On a categorical framework for classifying C*-dynamics up to cocycle conjugacy
Gabor Szabo
TL;DR
This paper develops a categorical framework for classifying C*-dynamics up to cocycle conjugacy, extending known functors and establishing intertwining results to facilitate classification efforts.
Contribution
It introduces a new categorical approach to C*-dynamics classification, generalizing intertwining techniques and allowing for inductive limits and extended functors.
Findings
Categorical framework for G-C*-algebras with cocycle morphisms
Extension of functors to the new categorical setting
Generalization of intertwining results for classification
Abstract
We provide the rigorous foundations for a categorical approach to the classification of C*-dynamics up to cocycle conjugacy. Given a locally compact group , we consider a category of (twisted) -C*-algebras, where morphisms between two objects are allowed to be equivariant maps or exterior equivalences, which leads to the concept of so-called cocycle morphisms. An isomorphism in this category is precisely a cocycle conjugacy in the known sense. We show that this category allows sequential inductive limits, and that some known functors on the usual category of -C*-algebras extend. After observing that this setup allows a natural notion of (approximate) unitary equivalence, the main aim of the paper is to generalize the fundamental intertwining results commonly employed in the Elliott program for classifying C*-algebras. This reduces a given classification problem for C*-dynamicsâŠ
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On a categorical framework for classifying \cstar-dynamics up to cocycle conjugacy
GĂĄbor SzabĂł
KU Leuven, Department of Mathematics, Celestijnenlaan 200b box 2400, -..-B-3001 Leuven, Belgium
Abstract.
We provide the rigorous foundations for a categorical approach to the classification of \cstar-dynamics up to cocycle conjugacy. Given a locally compact group , we consider a category of (twisted) -\cstar-algebras, where morphisms between two objects are allowed to be equivariant maps or exterior equivalences, which leads to the concept of so-called cocycle morphisms. An isomorphism in this category is precisely a cocycle conjugacy in the known sense. We show that this category allows sequential inductive limits, and that some known functors on the usual category of -\cstar-algebras extend. After observing that this setup allows a natural notion of (approximate) unitary equivalence, the main aim of the paper is to generalize the fundamental intertwining results commonly employed in the Elliott program for classifying \cstar-algebras. This reduces a given classification problem for \cstar-dynamics to the prevalence of certain uniqueness and existence theorems, and may provide a useful alternative to the EvansâKishimoto intertwining argument in future research.
2010 Mathematics Subject Classification:
46L55
This work was supported by the start-up grant STG/18/019 of KU Leuven, the research project C14/19/088 funded by the research council of KU Leuven, and the research project G085020N funded by the Research Foundation Flanders (FWO)
Contents
Introduction
The concept of noncommutative dynamics is deeply rooted and ubiquitous within the subject of operator algebras, arguably because of the interesting ways in which groups can act on noncommutative structures. On the one hand, elementary models of quantum mechanics conceptualize the passage of time in a physical system via an action of the real numbers on a suitable operator algebra of observables. On the other hand, every classical topological dynamical system, say consisting of a locally compact group acting on a locally compact space , can be considered as a \cstar-dynamical system and then studied with the aid of operator algebraic methods. The overarching concepts of \cstar-dynamics and -dynamics unify these ideas in one neat package.
Noncommutative dynamics have been taking center stage on the -side ever since the development of the fundamental structure theory of factors, notably including TomitaâTakesaki theory and the work of ConnesâHaagerup for injective -algebras. The structure of actions of (discrete) amenable groups on injective factors has subsequently been unraveled by the combined work of many researchers. Through groundbreaking research spear-headed by Popa, various superrigidity phenomena were discovered in the context of -dynamics for actions of nonamenable groups, which stand in great contrast to the amenable setting via results such as the ConnesâFeldmanâWeiss theorem.
In a similar fashion and in part motivated by the above, \cstar-algebraists have long been interested to either apply or explore the structure of \cstar-dynamical systems. The focus has mostly been on two natural but opposite extremes. One possibility is to consider certain topological dynamical systems and investigate the crossed products , either for their classifiability [76, 14, 60, 22, 95, 31, 81, 91, 32, 29, 33, 45, 46] or for the information that their isomorphism classes retain about the original dynamics [27, 23]. Another possibility is to start with certain group actions on simple \cstar-algebras and ask for their classification up to cocycle conjugacy, preferably using computable invariants such as -theory. The present work should be considered as a piece in the latter line of research.
In direct comparison to the -side, our understanding of \cstar-dynamics on simple \cstar-algebras is still somewhat underdeveloped. Nevertheless, the past attempts to classify automorphisms or group actions on simple \cstar-algebras have been both numerous and highly inventive. We shall give a brief (albeit incomplete) summary of the state of the art here. Many of the attempts to classify \cstar-dynamics have so far been underpinned by some kind of Rokhlin-type property. After ideas of this type have surfaced in work of Connes, a propotypical definition of sort made its way to \cstar-algebras via work of HermanâJones and HermanâOcneanu [26, 25]. After a few more applications of these ideas [7, 77, 4, 51], notably in the study of certain purely infinite \cstar-algebras, the correct version of the Rokhlin property for single automorphisms was pinned down and exploited to great success [49, 16, 15, 52, 53, 5, 54]. This line of research has been considerably extended and perfected to this day in subsequent work by many researchers to classify automorphisms on more general \cstar-algebras or actions of certain higher-rank groups [69, 70, 62, 44, 79, 37, 63, 64, 73, 38, 39]. The Rokhlin property for actions of finite groups was introduced by Izumi [34, 35], who also showcased its rigid behavior by giving a satisfactory classification theory up to conjugcy. Results of this type were subsequently found for not necessarily unital \cstar-algebras [71, 21, 72], actions of compact groups [30, 18, 19], and compact quantum groups [1]. The Rokhlin property for flows was introduced by Kishimoto in [50], who provided evidence why one should expect that these can be classified up to cocycle conjugacy [56, 6]. This was confirmed in my recent work [89], which was in part inspired by [61].
A related but not identical line of research relating to tensorial absorption of strongly self-absorbing dynamical systems was fleshed out and applied in [85, 84, 82, 83, 88, 86, 87]. This was in turn heavily based on pioneering work in [55, 37, 24, 65, 66] that exploited ideas of this nature long before. Related recent results include [58, 59, 80, 20].
The purpose of the present work can be seen as an attempt to reconsider and improve on the overarching framework and some of the elementary analytic methods involved in the classification of group actions up to cocycle conjugacy. As motivation, let us recall a simple fact that has fundamental consequences for the Elliott classification program of \cstar-algebras:
Theorem** (cf. [78, Corollary 2.3.4] and [13]).**
Let and be two separable \cstar-algebras. Suppose that and are two (extendible)111To be precise, one is faced with a choice here for the original proof in [78] to be correct. Either one has to assume that the involved -homomorphisms are extendible (see subsection 1.1) or one has to use the notion of approximate innerness involving unitaries in the smallest unitizations of and , respectively, instead of their multiplier algebras. The latter is suggested on the Errata page for [78]; cf. http://web.math.ku.dk/~rordam/Encyclopaedia.html. -homomorphisms such that the compositions and are approximately inner endomorphisms on and , respectively. Then and are approximately unitarily equivalent to mutually inverse isomorphisms between and .
The utility of this statement comes from the fact that it implicitly gives us a roadmap on how to classify a given category of separable \cstar-algebras by a functorial invariant ââ. The first step is to establish a uniqueness theorem for the invariant, which typically asserts that two (sufficiently non-trivial) -homomorphisms with are approximately unitarily equivalent if . The second step is to establish an existence theorem for the invariant, which typically asserts that for , every suitable arrow lifts to a -homomorphism . The third step is to tie it all together with the above theorem for : Combing the first two steps, an invertible arrow gives rise to two -homomorphisms as required by the theorem above, which implies that and are isomorphic. This framework for classification has been at the heart of the Elliott program since the early BratteliâElliott classification [3, 12] of AF algebras (with the ordered -group playing the role of ââ), and has essentially remained unchanged notwithstanding the staggeringly increasing depth behind the modern versions of the correct invariant as well as the uniqueness and existence theorems.
Despite the importance of attempting to classify \cstar-dynamical systems up to cocycle conjugacy [36], there is still no clear-cut analog of this roadmap in general. For compact acting groups, it is relatively straightforward to come up with some suitable generalization of the statement in the above theorem, and to come up with analogous uniqueness and existence theorems to classify actions up to genuine conjugacy. In fact this viewpoint has been leveraged to classify actions with the Rokhlin property in [21, 1].
For non-compact acting groups, on the other hand, it is much less clear what an analogous roadmap for classification would be from consulting the literature. In light of some very elementary examples [36, Section 2.2], it is clear that classification up to genuine conjugacy on sufficiently noncommutative \cstar-algebras is not feasible using a computable invariant, and has to be weakened to cocycle conjugacy instead. The usual way to classify actions of groups like or , as pioneered by Kishimoto, follows a somewhat different path to the above: Instead of the uniqueness theorem, one needs to bother with establishing a cohomology-vanishing statement for the group actions under consideration. Instead of the existence theorem, one needs to determine when two actions on the same \cstar-algebra are approximate cocycle perturbations of each other. Instead of the intertwining theorem above, one then combines these pieces with a fairly involved EvansâKishimoto intertwining argument (cf. [16]), which directly gives one cocycle conjugacy between two dynamical systems. One may certainly argue that this runs somewhat parallel to the three step recipe sketched in the context of classifying \cstar-algebras, but the analogy only goes so far. Despite being an undoubtedly ingenious method, the EvansâKishimoto intertwining argument does not shed much light on the category of \cstar-dynamics being classified as a whole, but rather helps one to compare cocycle conjugacy classes directly. In a way, the assumptions entering into the EvansâKishimoto intertwining argument to relate two actions should (in general) be seen as extremely strong by design, in order to ensure that some kind of existence proof for both and and the uniqueness proof for their compositions is taken care of in one fell swoop. It is therefore not surprising that classification via this method gets increasingly difficult to implement with more complicated acting groups, as the necessary assumptions for performing the intertwining argument appear to be more and more difficult to handle; see [38, 39] for recent state of the art results that serve as a good example for this phenomenon.
The subject of this paper is to provide a categorical framework in which Elliottâs approach to classification carries over directly to the context of \cstar-dynamics. The obvious first obstacle is given by the fact that the category of -\cstar-algebras, with being a locally compact group, is usually equipped with the genuine equivariant -homomorphisms as morphisms. The naive way to overcome it is to add the least amount of extra information to the category as to include the language necessary to describe cocycle conjugacy:
Definition \thedefiiCT.
Let be a locally compact group and let and be two actions on \cstar-algebras. A cocycle representation from to is a pair consisting of a -homomorphism and a -cocycle satisfying for all . If , we call the pair a cocycle morphism from to .
