text stringlengths 128 2.05k |
|---|
The metric monoid (resp. group) is proper when all its closed balls are compact. Definition 2.5 (metric monoid) morphism [MATH] is a map such that: |
1. [MATH] maps the identity element of [MATH] to the identity element of [MATH] 2. [MATH] 3. [MATH] is continuous. Remark 2.6 A proper metric space is always complete and separable. |
We now formally define the objects of the space under consideration in this paper: Lipschitz dynamical systems. Notation 2.7 Let [MATH] and [MATH] be two quantum compact metric spaces. If [MATH] is a unital positive linear map, then: |
[EQUATION] By definition, [MATH] if and only if [MATH] is a Lipschitz linear map. Definition 2.8 Let [MATH] be a permissible function. A Lipschitz dynamical [MATH] -system |
[MATH] is a [MATH] -quantum compact metric space [MATH] and a proper monoid [MATH] , together with an action [MATH] by positive unital maps (i.e. a morphism from [MATH] to the monoid of positive linear maps) such that: |
1. [MATH] is strongly continuous: for all [MATH] and [MATH] , we have: [EQUATION] 2. [MATH] is locally bounded: for all [MATH] there exist [MATH] and a neighborhood [MATH] of [MATH] in [MATH] such that if [MATH] then [MATH] |
Lipschitz [MATH] -dynamical [MATH] -system [MATH] is a Lipschitz dynamical system where [MATH] is a proper group and [MATH] is a Lipschitz unital *-automorphism for all [MATH] |
The class of Lipschitz dynamical systems include various sub-classes of interest, from group actions by full quantum isometries, to actions by completely positive maps, to actions by Lipschitz automorphisms or even unital endomorphisms. |
There is a natural manner to combine the notions of Lipschitz morphisms and Lipschitz morphism of proper monoids into a notion of morphism for Lipschitz dynamical systems. For our purpose, we will focus on what it means for two such systems to be considered the same system, i.e. our notion of isomorphism. |
Definition 2.9 Let [MATH] and [MATH] be two Lipschitz dynamical systems. An equivariant quantum full isometry [MATH] is given by a full quantum isometry [MATH] and a monoid isometric isomorphism [MATH] such that for all [MATH] |
[EQUATION] The construction of the covariant propinquity begins with the definition of a monoid-adapted Gromov-Hausdorff distance. We define our distance between two proper metric monoids [MATH] and [MATH] by measuring how far a given pair of maps [MATH] and [MATH] is from being an isometric isomorphism and its inverse... |
Notation 2.10 For a metric space [MATH] [MATH] and [MATH] , the closed ball in [MATH] centered at [MATH] , of radius [MATH] , is denoted as [MATH] , or simply [MATH] If [MATH] is a metric monoid with identity element [MATH] , and if [MATH] , then [MATH] is denoted as [MATH] |
Definition 2.11 Let [MATH] and [MATH] be two metric monoids with respective identity elements [MATH] and [MATH] . An [MATH] -local [MATH] -almost isometric isomorphism |
[MATH] , for [MATH] and [MATH] , is an ordered pair of maps [MATH] and [MATH] such that for all [MATH] [EQUATION] and [EQUATION] |
The set of all [MATH] -local [MATH] -almost isometric isomorphism is denoted by: [EQUATION] Convention 2.12 Write [MATH] for the restriction of a function [MATH] to some subset [MATH] of its domain. Let [MATH] and [MATH] with [MATH] for some [MATH] and [MATH] . For any [MATH] , we will simply write [MATH] to mean: |
[EQUATION] Moreover, if [MATH] then we may as well assume that [MATH] and [MATH] are defined on [MATH] and [MATH] , respectively, by choosing any extension of [MATH] and [MATH] , since it does not affect the local almost isometry property. |
We record that almost isometries behave well under composition. Lemma 2.13 Let [MATH] [MATH] and [MATH] be three metric monoids with respective identity elements [MATH] [MATH] and [MATH] |
If: [EQUATION] for some [MATH] , then: [EQUATION] Our covariant Gromov-Hausdorff distance over the class of proper metric monoids is then defined along the lines Gromov’s distance. |
Definition 2.14 The Gromov-Hausdorff monoid distance [MATH] between two proper metric monoids [MATH] and [MATH] is given by: [EQUATION] |
We indeed prove in Theorem 2.15 For any proper metric monoids [MATH] [MATH] and [MATH] 1. [MATH] 2. [MATH] 3. [MATH] 4. If [MATH] if and only if there exists a monoid isometric isomorphism from [MATH] to [MATH] |
