aa r X i v : . [ m a t h . OA ] A p r STRONGLY SELF-ABSORBING C ∗ -DYNAMICALSYSTEMS, III GÁBOR SZABÓ
Abstract.
In this paper, we accomplish two objectives. Firstly, weextend and improve some results in the theory of (semi-)strongly self-absorbing C ∗ -dynamical systems, which was introduced and studied inprevious work. In particular, this concerns the theory when restrictedto the case where all the semi-strongly self-absorbing actions are as-sumed to be unitarily regular, which is a mild technical condition. Thecentral result in the first part is a strengthened version of the equivari-ant McDuff-type theorem, where equivariant tensorial absorption canbe achieved with respect to so-called very strong cocycle conjugacy.Secondly, we establish completely new results within the theory. Thismainly concerns how equivariantly Z -stable absorption can be reducedto equivariantly UHF-stable absorption with respect to a given semi-strongly self-absorbing action. Combining these abstract results withknown uniqueness theorems due to Matui and Izumi-Matui, we obtainthe following main result. If G is a torsion-free abelian group and D is one of the known strongly self-absorbing C ∗ -algebras, then stronglyouter G -actions on D are unique up to (very strong) cocycle conjugacy.This is new even for Z -actions on the Jiang-Su algebra. Contents
Introduction 11. Preliminaries 42. Strengthened McDuff-type theorem 83. Optimal McDuff-type theorem for compact groups 114. Reduction to subgroups 135. Reducing Z -stable absorption to UHF-stable absorption 166. Application to actions on strongly self-absorbing C ∗ -algebras 19References 21 Introduction
This is a further continuation of my previous papers [26, 25], which in-troduced and studied (semi-)strongly self-absorbing C ∗ -dynamical systems.The motivation for studying such objects comes from the fundamental im-portance of strongly self-absorbing C ∗ -algebras [28] in the Elliott program.For a more detailed description of this motivation and some history of theclassification of group actions on C ∗ -algebras and W ∗ -algebras, the reader Mathematics Subject Classification.
Supported by:
SFB 878
Groups, Geometry and Actions and EPSRC grant EP/N00874X/1. is referred to the introductions of the previous papers [26, 25] and the ref-erences therein. A survey article [9] by Izumi on these topics is especiallynoteworthy for anyone interested in the classification problem for group ac-tions on operator algebras.The first [26] of the previous papers provided an equivariant McDuff-typetheorem characterizing equivariant tensorial absorption of (semi-)stronglyself-absorbing actions, generalizing classical results of Rørdam [20, Chapter7, Section 2], Toms-Winter [28] and Kirchberg [13]. The second paper [25]generalized some other classical results about strongly self-absorbing C ∗ -algebras to the equivariant context, such as a stronger uniqueness theoremfor certain equivariant ∗ -homomorphisms by Dadarlat-Winter [4] and per-manence properties for the class of C ∗ -algebras absorbing a fixed stronglyself-absorbing C ∗ -algebra. In [25], the more sophisticated results could onlybe proved for semi-strongly self-absorbing actions that are unitarily regu-lar. Simply put, this is an equivariant analog of the C ∗ -algebraic propertythat the unitary commutator subgroup is in the connected component ofthe unit. For a semi-strongly self-absorbing G -action γ , unitary regularityhas been shown to be equivalent to the statement that the separable, γ -absorbing G -C ∗ -dynamical systems are closed under equivariant extensions;see [28, Section 4] and [13, Section 4] for the corresponding classical results.At present, it is open whether semi-strongly self-absorbing actions are au-tomatically unitarily regular. However, γ is unitarily regular if it is equiv-ariantly Z -stable, which in turn is often automatic for discrete amenableacting groups, but not in general. In particular, the equivariant analog ofthe main result of [29] is not true in general; see [25, 5.4]. Apart from consid-ering equivariant extensions, it is a theme throughout [25] that for unitarilyregular and semi-strongly self-absorbing actions, statements involving cer-tain approximations by sequences can be smoothed out and strengthened toapproximations by continuous paths.This is pursued further within the first half of this paper, where we im-prove the equivariant McDuff-type theorem from [26] in the unitarily regularcase. Namely, we show that for a unitarily regular and semi-strongly self-absorbing action γ : G y D and another action α : G y A on a separableC ∗ -algebra, the McDuff condition implies that α and α ⊗ γ are very