Notice that in the special case above, we simply recover what it means for the -homomorphism to be equivariant with respect to and . On the other hand, if and , we recover what it means for two -actions on the same \cstar-algebra to be exterior equivalent via the cocycle . In other words, the usual category of -\cstar-algebras is enlarged by declaring that exterior equivalences are also morphisms. It turns out that there is a natural way to define a composition between cocycle morphisms222Similarly to the previous footnote, one is again faced with two possibilities here, depending on whether one prefers to work with unitaries in unitizations or multiplier algebras. Both perspectives merit investigation and will be pursed in the paper.; see subsection 1.3. This enables one to define a larger category of -\cstar-algebras, which we choose to call the cocycle category. Anybody who is familiar with the concept of cocycle conjugacy will certainly not be taken by surprise by the fact that in this category, an isomorphism between -\cstar-algebras is precisely a cocycle conjugacy in the known sense.
The rigorous introduction to this framework along with some elementary examples and observations will occupy most of the first section. The level of generality in the main body of the paper is higher than indicated up to this point, because this framework makes sense and will be developed in the context of twisted -\cstar-dynamical systems Ă la BusbyâSmith [8, 74]. It is noteworthy that the categorical framework pursued here is not entirely new. A similar framework was conceived in a higher category approach to (twisted) \cstar-dynamics due to BussâMeyerâZhu [9] and in the context of crossed product dualities by KaliszewskiâOmlandâQuigg [42], although ultimately used for different purposes than here. In the terminology of [9], a weakly equivariant map or transformation is precisely a cocycle morphism with the extra assumption that the map is non-degenerate. Looking back even further, one may argue for that traces of this idea can be found in Kishimotoâs work [57], where in the context of a flow , the cocycle conjugacies have been studied as core symmetries and identified as an interesting invariant.
The second section deals with inductive limits. We show that in the cocycle category, sequential inductive limits always exist, and can be constructed explicitly in the language of the ordinary \cstar-algebra inductive limit construction. We furthermore discuss topologies on the Hom-sets, which for pairs as in Introduction will usually boil down to the point-norm topology in the first variable, and the uniform strict topology over compact subsets of in the second variable. This gives rise to a suitable notion of approximate unitary equivalence; cf. subsection 2.2 and subsection 2.2:
Definition \thedefiiCT.
Let be two non-degenerate cocycle morphisms. We say is approximately unitarily equivalent to , if there exists a net such that
[TABLE]
In the special case where and , we say that is approximately inner.
The third and fourth sections are concerned with providing detailed generalizations of the elementary intertwining results commonly employed in the classification program of \cstar-algebras, in particular utilizing approximate unitary equivalence. This part of the paper can be seen as a variant of some early sections in [78] for \cstar-dynamics. There are several layers of generality treated in this context, but we shall for now cherry-pick the following result from the third section to illustrate the parallels to the theorem above; the following is a special case of section 3:
Theorem \thetheoremiCT.
Let and be two separable -\cstar-algebras. Suppose that and are two non-degenerate cocycle morphisms such that the compositions and are approximately inner on and , respectively. Then and are approximately unitarily equivalent to mutually inverse cocycle conjugacies between and .
In a nutshell, this means that the framework discussed here comes equipped with a clear-cut roadmap for classification of \cstar-dynamics up to cocycle conjugacy, involving uniqueness and existence theorems just like explained before in the context of the Elliott program. In some cases, this may provide a useful alternative to classification via the EvansâKishimoto intertwining method. A good example of this is the homotopy rigidity result [83, Theorem F]333We note that the current published version of this result [83, Corollary 6.5] mistakenly claims in its conclusion the conjugacy of two actions instead of cocycle conjugacy. The correct statement is proved for example via section 3, whereas the published version tried to appeal to an erronous version of this intertwining theorem from [85], which has since then been corrected in [90]. for outer actions of amenable groups on Kirchberg algebras, which to my knowledge nobody can currently prove via EvansâKishimoto intertwining. In fact, the framework developed here and the intertwining results will be a crucial ingredient in forthcoming work joint with James Gabe, in which we classify outer actions of discrete amenable groups on Kirchberg algebras via equivariant Kasparov theory. The content of the last section can be seen as preliminary evidence why the framework appears to be useful for extracting information about \cstar-dynamics from -theoretical data, which will be fleshed out further in our forthcoming work with James Gabe. Our approach to classification of \cstar-dynamics will as a result substantially differ from previous work in this area, such as for example the recent articles [38, 39] handling the case of poly- groups based on variants of the EvansâKishimoto intertwining method.
We note that the main result of [13] has been an apparent template for Introduction, and in fact the former directly implies the latter under the assumpion that both and are unital. However, the assumptions in the main result of [13] no longer apply in any obvious way to the non-unital case. Since we are also interested in comparing more general inductive limits via intertwining arguments, which is not handled in [13], we will build the intertwining theory in our context from scratch.
In the fourth section, we will also deal with some one-sided intertwining arguments, as well as prove a version of Introduction involving asymptotic unitary equivalence; see subsection 4.1.444To the best of my knowledge, the non-dynamical version of this fact is folklore but has so far remained unpublished. In the fifth section we briefly revisit the concept of strong self-absorption for \cstar-dynamical systems, which turns out to be nicely expressible in the framework of this paper. The main result of that section is a stronger version of the existing equivariant McDuff-type theorems [85, 82] for non-unital \cstar-algebras (see section 5), which holds under the assumption of equivariant JiangâSu stability, and arises as a straightforward consequence of an intertwining argument in the preceding section. In the sixth section, we take a look at -equivariant Kasparov theory, and show that it naturally extends to a (bi-)functor on the cocycle category of -\cstar-algebras. This holds true both in Kasparovâs original Fredholm picture [43], as well as the CuntzâThomsen picture, which was in fact conceptualized in Thomsenâs original work [92] by using cocycle representations in the sense of Introduction. We also take the opportunity to show that a certain continuity assumption in the definition of an equivariant Cuntz pair describing elements in a -group is redundant; cf. section 6.
Acknowledgement. The key external impulse to pursue this line of research came to me while I was giving a lecture series at the NCGOA 2018 in MĂŒnster, the subject of which boiled down to the classification of group actions via the EvansâKishimoto intertwining method. During the lengthy discussion of its technical details, George Elliott made several suggestive and open-ended remarks about the framework as a whole, insisting (or perhaps speculating) that it ought to be possible to intertwine at the level of the morphisms connecting two \cstar-dynamics. To make a long story short, this paper can in large part be understood as a rigorous positive answer to said suspicion, and I would like to thank George Elliott for voicing it.
1. Introduction to the cocycle category
1.1. Preliminaries
Throughout, will denote a second-countable, locally compact group. The capital letters will usually denote \cstar-algebras. For elements and in some \cstar-algebra, we frequently write as short-hand for . The multiplier algebra of is denoted as , whereas its unitary group will be denoted by if is unital. We define , where is the \cstar-algebra arising from adding a unit to . One has a canonical isomorphism of groups in case is unital, even though the two sets are not equal. Normal alphabetical letters such as will be frequently used for unitary elements in \cstar-algebras. For a given unitary or , we denote by the inner automorphism given by . Greek letters such as are used for point-norm continuous maps . Most of the time these are either group actions or part of a twisted action on . Fraktur letters such as are used for strictly continuous maps , and usually denote the 2-cocycles belonging to a twisted action on . Double-struck letters such as are used for strictly continuous maps . In practice, these are often 1-cocycles with respect to a -action or more generally are part of a cocycle morphism, as defined in the main body of the article.
Definition \thedefiCT.
Let be a locally compact group. A twisted action of on a \cstar-algebra is a pair for a point-norm continuous map and a strictly continuous map satisfying
[TABLE]
and
[TABLE]
for all . We commonly write . In this context, the map is referred to as a 2-cocycle, and the triple is referred to as a twisted -\cstar-algebra.
In the case that is norm-continuous and takes values in , we say that is a gently twisted action and call the triple a gently twisted -\cstar-algebra.
Remark \theremCT.
In the special case , we recover what it means for to be a -action, and the pair is then called a -\cstar-algebra, which we will identify with the triple when we view it as a (gently) twisted -\cstar-algebra in the canonical way.
Let be a twisted -\cstar-algebra as in the above definition. Sometimes it will be important to keep track of the triple , which in general is not a twisted -\cstar-algebra in the above sense. The canonical induced map is only point-strictly continuous instead of point-norm continuous, but all the other axioms of a twisted -\cstar-algebra make sense for this triple.
In what follows, we need to recall a few basic concepts from the theory of twisted actions; see [74].
Remark \theremCT.
Let be a twisted -\cstar-algebra. Recall that the (maximal) twisted crossed product comes with a covariant representation of , i.e., a non-degenerate -homomorphism and a strictly continuous map such that
[TABLE]
for all . This covariant representation has the universal property that every other one factors through it. To be more precise, whenever is another \cstar-algebra and the pair and forms another (non-degenerate) covariant representation into , then there exists a unique non-degenerate -homomorphism such that and .
Definition \thedefiCT.
Let be a twisted action. Given any strictly continuous map with , we may define a new twisted action via and for all . If there exists a map such that , then the 2-cocycle is called a coboundary, and it means that is equivalent to a genuine action in the sense below.
Definition \thedefiCT.
Two twisted actions on the same \cstar-algebra are called exterior equivalent, if there exists a strictly continuous map with such that .
Remark \theremCT.
In light of subsection 1.1, if we restrict subsection 1.1 to genuine actions , then we write to mean , which automatically forces to be an -1-cocycle (or briefly -cocycle), and to be the associated cocycle perturbation. That is, we have for all .
Definition \thedefiCT.
A -homomorphism is called extendible, if for any increasing approximate unit , the net converges strictly to a projection in .555In particular, if is non-degenerate, then it is extendible. In this case factorizes through , where it becomes a non-degenerate -homomorphism , which uniquely extends to a -homomorphism from the multiplier algebra that becomes strictly continuous on the unit ball.
The map given by
[TABLE]
is then a well-defined, strictly continuous homomorphism between unitary groups. It satisfies the formula for all and , which we will frequently use without mention. If happens to be non-degenerate, then it extends to a unital -homomorphism , in which case we can write instead of .
Remark \theremCT.
To avoid confusion, here is a word of caution for the reader. Depending on context, we may use the similarly looking expression to denote the unital -homomorphism induced by a -homomorphism . Under this notation, if is assumed to be extendible and is a unitary in , then the unitary multiplier in arising from in agrees with the multiplier in in the sense of the definition above. In particular, these two pieces of notation, which could a priori mean different things, always yield the same elements in whenever both are defined.
Remark \theremCT.
If and are two extendible -homomorphisms with corresponding projections and , then one may compose them via
[TABLE]
and thus write . This is clearly again an extendible -homomorphism with respect to the corner spanned by in . It follows moreover that as group homomorphisms . We will use all this without further mention.
1.2. Cocycle representations and morphisms
Definition \thedefiCT.
Let and be two twisted actions on \cstar-algebras.
- (i)
A cocycle representation from to is a pair , where is an extendible -homomorphism and is a strictly continuous map such that
[TABLE]
and
[TABLE]
for all . 2. (ii)
A cocycle morphism from to is a cocycle representation as above, with the additional requirement that . We will write . The set of all such pairs will be denoted by . 3. (iii)
Suppose that and are both gently twisted actions. A proper cocycle morphism from to is a pair , where is an arbitrary -homomorphism, is a norm-continuous map taking values in , and such that the conditions (e1.1) and (e1.2) hold, with ââ being replaced by ââ. The set of all such pairs will be denoted by .
