [0 , ∞ ) be a locally bounded function. Let ( A n , L n , G n , δ n , α n ) n ∈ N be a sequence of Lipschitzdynamical F -systems and ( ε n ) n ∈ N a sequence of positive real numbers such that forall n ∈ N , there exists ε n > and ( ς n , κ n ) ∈ UIso ε n (cid:16) G n → G n +1 (cid:12)(cid:12)(cid:12) ε n (cid:17) and:(1) P ∞ n =0 ε n < ∞ ,(2) for all n ∈ N and g ∈ G n : g n = e n if n < N , g n = g if n = N , g n = ς n ( g n − ) if n > N n ∈ N ∈ R (( G n , δ n ) n ∈ N ) ,(3) ∀ n ∈ N Λ ∗ (( A n , L n ) , ( A n +1 , L n +1 )) < ε n ,(4) L n ◦ α gn D ( δ n ( e n , g )) L n ,(5) for all ε > , there exists ω > and N ∈ N such that if n > N and if g, h ∈ G n with δ n ( g, h ) < ω , then mkD L n (cid:0) α gn , α hn (cid:1) < ε . Then there exists a Lipschitz dynamical F -system ( A , L A , G, δ, α ) such that: lim n →∞ Λ cov (( A n , L n , G n , δ n , α n ) , ( A , L A , G, δ, α )) = 0 .Moreover, if for all n ∈ N , the action α n is by *-endomorphisms (resp. full quantumisometries, when G n is a group for all n ∈ N ), then so also is the action α . We conclude this section with an example of an explicit covariant convergence.We note that Theorem (3.14) provides many examples of convergent subsequencesfor the covariant propinquity, arising from convergent sequences for the dual propin-quity; however these convergent subsequences arise from an implicit construction.As may be expected, it is helpful to return to the notion of a bridge whenworking with the covariant propinquity. Let there be given two Lipschitz C*-dynamical systems ( A , L A , G, δ G , α ) and ( B , L B , H, δ H , β ) , and let us start with abridge γ = ( D , x, π A , π B ) from A to B . Let there also be given some pair ( ς, κ ) ∈ UIso ε (cid:18) ( G, δ G ) → ( H, δ H ) (cid:12)(cid:12)(cid:12)(cid:12) ε (cid:19) .Now, there is a natural means to define a new seminorm from this data, whichincorporates the actions into the bridge seminorm, by setting, for all ( a, b ) ∈ A ⊕ B : bn γ,ς, κ ( a, b ) = max (cid:26) bn γ (cid:16) α g ( a ) , β ς ( g ) ( b ) (cid:17) , bn γ (cid:16) α κ ( h ) ( a ) , β h ( b ) (cid:17) : g ∈ G (cid:20) ε (cid:21) , h ∈ H (cid:20) ε (cid:21) (cid:27) .We can then adjust the notion of the reach of a bridge using our modified bridgeseminorm to prove an analogue [30, Proposition 4.5] of Theorem (2.9). Using sucha technique, we can prove: Theorem 3.18 ([30]) . Let ℓ be a continuous length function on T d . Let ( G n ) n ∈ N be a sequence of closed subgroups of T d converging to T d for the Hausdorff distanceinduced by ℓ on the closed subsets of T d .Let ( σ n ) n ∈ N be a sequence of multipliers of Z d converging pointwise to σ andsuch that σ n is the lift of a multiplier of the Pontryagin dual c G n of G n .For all n ∈ N , denote by α n the dual action of G n on C ∗ ( c G n , σ n ) and by α thedual action of T d on the quantum torus C ∗ ( Z d , σ ) , and consider the L-seminorms L n and L induced by Example (1.5).Denote by Λ cov the covariant propinquity for the class of all Leibniz LipschitzC*-dynamical systems, and identify, for the sake of simplicity, the distance inducedby ℓ and ℓ itself. Then lim n →∞ Λ cov (cid:16)(cid:16) C ∗ ( c G n , σ n ) , L n , G n , ℓ, α n (cid:17) , (cid:0) C ∗ ( Z d , σ ) , L , T d , ℓ, α (cid:1)(cid:17) = 0 . References
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