In particular, [MATH] is a metric up to metric group isometric isomorphism on the class of proper metric groups. Moreover, if [MATH] is the pointed Gromov-Hausdorff distance on proper metric spaces, and if [MATH] and [MATH] are the respective identity elements of [MATH] and [MATH] , then: |
[EQUATION] It will also be helpful to recall from the following properties of almost isometries. Lemma 2.16 Let [MATH] [MATH] be two metric monoids and [MATH] [MATH] . If [MATH] then for all [MATH] , if [MATH] then: |
1. [MATH] 2. [MATH] 3. [MATH] 4. [MATH] 5. [MATH] The construction of the covariant propinquity begins with generalizing the notion of a tunnel between quantum compact metric spaces, as defined in |
for our construction of the Gromov-Hausdorff propinquity, to our class of Lipschitz dynamical systems. Notably, the needed changes are minimal. |
Definition 2.17 Let [MATH] and [MATH] be a permissible function. Let [MATH] and [MATH] be two Lipschitz dynamical [MATH] -systems. Let [MATH] and [MATH] be the identity elements of [MATH] and [MATH] respectively. A [MATH] -covariant [MATH] -tunnel |
[EQUATION] from [MATH] to [MATH] is given by [EQUATION] an [MATH] -quantum compact metric space [MATH] , and two quantum isometries [MATH] and [MATH] |
Remark 2.18 If [MATH] is an [MATH] -covariant tunnel then it is also an [MATH] -covariant tunnel for any [MATH] Remark 2.19 If [MATH] is a covariant tunnel from [MATH] to [MATH] , then [MATH] is a tunnel from [MATH] to [MATH] in the sense of |
. We also note that covariant tunnels are not constructed using a Lipschitz dynamical systems. They only involve an almost isometric isomorphism. |
The covariant propinquity is defined from certain quantities associated with covariant tunnels. These quantities do not depend on the quasi-Leibniz inequality. We now give their definitions, as they will be helpful with our current work. |
Notation 2.20 Let [MATH] be a positive unital linear map between two unital C*-algebras [MATH] and [MATH] . We denote the dual map [MATH] by [MATH] |
Notation 2.21 If [MATH] is a metric space, then the Hausdorff distance defined on the space of the closed subsets of [MATH] is denoted by [MATH] . In case [MATH] is a normed vector space and [MATH] is the distance associated with some norm [MATH] , we write [MATH] for [MATH] |
Definition 2.22 16 , Definition 2.11] Let [MATH] and [MATH] be two Lipschitz dynamical systems. The extent [MATH] of a covariant tunnel [MATH] from [MATH] to [MATH] is given as: |
[EQUATION] Definition 2.23 Let [MATH] . Let [MATH] and [MATH] be two Lipschitz dynamical systems. The [MATH] -reach [MATH] of a [MATH] -covariant tunnel [MATH] from [MATH] to [MATH] is given as: |
[EQUATION] The magnitude of a covariant tunnel summarizes all the data computed above. Definition 2.24 Let [MATH] . The [MATH] -magnitude |
[MATH] of a [MATH] -covariant tunnel [MATH] is the maximum of its [MATH] -reach and its extent: [EQUATION] We then define the covariant propinquity between Lipschitz dynamical systems as follows. While there are many appropriate choices for a class of tunnel used in the following definition as discussed in |
, we will focus on the class of all covariant [MATH] -tunnels. Thus, for a permissible function [MATH] and for any [MATH] , and for any two Lipschitz dynamical systems [MATH] and [MATH] , we denote the class of all [MATH] -covariant [MATH] -tunnels from [MATH] to [MATH] by: |
[EQUATION] Definition 2.25 Let [MATH] be a permissible function. For [MATH] two Lipschitz dynamical [MATH] -system, the covariant [MATH] -propinquity |
[MATH] is defined as: [EQUATION] In , we prove that [MATH] is indeed a metric up to equivariant full quantum isometry. Theorem 2.26 |
Let [MATH] be a permissible function. If [MATH] and [MATH] in are two Lipschitz dynamical [MATH] -systems, then: [EQUATION] if and only if there exists a full quantum isometry [MATH] and an isometric isomorphism of monoids [MATH] such that: |
[EQUATION] i.e. [MATH] and [MATH] are isomorphic as Lipschitz dynamical systems. Moreover, [MATH] satisfies the triangle inequality and is symmetric in its arguments, so it defines a metric on the class of Lipschitz dynamical [MATH] -systems up to equivariant full quantum isometries. |