We will refer to (e1.1) as the equivariance condition and to (e1.2) as the cocycle identity for the pair . In the special case that is the trivial map, we identify with the pair and say that
[TABLE]
is an equivariant -homomorphism.
Remark \theremCT.
In the above definition, the equivariance condition implies due to the fact that we have , and .
Remark \theremCT.
If we restrict subsection 1.2 to the case of genuine -\cstar-algebras and , then a cocycle representation
[TABLE]
is a pair such that is a -cocycle in the ordinary sense by (e1.2), and (e1.1) means that the corresponding cocycle perturbation turns into an --equivariant -homomorphism. In this particular case, we may sometimes call such a pair a cocycle representation even if is not assumed to be extendible. Although this definition is too weak to define a category as we will do below, it is nevertheless relevant in the description of -equivariant -theory as treated in the last section. The analogous observation as above is true for cocycle morphisms to , and being a proper cocycle morphism additionally means that is a (norm-continuous) cocycle in .
Example \theexampleCT.
Given a twisted action , a covariant representation into in the sense of subsection 1.1 is just a non-degenerate cocycle representation .
Example \theexampleCT.
Let be a twisted action. For any strictly continuous map with , we obtain (by definition) a cocycle morphism
[TABLE]
We refer to such an example as an exterior equivalence. If is norm-continuous and takes values in , then we call this a proper exterior equivalence. In particular, if is a genuine action and is an -cocycle, then
[TABLE]
is an exterior equivalence, and it is moreover a proper exterior equivalence precisely when is a norm-continuous map with values in .666We will see later in section 6 that norm-continuity of is a redundant assumption that happens to hold automatically if it takes values in .
1.3. The cocycle category
Proposition \thepropCT.
Let , and be three twisted actions on \cstar-algebras.
- (i)
Suppose that
[TABLE]
are two cocycle representations. Then the composition, which we define as the pair
[TABLE]
defines a cocycle representation from to . This binary operation is associative. 2. (ii)
In this sense, the composition of two cocycle morphisms is again a cocycle morphism. 3. (iii)
Suppose that all three of them are gently twisted actions. If
[TABLE]
are two proper cocycle morphisms, then the composition formula above, with ââ replaced by ââ, defines a proper cocycle morphism.
Proof.
(i): Let us first verify the equivariance condition. Using the equivariance condition for both pairs and , one has for all that
[TABLE]
Let us now verify the cocycle identity. For all , we use equivariance and cocycle identities for our given cocycle representations to see that
[TABLE]
So the pair is indeed a cocycle representation from to .
In order to obtain associativity, let be a twisted action and another cocycle representation. Then evidently and
[TABLE]
for all , which by definition yields
[TABLE]
Part (ii) is trivial, and (iii) follows from the same computations if we replace all instances of ââ with ââ. â
Definition \thedefiCT.
We define the following categories:
- (i)
The twisted cocycle category is defined as the category with objects being the twisted -\cstar-algebras, and the morphisms from to being the cocycle morphisms
[TABLE]
Composition of morphisms in is defined via the operation in subsection 1.3. On any object , the identity morphism in this category is given by , which we will sometimes abbreviate as if there is no cause for confusion. 2. (ii)
The cocycle category is the full subcategory of with objects being all the genuine -\cstar-algebras . 3. (iii)
The twisted proper cocycle category has the gently twisted -\cstar-algebras as objects, and proper cocycle morphisms as morphisms. The full category given by all genuine -\cstar-algebras is called the proper cocycle category and is denoted .
Remark \theremCT.
Any cocycle morphism can always be expressed as the composition
[TABLE]
of a genuine equivariant -homomorphism and an exterior equivalence.
Remark \theremCT.
A cocycle morphism is invertible in the category if and only if is an isomorphism between \cstar-algebras. The inverse is then given by the formula
[TABLE]
We will refer to an invertible cocycle morphism as a cocycle conjugacy.
As the terminology suggests, there exists a cocycle conjugacy between the twisted -\cstar-algebras and if and only if the twisted actions and are cocycle conjugate in the ordinary sense.
In an analogous fashion, we define a proper cocycle conjugacy between gently twisted -\cstar-algebras as an isomorphism in the category . This is a stronger requirement than cocycle conjugacy in the sense that exterior equivalence has to be implemented by unitaries coming directly from the (unitized) underlying \cstar-algebra rather than its multiplier algebra.
Remark \theremCT.
Let be a genuine action, and a twisted action. If there exists a cocycle representation , then the cocycle identity (e1.2) immediately implies that is a coboundary, i.e., is exterior equivalent to a genuine action. Note that this is the case even when .
The reverse statement is not true, on the other hand, as there generally exist many cocycle representations from to , which are nothing but covariant representations, provided that is separable.
Example \theexampleCT.
Let be a twisted action. Then any unitary element gives rise to the cocycle conjugacy
[TABLE]
If it is clear from context that we are making statements within the category , then we simply write \operatorname{Ad}(v):=\big{(}\operatorname{Ad}(v),v\alpha_{\bullet}(v)^{*}\big{)}.
Proof.
Equivariance follows from
[TABLE]
and the cocycle identity follows from
[TABLE]
So indeed always defines a cocycle morphism. â
Definition \thedefiCT.
Let be a twisted action. We call a cocycle morphism inner, if it is of the form in the sense of subsection 1.3 for some unitary .
Example \theexampleCT.
Let be a gently twisted action. Then for any unitary , the cocycle conjugacy
[TABLE]
as defined above becomes a proper cocycle conjugacy.
Remark \theremCT.
Let and be two twisted actions. If is a cocycle representation and is a unitary, then the composition with its inner cocycle automorphism is given as
[TABLE]
Proposition \thepropCT.
Let be a twisted action. Then the inner cocycle automorphisms of form a normal subgroup in the cocycle automorphism group of .
Proof.
Let be a cocycle conjugacy and . Then
[TABLE]
â
1.4. Functoriality of crossed products
Definition \thedefiCT.
Let and be two twisted actions on \cstar-algebras. We denote by the universal covariant representation of , and by the universal covariant representation of , which appear in the definition of their respective twisted crossed products. For a non-degenerate cocycle representation , we define the non-degenerate -homomorphism between the (full) twisted crossed products via
[TABLE]
Notice that there is indeed a unique such -homomorphism by the universal property of , as these formulas define a covariant representation into . If is a cocycle morphism into , then the image of is contained in .
Proposition \thepropCT.
Let
[TABLE]
be two non-degenerate cocycle representations between twisted -\cstar-algebras. Then the construction in subsection 1.4 is compatible with respect to compositions, i.e., one has
[TABLE]
In particular, if we restrict to non-degenerate maps, then the (twisted) crossed product construction is functorial on the (twisted) cocycle category .
Proof.
Let us denote
[TABLE]
Then we can see directly at the level of covariant representations that
[TABLE]
and
[TABLE]
for all . By the universal property of with respect to covariant representations, this forces and finishes the proof. â
2. Inductive limits
Notation \thenotaCT.
Let be a sequence of \cstar-algebras and a sequence of -homomorphisms. If we view this as an inductive system so that we can define the inductive limit , we will denote
[TABLE]
For notational convenience we also set . Furthermore the universal map from the -th building block into the limit will be denoted as
[TABLE]
If every connecting map is extendible, then so is every -homomorphism of the form or . Analogously, if
[TABLE]
is an inductive system of cocycle morphisms between twisted -\cstar-algebras, we introduce the symbolic notation for , and
[TABLE]
provided that is the inductive limit in the category . The fact that this always exists is the subject of the next proposition.
Proposition \thepropCT.
Sequential inductive limits exist in the category .
To be more precise, let
[TABLE]
be an inductive system of cocycle morphisms between twisted -\cstar-algebras. Then the inductive limit in is of the form , where is the ordinary inductive limit \cstar-algebra. The universal cocycle morphisms
[TABLE]
are of the form
[TABLE]
where
[TABLE]
are strictly continuous maps defined inductively via and for and .
Proof.
As all maps are assumed to be extendible, this will be the case for as well.
For each , we have . By induction this translates to the condition
[TABLE]
for all and . By the universal property of the limit , there is a unique (point-norm continuous) map
[TABLE]
For each , we have
[TABLE]
By induction, this translates to
[TABLE]
for all and . Consider the strictly continuous map via .
We claim that the pair defines a twisted action on . For every and , we have
[TABLE]
Since is arbitrary, this yields . Moreover, as is defined so that satisfies and for all , this immediately forces the map to satisfy the 2-cocycle identity. This verifies our claim that the triple becomes a twisted -\cstar-algebra.
Let us now verify that are well-defined cocycle morphisms. We have arranged by construction that
[TABLE]
and
[TABLE]
In particular,
[TABLE]
So we see that the pair \big{(}\varphi_{n,\infty},\varphi_{n,\infty}^{+}(\mathbb{U}^{(n-1)*}_{\bullet})\big{)} yields a well-defined cocycle morphism
[TABLE]
Moreover it holds for all that
[TABLE]
Finally, we claim that together with the family of morphisms satisfies the universal property of the inductive limit in the category . Suppose that is a twisted -\cstar-algebra and that for each , we are given a cocycle morphism
[TABLE]
such that
[TABLE]
holds for all . Then the universal property of the \cstar-algebraic limit gives us a unique -homomorphism with for all . From the composition formula above it follows inductively that
[TABLE]
Setting , we claim that yields a cocycle morphism from to . Indeed, we get equivariance from
[TABLE]
As is arbitrary, this implies for all . The cocycle identity again follows immediately from the fact that is genuinely equivariant and was subject to the cocycle identity.
So is indeed a cocycle morphism. It follows from construction that for all , which finishes the proof. â
Proposition \thepropCT.
Sequential inductive limits exist in the category . More specifically, for any inductive system in the (twisted) proper cocycle category , the same construction as in section 2, with all instances of ââ replaced by ââ, yields the limit in the category .
Proof.
We can read this off immediately from the proof of section 2. If are proper cocycle morphisms, then evidently is a norm-continuous map with values in for all , and hence the universal morphisms (\varphi,\mathbbmss{u})_{n,\infty}=\big{(}\varphi_{n,\infty},\varphi_{n,\infty}^{\dagger}(\mathbb{U}^{(n-1)*})\big{)} are also proper cocycle morphisms. Every other part of the proof can be copied verbatim. â
Remark \theremCT.
From the proof of section 2 we can also deduce that if for all , then . In fact it suffices to assume ; cf. subsection 1.3. In particular, the inductive limit construction respects the subcategory of genuine -\cstar-algebras. For the same reason, the proper cocycle category is closed under inductive limits.
2.1. The topology on Hom-sets
Definition \thedefiCT.
Let and be two twisted actions on \cstar-algebras. We equip the possible sets of arrows between them with the following uniform topologies:
- (i)
On \mathrm{CoMor}\big{(}(\alpha,\mathfrak{u}),(\beta,\mathfrak{v})\big{)}, we consider the topology generated by the family of pseudometrics defined via
[TABLE]
for compact sets , , , and two elements (\psi,\mathbbmss{v}),(\varphi,\mathbbmss{u})\in\mathrm{CoMor}\big{(}(\alpha,\mathfrak{u}),(\beta,\mathfrak{v})\big{)}. 2. (ii)
Suppose that both and are gently twisted actions. On \mathrm{CoMor}_{\mathfrak{p}}\big{(}(\alpha,\mathfrak{u}),(\beta,\mathfrak{v})\big{)}, we consider the topology generated by the family of pseudometrics defined via
[TABLE]
for compact sets and , and (\psi,\mathbbmss{v}),(\varphi,\mathbbmss{u})\in\mathrm{CoMor}_{\mathfrak{p}}\big{(}(\alpha,\mathfrak{u}),(\beta,\mathfrak{v})\big{)}.