stronglycocycle conjugate. This means that they are conjugate modulo a cocyclethat can be approximated by a continuous path of coboundaries starting atthe unit; see Definition 1.4(iv) and Theorem 2.2. In the case of compactacting groups, one moreover gets that α and α ⊗ γ are in fact conjugate; seeTheorem 3.3.In the second half of the paper, some new results are obtained withinthe theory of semi-strongly self-absorbing actions. In section 4, we provethat for an action, the property of being semi-strongly self-absorbing canbe detected by considering the restrictions with respect to an exhaustingsequence of open subgroups of the acting group. The same holds for theproperty of tensorially absorbing a given semi-strongly self-absorbing ac-tion; see Theorem 4.6. This arises as a fairly straightforward consequenceof the characterizations of these properties as approximate ones in previouswork, combined with reindexation arguments. In section 5, we prove that TRONGLY SELF-ABSORBING C ∗ -DYNAMICAL SYSTEMS, III 3 under the assumption of equivariant Z -stability, such properties can further-more be detected after stabilizing with the trivial actions on UHF algebrasof infinite type; see Theorem 5.6. In particular, this provides a way to re-duce the classification of certain group actions on strongly self-absorbingC ∗ -algebras to the classification of their UHF-stabilizations. This is some-what reminiscent of the main thrust of the methods developed by Winter in[30], which gave great impulse to the Elliott program, albeit the techniquesdeveloped in this paper have a much more narrow range of applicability incomparison.In section 6, these abstract results are then applied in combination withsome known classification results to obtain the following uniqueness theoremfor pointwise strongly outer actions on strongly self-absorbing C ∗ -algebras.This constitutes the main result of the paper. Theorem.
Let D be a strongly self-absorbing C ∗ -algebra satisfying the UCT.Let G be a countable, torsion-free abelian group. Then any two pointwisestrongly outer G -actions on D are very strongly cocycle conjugate. We remark that, on a conceptual level, this type of result resembles Oc-neanu’s uniqueness theorem (see [18]) for outer actions of amenable groupson the hyperfinite II -factor. So in a sense, if one regards a strongly self-absorbing as a close C ∗ -algebra analog of the hyperfinite II -factor, onemight call the above an Ocneanu-type uniqueness theorem.Results of Matui [14, 15] and Izumi-Matui [11] have previously shown thatthe above is true for Z d -actions on all the known strongly self-absorbing C ∗ -algebras except for the Jiang-Su algebra Z . Sato [22] has shown such auniqueness for Z -actions on Z , and Matui-Sato [16] have extended this alsoto Z -actions on Z . We note that the uniqueness for actions of the Kleinbottle group Z ⋊ − Z is also known by further work of Matui-Sato [17] onUHF algebras as well as Z ; this was the first classification result for actionsof non-abelian infinite groups on stably finite C ∗ -algebras. Curiously, theknown methods for showing a uniqueness result as above get increasinglydifficult to implement with increasingly complicated acting groups, and eventhe uniqueness for pointwise strongly outer Z -actions on Z has previouslybeen open.Our main result essentially follows from three main ingredients: firstly, theknown uniqueness for Z d -actions on UHF-stable strongly self-absorbing C ∗ -algebras mentioned above, which forms the basis of our argument; secondly,the reduction theorems proved in sections 4 and 5 based on the abstracttheory of semi-strongly self-absorbing actions; and thirdly a result of Matui-Sato [17, 4.11] asserting that pointwise strongly outer actions like above areautomatically equivariantly Z -stable. See also a more recent paper [23] ofSato for a much more general Z -stability result.It seems natural to expect that the known uniqueness results for Z d -actions from [14, 15, 11] could be reproved abstractly within the commonframework of semi-strongly self-absorbing actions, and without requiring theUCT assumption. It also seems plausible that this should in fact be pos-sible for not necessarily abelian acting groups. For example, a uniquenessfor actions like above seems feasible for (local) poly- Z groups, consider-ing an unpublished result of Izumi-Matui [10]. Considering moreover the GÁBOR SZABÓ KK -theoretically rigid situation for torsion-free amenable group actions onstrongly self-absorbing C ∗ -algebras showcased in [24, 4.12 and 4.17], I wouldgo as far as to conjecture the following Ocneanu-type uniqueness, which shallbe pursued in subsequent work: Conjecture.