This paper is concerned with the question of the completeness of the covariant propinquity on certain classes of Lipschitz dynamical systems. We begin with the matter of completeness for [MATH] |
Cauchy sequences of proper monoids for [MATH] An interesting problem arises when studying the completeness of [MATH] : given a Cauchy sequence for [MATH] , the construction of a potential limit guided by the completeness of the Gromov-Hausdorff distance may not be a topological monoid in general, without assuming some ... |
We begin with a definition which we will use to capture the equicontinuity of right translations for a sequence of proper monoids. |
Notation 3.1 We write: [EQUATION] Definition 3.2 Let [MATH] be a sequence of proper monoids. The set of regular sequences [MATH] is: |
[EQUATION] While it is unclear in general how large the set of regular sequences associated to a sequence of proper monoids may be, it is always a monoid. |
Lemma 3.3 [MATH] is a monoid for the pointwise multiplication. Proof. First, we note that the sequence [MATH] of the identity elements of [MATH] is regular. |
Let [MATH] and [MATH] be regular sequences. First note that since for all [MATH] , the metric [MATH] is left invariant, we have for all [MATH] |
[EQUATION] and thus [MATH] is bounded since [MATH] and [MATH] are. Let [MATH] . There exists [MATH] and [MATH] such that if [MATH] and if [MATH] , and if [MATH] then [MATH] . Now, there exists [MATH] and [MATH] such that if [MATH] , if [MATH] and if [MATH] then [MATH] . Hence if [MATH] and if [MATH] and if [MATH] then: |
[EQUATION] Hence [MATH] is regular. Thus [MATH] is closed under pointwise product, hence it is a monoid, as the multiplication is easily checked to be associative. |
We now can prove our theorem on convergence of Cauchy sequences for our metric [MATH] . If [MATH] is a Cauchy sequence of proper monoids for [MATH] , then there exists a subsequence [MATH] such that: |
[EQUATION] We will work with such subsequences in the next result. Theorem 3.4 Let [MATH] be a sequence such that for all [MATH] , there exists [MATH] and: |
[EQUATION] such that: 1. [MATH] 2. for all [MATH] and [MATH] [EQUATION] then there exists a proper monoid [MATH] such that [MATH] |
Proof. Without loss of generality, we can actually assume that [MATH] (by simply truncating our original sequence). It will be helpful to define [MATH] and similarly [MATH] for [MATH] . We also set [MATH] and [MATH] for all [MATH] and [MATH] and [MATH] and [MATH] are set to the identity of [MATH] . By Lemma ( 2.13 ), w... |
[EQUATION] Let: [EQUATION] We first note that for all [MATH] and [MATH] , we have [MATH] by assumption. Moreover if [MATH] and [MATH] , and if we write [MATH] as [MATH] , then: |
[EQUATION] Hence [MATH] and in particular, [MATH] is not empty. We also define the equivalence relation on [MATH] by: [EQUATION] |
We set [MATH] and we set [MATH] the canonical surjection. We now define, for all [MATH] [EQUATION] The function [MATH] is a pseudo-metric on [MATH] |
We note that [MATH] if and only if [MATH] . Consequently, [MATH] induces a metric on [MATH] which we denote as [MATH] We turn to the matter of defining a multiplication on [MATH] . First, we prove that [MATH] is closed under pointwise multiplication. let [MATH] . By Lemma ( 3.3 ), the sequence [MATH] is regular. Moreov... |
[EQUATION] Let [MATH] . Since [MATH] is regular, there exists [MATH] and [MATH] such that for all [MATH] , if [MATH] , and if [MATH] , then [MATH] |
Let [MATH] such that [MATH] . Let [MATH] such that for all [MATH] and for all [MATH] , we have [MATH] . Let [MATH] such that [MATH] if [MATH] and [MATH] . We note that if [MATH] and [MATH] , by Assertion (1) of Lemma ( 2.16 ): |
[EQUATION] Using Definition ( 2.11 ), if [MATH] and [MATH] , then since [MATH] [MATH] , and [MATH] is an [MATH] -local [MATH] -almost isometry from [MATH] to [MATH] , we conclude: |
[EQUATION] Hence [MATH] Our next step is to prove that the pointwise product of equivalent sequences are again equivalent. Let [MATH] [MATH] [MATH] and [MATH] be four elements of [MATH] such that [MATH] and [MATH] . Let [MATH] . Since [MATH] is regular, there exists [MATH] and [MATH] such that if [MATH] , if [MATH] , a... |