Lemma \thelemmaCT.
Let and be two twisted actions on \cstar-algebras. Let (\varphi^{\lambda},\mathbbmss{u}^{\lambda})\in\mathrm{CoMor}\big{(}(\alpha,\mathfrak{u}),(\beta,\mathfrak{v})\big{)} be a net.
- (i)
* satisfies the Cauchy criterion with respect to every pseudometric in subsection 2.1(i)** if and only if the nets*
[TABLE]
are Cauchy for all , , , the latter two uniformly over compact sets . 2. (ii)
* converges to if and only if in the point-norm topology, and in the strict topology, uniformly over compact sets . In particular, the topology from subsection 2.1(i)** is complete.*
Proof.
Let us make an intermediate observation: If is a cocycle morphism, then it follows from the cocycle identity(e1.2) applied to that
[TABLE]
for all .
(i): The âifâ part is true by definition, so let us show the âonly ifâ part. Let (\varphi^{\lambda},\mathbbmss{u}^{\lambda})\in\mathrm{CoMor}\big{(}(\alpha,\mathfrak{u}),(\beta,\mathfrak{v})\big{)} be a net that satisfies the Cauchy criterion with respect to every pseudometric above. Then by definition, we can see that is a Cauchy net in for every .
Now let be a fixed element and a compact set. Evidently, the net satisfies the Cauchy criterion uniformly over . We want to show that this is also the case for the net . We consider the compact set defined as
[TABLE]
Let . Since satisfies the Cauchy criterion with respect to the pseudometric , let us choose such that for all , one has d_{0,\mathcal{F}^{B},K^{-1}}\big{(}(\varphi^{\lambda_{1}},\mathbbmss{u}^{\lambda_{1}}),(\varphi^{\lambda_{2}},\mathbbmss{u}^{\lambda_{2}})\big{)}<\varepsilon. Fix such and . Then
[TABLE]
(ii): The âifâ part is true by definition of the pseudometrics in subsection 2.1(i). For the âonly ifâ part, assume that indeed converges to . Then due to (i), the net is Cauchy in point-norm, and the nets and are Cauchy for all and , uniformly over compact sets . Since the set of -homomorphisms is complete in the point-norm topology and the unitary group is complete in the strict topology, it follows that converges in point-norm to a -homomorphism , and pointwise-strictly converges to a strictly continuous map . Since multiplication on is (jointly) strictly continuous on bounded sets, it follows immediately that the pair satisfies conditions (e1.1) and (e1.2), turning it into a cocycle morphism. Since all the topologies under consideration are Hausdorff, it follows that this limit must be equal to the pair . â
Lemma \thelemmaCT.
Let , and be three twisted actions on \cstar-algebras. Then the composition map
[TABLE]
given by
[TABLE]
is jointly continuous.
Proof.
Suppose that and are two convergent nets (indexed over the same directed set) with respective limits and . With respect to the ordinary composition of -homomorphisms, it is evident that
[TABLE]
holds in the point-norm topology. In order to show the claim, we have to show that also
[TABLE]
holds for all and compact sets .
Using the fact that is strictly continuous and that converges to in the point-norm topology, we indeed obtain for every compact set and that
[TABLE]
â
2.2. Approximate unitary equivalence
Definition \thedefiCT.
Let and be two twisted actions on \cstar-algebras. Let
[TABLE]
be a cocycle representation and let be a pair consisting of a -homomorphism and a strictly continuous map .
- (i)
The pairs and are unitarily equivalent, if there exists a unitary such that
[TABLE]
If this is the case, then automatically becomes another cocycle representation. We write (\varphi,\mathbbmss{u})\hskip 1.42262pt\sim_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\psi,\mathbbmss{v}). If one can in fact choose , then we say that and are properly unitarily equivalent. 2. (ii)
We say that is an approximate unitary conjugate of , if there exists a net of unitaries such that
[TABLE]
and
[TABLE]
for all , and every compact set . We denote this relation by (\psi,\mathbbmss{v})\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\varphi,\mathbbmss{u}). It follows that is automatically a cocycle representation. If is in fact a cocycle morphism into , then so is , and we have that the net converges to in the topology from subsection 2.1(i). We furthermore say that and are approximately unitarily equivalent, written (\varphi,\mathbbmss{u})\hskip 1.99168pt\approx_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\psi,\mathbbmss{v}), if (\psi,\mathbbmss{v})\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\varphi,\mathbbmss{u}) and (\varphi,\mathbbmss{u})\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\psi,\mathbbmss{v}). 3. (iii)
The pairs and are called properly approximately unitarily equivalent, if there exists a net of unitaries such that
[TABLE]
for all , and
[TABLE]
for every compact set . We write (\varphi,\mathbbmss{u})\hskip 2.84526pt\approx_{\makebox[0.0pt]{\footnotesize\hskip 7.8245pt\mathrm{pu}}}\hskip 5.69054pt(\psi,\mathbbmss{v}). If and are gently twisted actions and and are proper cocycle morphisms, then this means that the net converges to in the topology from subsection 2.1(ii).
Remark \theremCT.
The notion above in subsection 2.2(ii) warrants a word of caution that does not apply in (iii). It would be tempting to guess that the relation (\psi,\mathbbmss{v})\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\varphi,\mathbbmss{u}) ought to imply (\varphi,\mathbbmss{u})\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\psi,\mathbbmss{v}), or even that should imply , but this is false in general. This is due to the fact that if one equips (the unit ball of) a multiplier algebra with a metric inducing the strict topology, it can happen that inner automorphisms severely distort this metric. The following idea yields many possible counterexamples to the above. Consider , , and equip these with the trivial -actions. If we assume that we are dealing with unital cocycle representations , then these are nothing but strictly continuous unitary representations of on the Hilbert space . For the purpose of this remark let us denote the trivial one just by the symbol . It is clear that if is any unitary representation, then \mathbbmss{v}\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt\mathbf{1} implies . On the other hand, for any unitary representation , we claim that \mathbf{1}\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt\mathbbmss{u}. Indeed, if is an orthonormal basis, simply consider the sequence of unitaries given by
[TABLE]
Due to the fact that unitaries in the range of are compact perturbations of the unit, it is immediate that strictly and uniformly over compact subsets of . For familiar examples of groups such as , it is of course possible to choose such a map to be injective, hence we see that the relation \hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt is not at all symmetric in general.
In conclusion, in order to avoid logical pitfalls and to ensure that one is indeed working with an equivalence relation, one should distinguish \hskip 1.99168pt\approx_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt and \hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt , unless a given set of assumptions forces the symmetry already. For example, if and is non-degenerate, then it indeed follows that also . This is a consequence of the following (heuristic) calculation for large enough :
[TABLE]
In summary, if we consider only non-degenerate cocycle morphisms, then the relations \hskip 1.99168pt\approx_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt and \hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt do indeed coincide.
Remark \theremCT.
If we restrict this general formalism to the special case of genuine -\cstar-algebras and equivariant -homomorphisms , then unitary equivalence in the above sense implemented by forces the identity . In other words, the unitary needs to be a fixed point under the action .
Furthermore, if a net implements approximate unitary equivalence between two non-degenerate equivariant -homomorphisms (but viewed as cocycle morphisms), then this implies that \big{(}v_{\lambda}-\beta_{g}(v_{\lambda})\big{)} strictly converges to zero uniformly over compact subsets of . In particular, we recover the concept of (approximate) -unitary equivalence for equivariant maps, which has been considered in prior work; see for example [85, Definition 1.15]. The main thrust of the next few (sub-)sections is that the resulting formalism allows us to obtain a natural framework for the Elliott intertwining machinery on (twisted) -\cstar-algebras.
Proposition \thepropCT.
The relation of (proper) approximate unitary equivalence behaves well with respect to compositions in the following sense.
- (i)
Suppose that
[TABLE]
for are two pairs of cocycle morphisms between twisted -\cstar*-algebras. If*
[TABLE]
then
[TABLE] 2. (ii)
Suppose that
[TABLE]
for are two pairs of proper cocycle morphisms between gently twisted -\cstar*-algebras. If*
[TABLE]
then
[TABLE]
Proof.
(i): Let be a net that witnesses the relation (\varphi_{2},\mathbbmss{u}^{(2)})\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\varphi_{1},\mathbbmss{u}^{(1)}) and a net witnessing (\psi_{2},\mathbbmss{v}^{(2)})\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\psi_{1},\mathbbmss{v}^{(1)}). We consider the product as directed sets, i.e., we equip it with the order precisely when and . After replacing these nets, if necessary, we may assume that they are both indexed by , i.e., via setting and .
Then it follows directly from subsection 2.1 that
[TABLE]
In other words, the net \big{(}v_{\lambda}\psi_{1}^{+}(u_{\lambda})\big{)}_{\lambda\in\Lambda} witnesses the relation (\psi_{2},\mathbbmss{v}^{(2)})\circ(\varphi_{2},\mathbbmss{u}^{(2)})\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\psi_{1},\mathbbmss{v}^{(1)})\circ(\varphi_{1},\mathbbmss{u}^{(1)}).
(ii): Since the involved objects are gently twisted actions and the involved morphisms are proper cocycle morphisms, it follows that we get well-defined unital cocycle morphisms
[TABLE]
The assumption
[TABLE]
is the same as
[TABLE]
and as above we may find nets and witnessing this fact. It follows from the first part that the net \big{(}v_{\mu}\psi_{1}^{\dagger}(u_{\lambda})\big{)}_{(\lambda,\mu)\in\Lambda_{1}\times\Lambda_{2}}\in\mathcal{U}(\mathbf{1}+C) witnesses proper approximate unitary equivalence between the compositions and . â
3. Two-sided Elliott intertwining
In order to motivate what will follow below, let us already deduce a non-trivial consequence of what we have so far:
Theorem \thetheoremCT.
Let and be gently twisted actions on separable \cstar-algebras. Suppose that
[TABLE]
are two proper cocycle morphisms such that (\psi,\mathbbmss{v})\circ(\varphi,\mathbbmss{u})\hskip 2.84526pt\approx_{\makebox[0.0pt]{\footnotesize\hskip 7.8245pt\mathrm{pu}}}\hskip 5.69054pt\operatorname{id}_{A} and (\varphi,\mathbbmss{u})\circ(\psi,\mathbbmss{v})\hskip 2.84526pt\approx_{\makebox[0.0pt]{\footnotesize\hskip 7.8245pt\mathrm{pu}}}\hskip 5.69054pt\operatorname{id}_{B}. Then there exist mutually inverse proper cocycle conjugacies
[TABLE]
with (\varphi,\mathbbmss{u})\hskip 2.84526pt\approx_{\makebox[0.0pt]{\footnotesize\hskip 7.8245pt\mathrm{pu}}}\hskip 5.69054pt(\Phi,\mathbb{U}) and (\psi,\mathbbmss{v})\hskip 2.84526pt\approx_{\makebox[0.0pt]{\footnotesize\hskip 7.8245pt\mathrm{pu}}}\hskip 5.69054pt(\Psi,\mathbb{V}).