Let D be a strongly self-absorbing C ∗ -algebra. Let G be acountable, torsion-free amenable group. Then any two pointwise stronglyouter G -actions on D are very strongly cocycle conjugate.Note that a uniqueness theorem like this usually fails already for finitegroups; see [7, 8] for range results of outer cyclic group actions on O . Sincethe computation of the equivariant KK -theory of an action via the Baum-Connes assembly map requires one to consider all the finite subgroups of theacting group, it is natural to expect that the above conjecture should failoutside the torsion-free case. Concerning non-amenable groups, actions areknown not to be rigid. On the one hand, for a given non-amenable group G , an argument in a paper of Jones [12] implies that for any finite stronglyself-absorbing C ∗ -algebra D , the noncommutative Bernoulli-shift on N G D does not absorb the trivial G -action on D ; this yields two pointwise stronglyouter G -actions on D that are not cocycle conjugate. On the other hand,a recent paper of Gardella-Lupini [5] shows that rigidity fails much morespectacularly upon assuming that G has property (T). Acknowledgement.
The work presented in this paper has benefitedfrom a visit to the Department of Mathematics at the University of Kyotoin January 2016, and I would like to express my gratitude to Masaki Izumifor the hospitality and support.1.
Preliminaries
Notation 1.1.
Unless specified otherwise, we will stick to the followingnotational conventions in this paper: • The symbol α is used for a continuous action α : G y A of a locallycompact group G on a C ∗ -algebra A . By slight abuse of notation, wewill also write α : G → Aut( M ( A )) for the unique strictly continuousextension. • For an action α : G y A , A α denotes the fixed-point algebra of A . • If (
X, d ) is some metric space with elements a, b ∈ X , then we write a = ε b as a shortcut for d ( a, b ) ≤ ε . • By a unitary path in a C ∗ -algebra A we shall understand a norm-continuous map from [0 ,
1] to U ( ˜ A ).First we recall the notion of 1-cocycles for actions on C ∗ -algebras andtheir cocycle perturbations. Note that we are adding a new refinement inthis paper, given by so-called asymptotic coboundaries, very strong exteriorequivalence and very strong cocycle conjugacy. Definition 1.2 (see [19, 3.2] and [26, 1.3, 1.6]) . Let α : G y A be an action.Consider a strictly continuous map w : G → U ( M ( A )).(i) w is called an α -1-cocycle (or just α -cocycle), if one has w g α g ( w h ) = w gh for all g, h ∈ G . In this case, the map α w : G → Aut( A ) given by TRONGLY SELF-ABSORBING C ∗ -DYNAMICAL SYSTEMS, III 5 α wg = Ad( w g ) ◦ α g is again an action, and is called a cocycle perturba-tion of α . Two G -actions on A are called exterior equivalent if one ofthem is a cocycle perturbation of the other.(ii) Assume that w is an α -1-cocycle. It is called an approximate cobound-ary, if there exists a sequence of unitaries x n ∈ U ( M ( A )) such that x n α g ( x ∗ n ) n →∞ −→ w g in the strict topology for all g ∈ G and uniformlyon compact subsets of G . Two G -actions on A are called strongly ex-terior equivalent, if one of them is a cocycle perturbation of the othervia an approximate coboundary.(iii) Assume that w is an α -1-cocycle. It is called an asymptotic cobound-ary, if there exists a strictly continuous path of unitaries x : [0 , ∞ ) →U ( M ( A )) with x = such that x t α g ( x ∗ t ) t →∞ −→ w g in the strict topol-ogy for all g ∈ G and uniformly on compact subsets of G . Two G -actions on A are called very strongly exterior equivalent, if one of themis a cocycle perturbation of the other via an asymptotic coboundary.Analogously, let us consider the generalization of these equivalence rela-tions to cocycle actions: Definition 1.3 (see [19, 3.1] for (i)) . Let ( α, u ) , ( β, w ) : G y A be twococycle actions.(i) The pairs ( α, u ) and ( β, w ) are called exterior equivalent, if there is astrictly continuous map v : G → U ( M ( A )) satisfying β g = Ad( v g ) ◦ α g and w ( s, t ) = v s α s ( v t ) u ( s, t ) v ∗ st for all g, s, t ∈ G .(ii) The pairs ( α, u ) and ( β, w ) are called strongly exterior equivalent,if there is a map v : G → U ( M ( A )) as in (i) such that there is asequence of unitaries x n ∈ U ( M ( A )) with x n α g ( x ∗ n ) n →∞ −→ v g in thestrict topology for all g ∈ G and uniformly on compact subsets of G .(iii) The pairs ( α, u ) and ( β, w ) are called very strongly exterior equivalent,if there is a map v : G → U ( M ( A )) as in (i) such that there is a strictlycontinuous path of unitaries x : [0 , ∞ ) → U ( M ( A )) with x = and x t α g ( x ∗ t ) t →∞ −→ v g in the strict topology for all g ∈ G and uniformly oncompact subsets of G .We recall several notions that describe how one can identify two cocycleactions on C ∗ -algebras. We note that condition (iv) below is a new defini-tion and a natural strengthening of the notion of strong cocycle conjugacyoriginally introduced by Izumi-Matui in [11]. Definition 1.4.