Now, there exists [MATH] such that if [MATH] then [MATH] . There exists [MATH] such that if [MATH] then [MATH] . Thus, we estimate that for [MATH] |
[EQUATION] Hence [MATH] . We therefore define [MATH] , for [MATH] , to be the equivalence class of [MATH] for any [MATH] such that [MATH] and [MATH] |
It is then easy to check that this operation is associative since the law is associative on each [MATH] for all [MATH] . Moreover, it is easy to check that the equivalence class of [MATH] , where [MATH] is the unit of [MATH] for each [MATH] is the identity of [MATH] |
Moreover, it is also immediate that the distance [MATH] on [MATH] is left-invariant for the multiplication thus defined since for all [MATH] , the distance [MATH] is left-invariant. |
We now turn to the continuity of the multiplication on [MATH] . Let [MATH] [MATH] [MATH] and [MATH] , such that [MATH] . By regularity, there exists [MATH] and [MATH] such that for all [MATH] , if [MATH] and [MATH] , then [MATH] . Let now [MATH] in [MATH] such that [MATH] . There exists [MATH] such that for all [MATH] ... |
[EQUATION] So [MATH] . Hence, the multiplication is jointly continuous at [MATH] . In fact, [MATH] is uniformly continuous for all [MATH] |
We now check that the closed balls in [MATH] for [MATH] are totally bounded. Let [MATH] . Let [MATH] . There exists [MATH] such that [MATH] and [MATH] |
Let [MATH] . Let [MATH] such that [MATH] . We thus have [MATH] . Thus there exists [MATH] [MATH] such that for all [MATH] , we have [MATH] . Note that: |
[EQUATION] Of course, [MATH] depends on [MATH] , a dependence which we now remove by changing our choice of a representative of [MATH] |
Let us therefore define [MATH] by setting [MATH] . By definition of [MATH] , we have: [EQUATION] Hence [MATH] . On the other hand, [MATH] by Assertion (4) of Lemma ( 2.16 ). |
Now, since [MATH] is proper, the closed ball [MATH] is compact, hence there exists a finite, [MATH] -dense subset [MATH] of this ball. We then note that there exists [MATH] such that [MATH] . Therefore by Assertion (4) of Lemma ( 2.16 ): |
[EQUATION] Hence [MATH] is totally bounded as desired. It then follows that the metric completion [MATH] of [MATH] for [MATH] is a proper metric space: if [MATH] then [MATH] lies inside the closure of [MATH] and thus it is totally bounded. As a totally bounded, closed subset of a complete metric space, we conclude [MAT... |
Moreover, for all [MATH] , the map [MATH] is uniformly continuous and thus, it admits a unique extension to [MATH] . We note that by continuity, for any [MATH] , and [MATH] , we have: |
[EQUATION] Now, let [MATH] . Let [MATH] . There exists [MATH] with [MATH] . Moreover, there exists [MATH] and [MATH] such that if [MATH] [MATH] and [MATH] then [MATH] . Hence, for all [MATH] with [MATH] , we have: |
[EQUATION] Thus [MATH] is uniformly continuous for any [MATH] , and thus it too admits a unique extension to [MATH] . We have defined a multiplication on [MATH] |
Now, for all [MATH] and for all [MATH] , using continuity, we obtain for any [MATH] [EQUATION] Hence [MATH] is left invariant by our newly defined multiplication on [MATH] |
Furthermore, let [MATH] . Let [MATH] . By uniform continuity of [MATH] , there exists [MATH] such that if [MATH] and [MATH] then [MATH] . Consequently, if [MATH] with [MATH] and if [MATH] then: |
[EQUATION] Therefore, our multiplication is indeed jointly continuous at every point of [MATH] and [MATH] is left-invariant for the multiplication. Therefore, [MATH] is a proper monoid, as desired. |
Our last step is to prove that [MATH] converges to [MATH] for [MATH] . To begin with, we denote [MATH] as [MATH] for all [MATH] , to keep our notations simple. We now define the other maps for our almost isometric isomorphisms. |
Let [MATH] . For any [MATH] , there exists [MATH] such that [MATH] . Writing [MATH] , we then set [MATH] . Of course, this definition depends on our choice function [MATH] |
Let [MATH] , and let [MATH] such that [MATH] . Note that [MATH] . So for all [MATH] , we note that: [EQUATION] Let [MATH] [MATH] , and [MATH] . We write [MATH] . Note that if [MATH] then: |
[EQUATION] By definition of [MATH] , there exists [MATH] such that for all [MATH] and for all [MATH] , we have [MATH] . Now for all [MATH] and [MATH] , we note that: |