Proof.
Let us argue that this arises as a special case of âTheorem 2â in [13, pp. 35â36]. Our underlying category is the full subcategory of the twisted proper cocycle category (cf. subsection 1.3) defined by the separable objects. The definition of \hskip 2.84526pt\approx_{\makebox[0.0pt]{\footnotesize\hskip 7.8245pt\mathrm{pu}}}\hskip 5.69054pt is based on a compatible notion of inner automorphisms which always form a normal subgroup in the automorphism group; see subsection 1.3 and subsection 1.3.
Let and be two gently twisted actions on separable \cstar-algebras. Choose a sequence of contractions that is dense in the unit ball, and an increasing sequence of compact sets with . Then the assignment
[TABLE]
defines a metric on that describes the topology in subsection 2.1(ii), and turns it into a Polish space. Moreover, we see that composition with an inner cocycle automorphism (from the left) is isometric. It is also clear that composition of proper cocycle morphisms is continuous in both variables with respect to the topology in subsection 2.1(ii); the proof of this is analogous to but much easier than subsection 2.1. In summary, Elliottâs theorem applies in this context and directly gives us the desired statement. â
Remark \theremCT.
For the rest of this section we will focus on establishing other versions or generalizations of the theorem above. Firstly, it is desirable to have an Elliott intertwining machinery in a more general context, at the very least concerning more general inductive limits like in [78, Corollary 2.3.3]. Unfortunately this is not an outcome of Elliottâs high-level abstract approach in [13], so we need to work this out explicitly. Secondly, it is desirable to also have an Elliott intertwining machinery in the non-proper picture. Although the completely analogous theorem to the above is true in that context, as we will show below, it is important to note that this does not arise as a direct consequence from Elliottâs theorem. The latter only applies if in every space of morphisms, there is a compatible metric that turns the composition map with an inner automorphism (from the left) into an isometric map. The discussion in subsection 2.2 shows that for some non-unital \cstar-algebras such as , there does not exist such a metric with respect to the topology in subsection 2.1(i).
Definition \thedefiCT (cf. [78, Definition 2.3.1]).
Let and be sequences of twisted actions on separable \cstar-algebras. Let
[TABLE]
and
[TABLE]
be sequences of cocycle morphisms, which we view as two inductive systems in the category . Adopt the notations for the maps and as in section 2.
Consider two sequences of cocycle morphisms
[TABLE]
and
[TABLE]
fitting into the (not necessarily commutative) diagram888To lighten notation, we shall only carry around the underlying \cstar-algebras and -homomorphisms to denote the objects and arrows in the diagram.
[TABLE]
Let us call diagram (e3.1) an approximate cocycle intertwining, if the following holds: There exist self-adjoint compact sets , , increasing compact sets and numbers satisfying999This list of approximate relations is supposed to specify that the composition is sufficiently close to and that is sufficiently close to with respect to the topology in subsection 2.1(i).
- (i)
for all ; 2. (ii)
for all ; 3. (iii)
for all , ; 4. (iv)
for all , ; 5. (v)
; 6. (vi)
and ; 7. (vii)
;
; 8. (viii)
and are dense for all ; 9. (ix)
.
Proposition \thepropCT (cf. [78, Proposition 2.3.2]).
Let the diagram (e3.1) describe an approximate cocycle intertwining. Consider the inductive limits
[TABLE]
and
[TABLE]
Then the formulas101010Here âs-limâ signifies that the limit is taken in the strict topology.
[TABLE]
[TABLE]
and
[TABLE]
[TABLE]
define mutually inverse cocycle conjugacies and .
Proof.
The fact that the formulas (e3.2)+(e3.4) define mutually inverse isomorphisms between the \cstar-algebras and is already proved in [78, Proposition 2.3.2].
We shall prove that is indeed a well-defined cocycle morphism. First we need to show that the sequence of maps described in (e3.3) converges strictly and uniformly over compact sets in . We note that by the definition of composition, we have for all that
[TABLE]
All three pairs in the second line of the equation are cocycle morphisms, and hence so is the pair on the first line. Fix one of the compact sets and an element . In light of (v), (viii) and subsection 2.1, it is enough to show that maps of the form
[TABLE]
satisfy the Cauchy criterion (over ) uniformly in .
For this we compute for all that
For convenience, let us define for , which defines another summable sequence of positive numbers as this was the case for by (ix). If we apply the -homomorphism to all of the above and multiply from the right with the unitary element \psi_{k+2,\infty}^{+}\big{(}\mathbbmss{y}^{(k+1)}_{g}\mathbb{V}^{(k+1)*}_{g}\big{)}, we obtain
[TABLE]
It now follows from the summability of the sequence that the sequence of maps in (e3.6) satisfies the Cauchy criterion uniformly over compact subsets of . Applying subsection 2.1 as indicated above, it follows that the pointwise strict limit in (e3.3) exists, and it follows that is a cocycle morphism from to . Since by definition of the limit twisted action on , we have for all , we immediately get the cocycle identity (e1.2) for the pair .
In order for to be a cocycle morphism, we still need to verify the equivariance condition (e1.1). This follows from the following calculation for all and , where we make use of the inductive limit construction from section 2:111111The limits in this calculation refer to the point-norm limit of maps.
Since is arbitrary, we obtain , which concludes the proof that is a cocycle conjugacy.
The analogous justification is valid for why the pair is a well-defined cocycle conjugacy arising from the limit of maps in (e3.4) and (e3.5). We will omit the detailed argument as it follows by simply repeating the arguments above but exchanging the roles of and .
Lastly, we need show that , which will imply that and are mutually inverse.
We compute from (e3.2)+(e3.5) and (i)+(ii)+(iii)+(viii) that
[TABLE]
Here all limits are with respect to the strict topology. This finishes the proof. â
Remark \theremCT.
Let us once more reflect on what the statement of section 3 means. Keeping in mind section 2, we recall from section 2 that the universal morphism is given by the pair and similarly is given by the pair . Moreover we have for all that
[TABLE]
and
[TABLE]
Then we have
[TABLE]
The second argument in this pair is given as the pointwise strict limit
[TABLE]
Using the topology on as in subsection 2.1, this leads to the limit formula
[TABLE]
Arguing in the other direction we can similarly express
[TABLE]
This better highlights the conceptual parallel between section 3 and [78, Proposition 2.3.2].
We can now use section 3 and apply it to get an refined analog of [78, Corollary 2.3.3].
Theorem \thetheoremCT.
Let be a second-countable, locally compact group. Let and be sequences of twisted actions on separable \cstar-algebras. Let
[TABLE]
and
[TABLE]
be sequences of cocycle morphisms, which we view as two inductive systems in the category . Denote by and the universal cocycle morphisms into the respective limit.
Consider two sequences of cocycle morphisms
[TABLE]
and
[TABLE]
fitting into the (not necessarily commutative) diagram
[TABLE]
Suppose that
[TABLE]
holds for all . Then there exist mutually inverse cocycle conjugacies and .
If we may in fact assume that
[TABLE]
and
[TABLE]
holds for all , then we can arrange the cocycle conjugacies to satisfy
[TABLE]
and
[TABLE]
for all .
Proof.
We show that this arises as an application of section 3. For this purpose we need to arrange for an approximate cocycle intertwining diagram in the sense of section 3. Choose any (decreasing) summable null sequence , for example . Let be any increasing sequence of compact sets with . Inductively, choose finite sets of elements and such that the inclusions
[TABLE]
are dense for all .
Set , and
[TABLE]
Using the assumption that
[TABLE]
we may find a unitary such that if we set , we have
[TABLE]
and
[TABLE]
for all . We set
[TABLE]
Here we note that by definition, so this factor is redundant in this particular step. Using the assumption that
[TABLE]
we may find a unitary such that if we set , we have
[TABLE]
and
[TABLE]
for all . We set
[TABLE]
Using the assumption that
[TABLE]
we may find a unitary such that if we set , we have
[TABLE]
and
[TABLE]
for all . We set
[TABLE]
We continue like this by induction. This allows us to find compact subsets and , unitaries and such that if we define and , then the diagram
[TABLE]
becomes an approximate cocycle intertwining in the sense of section 3. More precisely, every condition in section 3 except for (viii) is automatically ensured by the inductive choice of this data with respect to . As is clear from the above, we can make our choices so that
[TABLE]
holds for every . By the initial choice of the elements and as to satisfy (e3.7), this implies for every that the subsets
[TABLE]
are indeed dense, which verifies condition (viii).
By applying section 3, we obtain mutually inverse cocycle conjugacies and . As noted in section 3, we have for every that
[TABLE]
Now let us pass to the stronger assumption in the statement of the theorem, namely that (\theta_{n},\mathbbmss{y}^{(n)})\circ(\kappa_{n},\mathbbmss{x}^{(n)})\hskip 1.99168pt\approx_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\psi_{n},\mathbbmss{v}^{(n)}) and (\kappa_{n+1},\mathbbmss{x}^{(n+1)})\circ(\theta_{n},\mathbbmss{y}^{(n)})\hskip 1.99168pt\approx_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\varphi_{n},\mathbbmss{u}^{(n)}) holds for all . For each , we have that is unitarily equivalent to via . Hence for all , we obtain121212This is the only place where it is important that we assume the symmetric version of approximate unitary equivalence. The following step towards (\theta,\mathbbmss{y})\circ(\varphi,\mathbbmss{u})_{n,\infty}\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\psi,\mathbbmss{v})_{n+1,\infty}\circ(\theta_{n},\mathbbmss{y}^{(n)}) uses at the very least the relations (\kappa_{n+1},\mathbbmss{x}^{(n+1)})\circ(\theta_{n},\mathbbmss{y}^{(n)})\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\varphi_{n},\mathbbmss{u}^{(n)}) and (\psi_{n},\mathbbmss{v}^{(n)})\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\kappa_{n+1},\mathbbmss{x}^{(n+1)})\circ(\theta_{n},\mathbbmss{y}^{(n)}). The proof for (\kappa,\mathbbmss{x})\circ(\psi,\mathbbmss{v})_{n,\infty}\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\varphi,\mathbbmss{u})_{n,\infty}\circ(\kappa_{n},\mathbbmss{x}^{(n)}) need in turn the same relations in the reverse direction.
[TABLE]
If we compose these morphisms from the left with and let , it follows that
[TABLE]
Since was arbitrary, this proves the first half of the statement. The other half, namely that
[TABLE]
holds for all , follows in a completely analogous fashion, by exchanging the roles of and . â
The following further special case yields a natural setting which showcases what type of existence and uniqueness theorems need to be proved in order to classify (twisted) -\cstar-algebras up to cocycle conjugacy.
Corollary \thecorCT.
Let be a second-countable, locally compact group. Let and be two twisted actions on separable \cstar-algebras. Suppose that
[TABLE]
are two cocycle morphisms such that
[TABLE]
Then there exist mutually inverse cocycle conjugacies
[TABLE]
such that
[TABLE]
Proof.
We consider both twisted -\cstar-algebras in the assumption as trivial inductive limits
[TABLE]
and
[TABLE]
Then the (not necessarily commuting) diagram of cocycle morphisms
[TABLE]
fits the assumptions in section 3, so we can simply repeat the same argument as in its proof. Then the resulting mutually inverse cocycle conjugacies and arise as certain limits
[TABLE]
where (\zeta_{k},\mathbb{Y}^{(k)})\hskip 1.42262pt\sim_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\varphi,\mathbbmss{u}) and (\eta_{k},\mathbb{X}^{(k)})\hskip 1.42262pt\sim_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\psi,\mathbbmss{v}) for all . Hence (\Phi,\mathbb{U})\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\varphi,\mathbbmss{u}) and (\Psi,\mathbb{V})\hskip 1.99168pt\precapprox_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\psi,\mathbbmss{v}).131313Due to the trivial form of the inductive systems, the need for the symmetric assumption in section 3 disappears because the relevant step in the proof becomes vacuous. This finishes the proof. â
For completeness, we shall also record the analogous result to the above in the proper picture, which one can obtain as a direct consequence and which generalizes section 3.
Corollary \thecorCT.
Let be a second-countable, locally compact group. Let and be sequences of gently twisted actions on separable \cstar-algebras. Let
[TABLE]
and
[TABLE]
be sequences of proper cocycle morphisms, which we view as two inductive systems in the category . Denote by and the universal cocycle morphisms into the respective limit.
Consider two sequences of proper cocycle morphisms
[TABLE]
and
[TABLE]
fitting into the (not necessarily commutative) diagram
[TABLE]
Suppose that
[TABLE]
and
[TABLE]
holds for all . Then there exist mutually inverse and proper cocycle conjugacies and satisfying
[TABLE]
and
[TABLE]
for all .
Proof.
By definition of being a gently twisted action, we have canonical extensions to twisted actions and to the proper unitizations of the involved \cstar-algebras. Since all of the involved cocycle morphisms in the assumption are proper, they uniquely extend to unital cocycle morphisms
[TABLE]
Correspondingly, we may identify with the limit with respect to the inductive system given by the cocycle morphisms , and analogously is the limit with respect to . By definition of proper approximate unitary equivalence, our assumptions imply
[TABLE]
and
[TABLE]
for all . Therefore we are in the situation to apply section 3 to the (possibly non-commuting) diagram of unitized cocycle morphisms above. It follows that there exist mutually inverse cocycle conjugacies and satisfying
[TABLE]
and
[TABLE]
for all . (Keep in mind subsection 2.2 here.)
We first note that the -homomorphism sends to . Since is obviously closed under unitary equivalence inside , it follows that the -homomorphism also sends to . Since this is true for every , we conclude that has image in . Hence is the unique unital extension of the isomorphism . By exchanging the roles of and and repeating this argument, we likewise conclude that is the unique unital extension of the isomorphism .
Lastly, we note that for any pair of unitaries and , one has for any , since the automorphism fixes the scalar part of . By assumption, the map belonging to the composition takes values in , and so the same will be true for the composition . Since the map belonging to this composition is given by and the first factor is in , we hence conclude that for all . By exchanging the roles of and once again and repeating this argument, we likelywise conclude that takes values in .
In summary, we have that and are mutually inverse and proper cocycle conjugacies. The statement that
[TABLE]
and
[TABLE]
holds for all , follows from the above. A priori, the approximate unitary equivalence between these compositions is implemented by sequences of unitaries in and , respectively. By rescaling with their scalar parts, we may assume that the unitaries are in and , respectively, so we indeed get proper approximate unitary equivalence as desired. â
4. One-sided intertwining
4.1. Asymptotic unitary equivalence
Definition \thedefiCT.
Let and be two twisted actions on \cstar-algebras. Suppose that is separable and is -unital. Let
[TABLE]
be a cocycle representation and let be a pair consisting of a -homomorphism and a strictly continuous map .
- (i)
We say that is an asymptotic unitary conjugate of , if there exists a strictly continuous map such that
[TABLE]
and
[TABLE]
for all , and every compact set . As before, it follows that is automatically a cocycle representation. If is in fact a cocycle morphism into , then so is , and we have that the net converges to in the topology from subsection 2.1(i). Similar to before, we say that and are asymptotically unitarily equivalent, written (\varphi,\mathbbmss{u})\hskip 1.99168pt\approxeq_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\psi,\mathbbmss{v}), if they are asymptotic unitary conjugates of each other. 2. (ii)
The pairs and are properly asymptotically unitarily equivalent, if there exists a norm-continuous map such that
[TABLE]
for all , and
[TABLE]
for every compact set . We write (\varphi,\mathbbmss{u})\hskip 1.99168pt\approxeq_{\makebox[0.0pt]{\footnotesize\hskip 8.53581pt\mathrm{pu}}}\hskip 6.25958pt(\psi,\mathbbmss{v}). If and are gently twisted actions and and are proper cocycle morphisms, then this means that the net converges to in the topology from subsection 2.1(ii).
In this subsection we will characterize when a given (non-degenerate) cocycle morphism is asymptotically unitarily equivalent to a cocycle conjugacy, which will culminate in a continuous version of section 3.
Proposition \thepropCT.
Let
[TABLE]
be two cocycle morphisms between separable twisted -\cstar-algebras. If both and are asymptotically unitarily equivalent to cocycle conjugacies, then the composition is asymptotically unitarily equivalent to a cocycle conjugacy.
Proof.
This is a trivial consequence from the proof of subsection 2.2. If there exist cocycle conjugacies
[TABLE]
with
[TABLE]
then it easily follows from subsection 2.1 that also
[TABLE]
â
The following is a variant of [82, Proposition 2.1]. Although half of the proof is very similar, we shall give the full argument for completeness.
Proposition \thepropCT.
Let be a second-countable, locally compact group. Suppose that and are twisted actions on separable \cstar-algebras. Let be a cocycle morphism such that is injective and non-degenerate. Then is asymptotically unitarily equivalent to a cocycle conjugacy if and only if the following is true:
For all finite sets , , compact sets , and , there exists a (strictly continuous) unitary path with and satisfying
- âą
* for all ;*
- âą
\operatorname{dist}\big{(}z_{1}^{*}bz_{1},\varphi(A)\big{)}\leq\varepsilon* for all ;*
- âą
* for all .*
Remark**.**
Before we resume with the proof, let us contemplate the above non-degeneracy assumption on . I do not know if it is a necessary assumption, but it appears to be somewhat relevant for the âonly ifâ part, which is important for the application that follows afterwards. The difficulty essentially lies in arranging the statement in the last bullet point for all (and not just for ), which under the non-degeneracy assumption is possible below because it suffices to consider only . In particular, when there are no group actions involved, the assumption can be relaxed to saying that is extendible.
Proof.
Let us first show the âonly ifâ part. Suppose that is asymptotically unitarily equivalent to a cocycle conjugacy . Since is non-degenerate, this means (cf. subsection 2.2) that there exists a strictly continuous path of unitaries such that
[TABLE]
and
[TABLE]
for all compact sets and .
Now let be given as in the statement. As pointed out in the remark above, we assume to be non-degenerate, and hence we may substitute the element in the last bullet point for for . By the above, we may then choose a number such that
[TABLE]
and
[TABLE]
Then, choose a bigger number such that
[TABLE]
We set for . We claim that this unitary path satisfies the desired properties. We first observe for all , and that
[TABLE]
In summary, we obtain
[TABLE]
Applying (e4.1) twice, we clearly have
[TABLE]
Furthermore, we have for all that
[TABLE]
This shows that is indeed a unitary path with the desired properties.
Now let us show the âifâ part. Let and be dense sequences. Write for an increasing union of compact subsets . We are going to perturb with paths of unitaries step by step:
In the first step, choose some and with such that for all , we have
- âą
;
- âą
;
- âą
for all .
In the second step, choose and with such that for every we have
- âą
for ;
- âą
for ;
- âą
;
- âą
for all and .
Now assume that for some , we have found and satisfying for every that
[TABLE]
[TABLE]
[TABLE]
[TABLE]
for all and .
Then we can again apply our assumptions to find with and so that for every we have
for ; 2.
for ; 3.
for and ; 4.
for all and .
Carry on inductively. We define a norm-continuous path of unitaries via for with . This map is indeed continuous since every path starts at the unit. We obtain a continuous path of cocycle morphisms for via and . We claim that this converges to a cocycle conjugacy when we let .
By condition (e4.5), we can immediately observe that the net is Cauchy for all . Since the set was assumed to be dense, this implies that the net converges to some -homomorphism in the point-norm topology. Since was assumed to be injective, so is . In order to show that is an isomorphism, it suffices to show that it has dense image. Let be given. Then for , condition (e4.4) tells us that . At the same time, it follows from condition (e4.6) that
[TABLE]
This culminates in the estimate for all and . Since the set was assumed to be dense, it follows that has dense image and is therefore an isomorphism.
By condition (e4.7), we have that the assignment
[TABLE]
yields a Cauchy net for every and , with uniformity on compact subsets of . Since was assumed to be dense, it follows that this is true for any element in place of . By subsection 2.1, it follows that the strict limit
[TABLE]
exists, and the convergence is uniform over compact sets in . Hence is the desired cocycle conjugacy. Since is non-degenerate, it follows from subsection 2.2 that (\psi,\mathbbmss{v})\hskip 1.99168pt\approxeq_{\makebox[0.0pt]{\footnotesize\hskip 4.2679pt\mathrm{u}}}\hskip 2.84526pt(\varphi,\mathbbmss{u}). â
Lemma \thelemmaCT.
Let be a second-countable, locally compact group. Let be a sequence of twisted actions on separable \cstar-algebras. Let
[TABLE]
be a sequence of injective and non-degenerate cocycle morphisms with inductive limit . Suppose that for every , is asymptotically unitarily equivalent to a cocycle conjugacy. Then it follows that
[TABLE]
is asymptotically unitarily equivalent to a cocycle conjugacy.141414In this context remember from the construction of the limit that the -homomorphism is genuinely equivariant here, so we identify it with the cocycle morphism .
Proof.
We adopt the notation from section 2.
Let and be finite sets. Let be compact and . In order to obtain the claim, we are going to verify the condition in subsection 4.1 for . After making a small perturbation to , we may assume without loss of generality that there is some large enough and finite set such that . By subsection 4.1, the cocycle morphism
[TABLE]
is asymptotically unitarily equivalent to a cocycle conjugacy. By subsection 4.1, we hence find a unitary path satisfying
for all ; 2.
for all ; 3.
for all .
Set . Then we evidently have the two conditions
[TABLE]
and
[TABLE]
Furthermore, we have for all , and that
[TABLE]
In summary, we have verified the condition in subsection 4.1 for , which implies that it is indeed asymptotically unitarily equivalent to a cocycle conjugacy. â
The following should be seen as a continuous version of section 3. Although some version of this statement appears to be folklore in the case due to Kirchbergâs famous unpublished preprint [47], there does to my knowledge not exist a published proof even of that statement anywhere. The proof presented below is in turn inspired by Kirchbergâs original approach. As mentioned further above, when there are no group actions involved, one can relax the non-degeneracy assumption on the involved -homomorphisms to extendability.