[EQUATION] We then estimate, for all [MATH] [EQUATION] Now, let [MATH] and write [MATH] and [MATH] . If [MATH] then: [EQUATION] Thus [MATH] . In particular, if [MATH] then: |
[EQUATION] and our proof is concluded. We emphasize that a priori, a Cauchy sequence of proper groups for [MATH] which meets the assumptions of Theorem ( 3.4 ) will indeed converge to a proper monoid, but maybe not to a group. In order to assure that the limit is indeed a group, a new assumption must be added to Theore... |
Lemma 3.5 Let [MATH] and [MATH] be two metric groups. Let [MATH] . If: [EQUATION] then for all [MATH] such that [MATH] , the following estimate holds: |
[EQUATION] Proof. We compute: [EQUATION] as desired. We thus get our result for convergence of Cauchy sequence of proper groups for [MATH] |
Corollary 3.6 If [MATH] is a Cauchy sequence of proper groups for [MATH] such that for all [MATH] there exists [MATH] and [MATH] such that for all [MATH] |
[EQUATION] then there exists a proper group [MATH] such that [MATH] Proof. First, let [MATH] and let [MATH] . By our assumption, there exists [MATH] and [MATH] such that if [MATH] , if [MATH] and [MATH] then [MATH] . Now, there exists [MATH] and [MATH] such that for all [MATH] and if [MATH] with [MATH] then [MATH] |
Thus for [MATH] and for all [MATH] with [MATH] , we conclude [MATH] . Hence: [EQUATION] So [MATH] is regular. We thus conclude that [MATH] |
Since [MATH] is Cauchy, up to extracting a subsequence, we can choose [MATH] such that [MATH] and [MATH] . Let: [EQUATION] We will use the notations and observations of the proof of Theorem ( 3.4 ). |
Let now [MATH] . First, note that the left invariance of the metric [MATH] for all [MATH] , the sequence [MATH] is bounded. Let [MATH] . There exists [MATH] and [MATH] such that if [MATH] and [MATH] with [MATH] then [MATH] . Now, there exists [MATH] such that if [MATH] and [MATH] then [MATH] , and thus [MATH] . There e... |
[EQUATION] Hence, [MATH] . It is now sufficient to observe, using the notations of the proof of Theorem ( 3.4 ), that for any [MATH] and [MATH] chosen in [MATH] and [MATH] then, for all [MATH] , there exists [MATH] and [MATH] such that if [MATH] and [MATH] for any [MATH] then [MATH] ; since there exists [MATH] such tha... |
Now, let [MATH] , and let [MATH] and [MATH] given as above. If [MATH] for some [MATH] and [MATH] then there exists [MATH] such that [MATH] and therefore, [MATH] , so [MATH] . Thus the inverse map is uniformly continuous on [MATH] and can be extended to [MATH] on which it is now easy to check, it is the inverse for the ... |
Of course, the multiplicative group [MATH] does not have a uniformly continuous inverse, so the assumption of Corollary ( 3.6 ) is strong, though not unreasonable, and it is useful in controlling the regularity condition of Theorem ( 3.4 ). We now discuss some other natural conditions under which the regularity conditi... |
Corollary 3.7 The metric [MATH] restricted to the class of proper monoids with bi-invariant metric is complete. Moreover, [MATH] restricted to the class of proper groups with bi-invariant metric is also complete. |
Proof. Let [MATH] be a Cauchy sequence for [MATH] such that for all [MATH] , the metric [MATH] is bi-invariant. There exists a subsequence [MATH] of [MATH] such that [MATH] . We immediately check that [MATH] , so we can apply Theorem ( 3.4 ) to conclude that [MATH] converges for [MATH] , and thus, as a Cauchy sequence ... |
Note last that if [MATH] is a proper group with [MATH] bi-invariant, then for all [MATH] we have [MATH] so the inverse map is an isometry, hence Corollary ( 3.6 ) applies. |
Another situation where the regularity condition in Theorem ( 3.4 ) can be handled, in principle, is when the right translations are Lipschitz. We just need to control the Lipschitz constant, rather than a whole modulus of continuity, so we can define the following: |
[EQUATION] where [MATH] is meant as the best Lipschitz constant for a function [MATH] between metric spaces. Now, convergence for [MATH] implies in particular that we can find almost isometric isomorphism which will meet our regularity condition in Theorem ( 3.4 ). |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.