Theorem \thetheoremCT.
Let be a second-countable, locally compact group. Let and be two twisted actions on separable \cstar-algebras. Suppose that
[TABLE]
are two non-degenerate cocycle morphisms such that
[TABLE]
Then there exist mutually inverse cocycle conjugacies
[TABLE]
such that
[TABLE]
Proof.
We consider the two compositions
[TABLE]
and
[TABLE]
Then evidently the following represents a commutative diagram of cocycle morphisms:
[TABLE]
Let us we form the stationary inductive limits
[TABLE]
and
[TABLE]
We consider the universal (equivariant) embeddings
[TABLE]
from each respective first building block into the limit. By the universal properties of both inductive limits, the commutative diagram induces mutually inverse cocycle conjugacies
[TABLE]
and
[TABLE]
such that and . By assumption, the cocycle morphism is asymptotically unitarily equivalent to an isomorphism, namely the identity map on . By subsection 4.1, it follows that is asymptotically unitarily equivalent to a cocycle conjugacy . Analogously it follows that is asymptotically unitarily equivalent to a cocycle conjugacy . Hence we conclude
[TABLE]
So is one of the desired cocycle conjugacies. For its inverse
[TABLE]
we similarly observe that
[TABLE]
This finishes the proof. â
Corollary \thecorCT.
Let be a second-countable, locally compact group. Let and be two gently twisted actions on separable \cstar-algebras. Suppose that
[TABLE]
are two proper cocycle morphisms such that
[TABLE]
Then is properly asymptotically unitarily equivalent to a proper cocycle conjugacy.
Proof.
Since the assumption involves gently twisted actions and proper cocycle morphisms, we may pass to the unitized \cstar-algebras and consider the unital cocycle morphisms
[TABLE]
and
[TABLE]
Since is properly asymptotically unitarily equivalent to , it follows that is asymptotically unitarily equivalent to . Analogously, the composition is asymptotically unitarily equivalent to . subsection 4.1 implies that is asymptotically unitarily equivalent to a cocycle conjugacy . Since sends to and to , the same follows for . Hence for a unique isomorphism . If is the unitary path witnessing the asymptotic equivalence between and , we may divide them by their scalar parts to ensure that . Hence is indeed properly asymptotically unitarily equivalent to the proper cocycle conjugacy . â
Remark \theremCT.
The results in this subsection so far are also true if âasymptotic unitary equivalenceâ is replaced with âapproximate unitary equivalenceâ. The only difference is that in subsection 4.1, the unitary path needs to be replaced by a single unitary, and the assumption is dropped; see also [85, Proposition 2.1].
4.2. From approximate morphisms to genuine morphisms
This subsection should be viewed as a dynamical generalization of [17, Section 4]; see also [75, Proposition 1.3.7] for a related statement.
Notation \thenotaCT.
In this subsection we will use the sequence algebra construction. That is, if is a \cstar-algebra, then its sequence algebra is denoted by
[TABLE]
If is a gently twisted action of a locally compact group, then we denote by the induced map into the automorphism group arising from applying componentwise. In general, will not be point-norm continuous. However, we may define the \cstar-subalgebra
[TABLE]
We therefore obtain a gently twisted action in the ordinary sense. The canonical inclusion as constant sequences is then equivariant. Given a gently twisted -\cstar-algebra , we call a proper cocycle morphism constant, if it factors through via the constant inclusion.
If is any map with , then it induces an endomorphism on via . Indeed, we see that for all , and hence induces an equivariant endomorphism on the gently twisted -\cstar-algebra .
Remark \theremCT.
Suppose that in the above situation, we are given a continuous map . Given any compact set , the restriction can be viewed as a single unitary
[TABLE]
It hence lifts to a sequence of unitaries \mathbbmss{u}^{(n)}\in\mathcal{U}\big{(}\mathbf{1}+\mathcal{C}(K,A)\big{)}. In other words, there is a sequence of continuous maps such that for all , the sequence represents .
Now let be an increasing sequence of compact sets with . For every , we can repeat the argument above and find a sequence of continuous maps such that it uniformly represents \mathbbmss{u}|_{K_{n}}\in\mathcal{U}\big{(}\mathbf{1}+\mathcal{C}(K_{n},B_{\infty})\big{)}. Then we have for all that
[TABLE]
holds for all . By induction, we can hence choose an increasing sequence such that
[TABLE]
So we see that the sequence of (partially defined) continuous maps on given by for is eventually defined on every compact subset of as , and represents the original map uniformly over compact subsets of .
Theorem \thetheoremCT.
Let be a second-countable, locally compact group. Let and be two gently twisted actions on \cstar-algebras. Suppose that is separable. Let
[TABLE]
be a proper cocycle morphism. Then the following are equivalent:
- (i)
* is properly unitarily equivalent to a constant proper cocycle morphism ;* 2. (ii)
For every map with , the two proper cocycle morphisms and are properly unitarily equivalent; 3. (iii)
Suppose that is a sequence of -linear maps lifting , and that is a sequence of partially defined, but eventually everywhere defined, maps from to lifting as in subsection 4.2.
Then for every , finite set , compact set and , there exists such that for every , there is a unitary such that
[TABLE]
Proof.
The implications (i)(ii) and (iii)(ii) are trivial.
(ii)(iii): We prove this by contradiction. Suppose that the claim does not hold. Then there exists , finite set , compact set and , such that for every there is such that no unitary satisfies the above property with in place of . Since the sequence of partial maps is eventually defined everywhere, we may assume without loss of generality (by choosing larger if necessary) that they are always defined on .
If we define via if and if , then evidently . By assumption is properly unitarily equivalent to . Find a unitary with , and represent it by a sequence of unitaries . We immediately get for all that
[TABLE]
Furthermore, we have
[TABLE]
But now we see that this yields a contradiction to our assumption.
(ii)+(iii)(i) Let be an increasing sequence of finite sets with dense union. Let be an increasing sequence of compact sets with . By inductively applying the statement in (iii), we may choose an increasing sequence of natural numbers with such that for every , the number satisfies the conclusion of (iii) in place of , with respect to in place of , in place of , in place of , and in place of . This allows us to find a sequence of unitaries such that
[TABLE]
Define the unitaries in for , and set .
Then evidently it follows for every that the sequence is Cauchy, and therefore has a limit in . Define via . Since as , it follows that is a -homomorphism. We have justified that the -homomorphism maps a dense subset into , which immediately implies that its entire range is in . We may therefore obtain a well-defined -homomorphism via for all .
For every compact set , the sequence of (partially defined) continuous maps n\mapsto\big{[}g\mapsto V_{n}^{*}\mathbbmss{u}^{(k_{n})}_{g}\beta_{g}(V_{n})\big{]} is eventually defined on and uniformly satisfies the Cauchy criterion in norm over . So we may define for all , which yields a continuous map into . For the unitary , this implies the equation for all . As both maps and are norm-continuous, we may conclude .
In summary, we get that is a proper cocycle morphism. By construction, it is properly unitarily equivalent to , which in turn is properly unitarily equivalent to by assumption. This finishes the proof. â
5. Strong self-absorption revisited
In this section we revisit the concept of strongly self-absorbing \cstar-dynamical systems [85, 84, 82] and make some observation based on the intertwining results obtained so far.
Definition \thedefiCT.
Let be a second-countable, locally compact group. Let be a separable unital \cstar-algebra and an action. Let be a separable \cstar-algebra and a twisted action. We say that strongly absorbs , if the equivariant embedding
[TABLE]
is approximately unitarily equivalent to a cocycle conjugacy.
Proposition \thepropCT.
Let be a second-countable, locally compact group. Let be a separable unital \cstar-algebra and an action. Let be a separable \cstar-algebra and a twisted action. Then strongly absorbs if and only if it strongly absorbs .
Proof.
Let us address the âonly ifâ part. We have that
[TABLE]
By assumption, strongly absorbs , which means that is approximately unitarily equivalent to a cocycle conjugacy. Hence the same is true for the connecting maps above. By subsection 4.1, it follows that the canonical embedding
[TABLE]
is also approximately unitarily equivalent to a cocycle conjugacy, which confirms the claim.
For the âifâ part, assume that strongly absorbs . Assume that is approximately unitarily equivalent to the cocycle conjugacy . Then
[TABLE]
is also a cocycle conjugacy. We moreover have
[TABLE]
Hence strongly absorbs . â
The following should be seen as a revised definition from [85].
Definition \thedefiCT (replacing [85, Definition 3.1]).
Let be a separable unital \cstar-algebra. Let be a second-countable, locally compact group, and let be an action. We say that is strongly self-absorbing, if strongly absorbs in the above sense.
Comparing the next proposition below with [85, Theorem 4.6] yields that the above definition turns out to coincide with the notion called âsemi-strongly self-absorbingâ in previous work. In hindsight this is hence the more natural concept compared to the one in [85, Definition 3.1], which was hinted at in its introduction but lacked hard evidence at the time.
Notation \thenotaCT.
For the rest of this section, we will use standard terminology involving central sequence algebras inspired by [48]. Let be a gently twisted action on a \cstar-algebra , and recall the terminology from subsection 4.2. Suppose that is an -invariant \cstar-subalgebra containing the set .
Then restrict to a genuine continuous action , which in turn induces a continuous action
[TABLE]
where . If is -unital, then always is a unital \cstar-algebra. If , then we abbreviate .
Recall that an action on a separable unital \cstar-algebra is said to have approximately -inner half-flip, if the two equivariant unital embeddings
[TABLE]
are approximately unitarily equivalent (as cocycle morphisms).
Proposition \thepropCT.
Let be an action on a separable, unital \cstar-algebra. The following are equivalent:
- (i)
* is strongly self-absorbing in the sense of section 5.* 2. (ii)
* has approximately -inner half-flip and there exists a unital equivariant -homomorphism from to .*
Proof.
(ii)(i): This follows directly from [85, Theorem 2.6]. Here note that the proof consisted in showing that the assumptions of [85, Proposition 2.1] (which coincides with the analog of subsection 4.1 referenced in subsection 4.1) hold for the equivariant embedding . Hence it is approximately unitarily equivalent to a cocycle conjugacy, which is what it means for to be strongly self-absorbing.
(i)(ii): If is strongly self-absorbing, then it follows by section 5 that is cocycle conjugate to . So the existence of some unital equivariant -homomorphism from to is trivial. In order to prove that has approximately -inner half-flip, we can proceed exactly as in the original proof by TomsâWinter in [94, Proposition 1.5]. Let be a cocycle conjugacy that is approximately unitarily equivalent to . Define the cocycle morphism on . Then
[TABLE]
Here we have used the equivalence between and from the fifth line onwards and applied the equivariant flip automorphism to it. This finishes the proof. â
We shall also revisit the equivariant McDuff-type theorem and prove a variant of [82, Theorem 2.2] intended for use in subsequent work, which requires a stronger assumption but yields a stronger statement.
Theorem \thetheoremCT.
Let be a second-countable, locally compact group. Let be a strongly self-absorbing action, and a gently twisted action on a separable \cstar-algebra. Suppose that is equivariantly -stable, i.e., is cocycle conjugate to for the JiangâSu algebra 151515See [41] for its introduction. In contrast to the redundance of -stability for strongly self-absorbing \cstar-algebras [96], note that there are examples of strongly self-absorbing actions that fail to be equivariantly -stable; see [84, Example 6.4].. Suppose that there exists a unital equivariant -homomorphism from to the central sequence algebra . Then the equivariant second-factor embedding
[TABLE]
is properly asymptotically unitarily equivalent to a proper cocycle conjugacy.
Proof.
This is almost the same as the proof of either [85, Theorem 3.7] or [82, Theorem 2.2], but with a slight additional twist, so we shall give it for completeness.
Since we assumed to be a gently twisted action, i.e., , we have that induces a genuine action on . Denote by
[TABLE]
the canonical (equivariant) surjection. We will use below in a crucial way that induces a surjection between the fixed point algebras with respect to the obvious action induced by on both sides, which follows from the fact that the kernel is a --ideal; see [84, Example 4.4 and Proposition 4.5].161616To be precise, only the case of genuine actions is treated in [84], but the general case is true with the same proof verbatim.
By assumption, we have an equivariant, unital -homomorphism from to \big{(}F_{\infty,\alpha}(A),\tilde{\alpha}_{\infty}\big{)}. Consider the canonical inclusions
[TABLE]
which define commuting \cstar-subalgebras. Since these inclusions are natural, they are equivariant with respect to the induced actions of , and . By assumption, it follows that we have a unital and equivariant -homomorphism
[TABLE]
satisfying and for all and .
Since is strongly self-absorbing, it follows from [84, Proposition 3.3] that there exist unitaries such that . As is equivariantly -stable, it is possible to find a unital copy that pointwise commutes with both and . By [40, Corollary 2.6], it follows that the unitary
[TABLE]
is homotopic to the unit. We observe
[TABLE]
for all and .
If we write as a finite product of exponentials with self-adjoints , we can lift these elements to self-adjoints in the fixed point algebra with the same norm, which can in turn be represented by bounded sequences of self-adjoints . These elements are approximately central relative to as , and satisfy for every compact set . We can therefore define a uniformly continuous sequence of maps z^{(n)}:[0,1]\to\mathcal{U}\big{(}\mathbf{1}+\mathcal{D}\otimes A\big{)} via . Then evidently , and the sequence represents .
We may finally observe the following properties:
- âą
for all ;
- âą
for all and ;
- âą
for every compact set .
We conclude that the unitized equivariant embedding
[TABLE]
satisfies the assumptions of subsection 4.1 when viewed as a unital cocycle morphism. In particular, it is asymptotically unitarily equivalent to a cocycle conjugacy. Since we may assume that the unitary path witnessing the equivalence is in and the resulting isomorphism will respect the scalar part on both sides, it follows that we obtain a proper cocycle conjugacy between and , which is properly asymptotically unitarily equivalent to . This finishes the proof. â
6. Remarks on -equivariant -theory
The aim of this section is to observe that -equivariant -theory [43] is functorial on the cocycle category of separable -\cstar-algebras. We note that there is little novelty behind this observation, as it is a fairly easy extension of the usual functoriality of with regard to equivariant -homomorphisms. Nevertheless, this may turn out to be a much more natural viewpoint in the future to exploit for classifying group actions up to cocycle conjugacy, so we include it here; cf. [68].
Moreover, as Thomsenâs description of -groups utilizes cocycle representations171717Throughout this section we will keep in mind subsection 1.2 and not necessarily assume that the map belonging to a cocycle representation between -\cstar-algebras is extendible, unless we need to compose them, such as in section 6. in his definition of equivariant Cuntz pairs â see [92, Section 3] â we shall describe the functoriality both in the common Fredholm picture and the CuntzâThomsen picture.
In this section we will assume familiarity of the reader with the standard (Fredholm) picture of -equivariant -theory and the standard associated terminology; see [2, Chapter 20] for details. The only major deviation is that we will denote for the equivariant -group associated to two actions and rather than , in order to avoid confusion when dealing with different actions on the same \cstar-algebra.
Notation \thenotaCT.
Let be an action on a \cstar-algebra, and suppose that is a -cocycle. We will denote by the Hilbert -module that is equal to as an ordinary Hilbert right--module, but is equipped with the continuous linear -action given by for all and . Note that under the canonical isomorphism of \cstar-algebras , the conjugation -action is simply given as the canonical extension of the cocycle perturbed action . Namely we can see that for all , , and , one has
[TABLE]
When we refer to the obvious Hilbert -module structure on , it is meant that we consider in this way.
Let be another action on a \cstar-algebra. Recall that for an equivariant -homomorphism between separable -\cstar-algebras, its associated element is the equivalence class of the equivariant Kasparov triple , where carries the obvious right Hilbert -module structure.
Notation \thenotaCT.
We denote by the category whose objects are separable -\cstar-algebras, and where morphisms from to are elements in . Then it is a tautological fact that the category of separable -\cstar-algebras, if equipped with non-degenerate equivariant -homomorphisms as the arrows, comes with an obvious natural transformation onto which is the identity map on objects, and which assigns an equivariant -homomorphism to its -element .
Definition \thedefiCT.
For a cocycle morphism between separable -\cstar-algebras, we associate the element represented by the Kasparov triple .181818This is a Kasparov triple precisely by the equivariance condition .
Remark \theremCT.
Obviously the construction in section 6 extends the one in section 6, which describes the case .
We may then come to our main observation:
Proposition \thepropCT.
For any two cocycle morphisms
[TABLE]
between separable -\cstar-algebras, if is non-degenerate, one has
[TABLE]
In particular, the assignment defines a natural transformation from the cocycle category onto which extends the one in section 6.
Proof.
In the special case at hand, the Kasparov triples representing our two factors are of a very special form, and hence it is easy to form their Kasparov product via considering a balanced tensor product; cf. [2, Example 18.4.2] or [67, Paragraph 2.4]. The left hand side of the equation is represented by the equivariant Kasparov triple given by
[TABLE]
We may identify via as a Hilbert right--module, and under this identification one has for all and . Furthermore the -action on is given by , which under this identification becomes g\cdot(\psi(b)c)=\psi(\mathbbmss{u}_{g}\beta_{g}(b))\mathbbmss{v}_{g}\gamma_{g}(c)=\psi^{+}(\mathbbmss{u}_{g})\mathbbmss{v}_{g}\gamma_{g}\big{(}\psi(b)c\big{)}, leading to the formula for the -action turning into a right Hilbert -module. Evidently the right hand side of the equation in our claim is represented by the same Kasparov triple, which shows the claim. â
Remark \theremCT.
Let be an action on a separable \cstar-algebra. It is well-known that for any -cocycle , its cocycle perturbation is naturally -equivalent to . Although this has been known to follow from abstract reasons due to being a stable functor, a more specific description of the equivalence has been given on various occasions. By consulting the beginning of [92, Section 3], one sees that the canonical equivalence in 191919In [92] this element is denoted , or rather the group homomorphism that is induced by multiplication from the right by this element, where is any other action on a separable \cstar-algebra. is simply given by in the sense of section 6. Recall from subsection 1.2 that within our formalism, the exterior equivalence is the canonical isomorphism in the category induced by the cocycle .
We shall also briefly review the CuntzâThomsen picture [10, 11, 28, 92] of , but we will somewhat adapt the terminology and notation for our purposes. The reader should consult [92, Section 3] for the theory we are about to recall.
Notation \thenotaCT.
We write for the \cstar-algebra of compact operators on some separable infinite-dimensional Hilbert space. For an action on a \cstar-algebra, let us write as short-hand and . Moreover we will abbreviate and .
Definition \thedefiCT.
Let and be two actions on separable \cstar-algebras. An -Cuntz pair consists of two cocycle representations
[TABLE]
such that one has
- âą
for all ;
- âą
for all ;
- âą
the map is norm-continuous on .
First let us use this opportunity to prove that the latter assumption about the cocycles is in fact redundant:
Proposition \thepropCT.
Let be an action on a non-unital \cstar-algebra.
- (i)
If is a unitary such that the coboundary takes values in , then is a norm-continuous map. 2. (ii)
Let be two strictly continuous -cocycles, and suppose that . Then the assignment defines a norm-continuous map on .
In particular, the last assumption in section 6 is redundant.
Proof.
(i): As one has for all , the assumption translates to . In other words, the image of in the corona algebra is a fixed point under the induced (algebraic) -action. By [93, Theorem 2.1], it follows that is norm-continuous, which implies our claim.
(ii): We may assume without loss of generality that , by replacing with and with inside if necessary. Let us choose sequences of isometries such that
[TABLE]
in the strict topology. For , we define the strict limits
[TABLE]
and note that both formulas define strictly continuous -cocycles. If we define
[TABLE]
then we see that holds for all .
Furthermore we have
[TABLE]
so the claim is equivalent to the assertion that is a norm-continuous map. This map takes values in . We observe again that by , the map takes values in . By strict continuity of , it suffices to show that is a norm-continuous map. Finally we observe for all that
[TABLE]
So the claim follows directly from (i). â
Definition \thedefiCT.
For two actions and on separable \cstar-algebras, let denote the set of all -Cuntz pairs, and let denote the subset of those pairs with , the degenerate Cuntz pairs.
Two elements and in are called homotopic, if there exists an -Cuntz pair which restricts to upon evaluation at , and restrict to upon evaluation at .
For any unital inclusion with generating isometries , one can perform the Cuntz addition for two -Cuntz pairs as
[TABLE]
which is independent of the choice of up to homotopy.
Finally, one may define an equivalence relation on whereby one has if there are such that and are homotopic. The quotient then becomes an abelian monoid via Cuntz addition.
Definition \thedefiCT.
Let and be actions on separable \cstar-algebras. For an -Cuntz pair , one may construct an --equivariant Kasparov triple as follows. One sets as a right Hilbert -module with grading . One defines via
[TABLE]
and via . We denote by its induced equivalence class in .
Notation \thenotaCT.
Let be an action on a separable \cstar-algebra. As is a stable functor in both variables, one has that the equivariant corner embedding is always a -equivalence, and we will denote its induced element by . Note that this defines a natural transformation on the category consisting of isomorphisms.
Theorem \thetheoremCT.
For any two actions and on separable \cstar-algebras, one has that is in fact a group, and the assignment defines a natural isomorphism onto .
Proposition \thepropCT.
For a cocycle morphism between separable -\cstar-algebras, the element is represented by any -Cuntz pair of the form
[TABLE]
So in particular one may choose .
Proof.
By section 6, the element described by such a Cuntz pair is represented by the equivariant Kasparov triple given by
[TABLE]
and
[TABLE]
As clearly takes values in , one may pass to the Kasparov triple without changing its -class; cf. [67, Paragraph 2.4]. Moreover we have . So is in fact represented by the triple . (We see here already that the specific choice of the cocycle does not matter.)
Next, we use (see [92, Section 2]) that the invertible element in is represented by the --equivariant Kasparov triple of the form . Here is viewed as a right Hilbert -module in the obvious way, and is defined by . Hence may be represented by the --equivariant Kasparov triple
[TABLE]
Here one has as a right Hilbert -module carrying the action
[TABLE]
and
[TABLE]
Hence we obtain a direct sum decomposition
[TABLE]
so the element is represented by , which is the same triple representing by definition. This shows our claim. â
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