Two cocycle actions ( α, u ) : G y A and ( β, w ) : G y B are called(i) conjugate, if there is an equivariant isomorphism ϕ : ( A, α, u ) → ( B, β, w ). In this case, we write ( α, u ) ∼ = ( β, w ).(ii) cocycle conjugate, if there is an isomorphism ϕ : A → B such that( ϕ ◦ α ◦ ϕ − , ϕ ◦ u ) is exterior equivalent to ( β, w ). In this case, wewrite ( α, u ) ≃ cc ( β, w ).(iii) strongly cocycle conjugate, if there is an isomorphism ϕ : A → B suchthat ( ϕ ◦ α ◦ ϕ − , ϕ ◦ u ) is strongly exterior equivalent to ( β, w ). Inthis case, we write ( α, u ) ≃ scc ( β, w ). GÁBOR SZABÓ (iv) very strongly cocycle conjugate, if there is an isomorphism ϕ : A → B such that ( ϕ ◦ α ◦ ϕ − , ϕ ◦ u ) is very strongly exterior equivalent to( β, w ). In this case, we write ( α, u ) ≃ vscc ( β, w ).If α and β are genuine actions, then we omit the 2-cocycles from this nota-tion. Definition 1.5 (see [13, 1.1] and [26, 1.7, 1.9, 1.10]) . Let A be a C ∗ -algebraand ( α, u ) : G y A a cocycle action of a locally compact group G .(i) The sequence algebra of A is given as A ∞ = ℓ ∞ ( N , A ) / n ( x n ) n | lim n →∞ k x n k = 0 o . There is a standard embedding of A into A ∞ by sending an elementto its constant sequence. We shall always identify A ⊂ A ∞ this way,unless specified otherwise.(ii) Suppose u = . Pointwise application of α on representing sequencesdefines a (not necessarily continuous) G -action α ∞ on A ∞ . Let A ∞ ,α = { x ∈ A ∞ | [ g α ∞ ,g ( x )] is continuous } be the continuous part of A ∞ with respect to α .(iii) For some C ∗ -subalgebra B ⊂ A ∞ , the (corrected) relative central se-quence algebra is defined as F ( B, A ∞ ) = ( A ∞ ∩ B ′ ) / Ann(
B, A ∞ ) . (iv) Suppose that B ⊂ A ∞ is an α ∞ -invariant C ∗ -subalgebra closed undermultiplication with the unitaries { u ( g, h ) } g,h ∈ G . Then the map α ∞ : G → Aut( A ∞ ) given by componentwise application of α induces a(not necessarily continuous) G -action ˜ α ∞ on F ( B, A ∞ ). Let F α ( B, A ∞ ) = { y ∈ F ( B, A ∞ ) | [ g ˜ α ∞ ,g ( y )] is continuous } be the continuous part of F ( B, A ∞ ) with respect to α .(v) In case B = A , we write F ( A, A ∞ ) = F ∞ ( A ) and F α ( A, A ∞ ) = F ∞ ,α ( A ). Notation 1.6 (see [25, 1.14]) . Let G be a second-countable, locally compactgroup. Let A be a C ∗ -algebra and α : G y A an action. For ε > K ⊂ G , define the closed set A αε,K = { a ∈ A | k α g ( a ) − a k ≤ ε for all g ∈ K } ⊂ A. If A is unital, then also consider U ( A αε,K ) = U ( A ) ∩ A αε,K and U ( A αε,K ) = n u (1) ∈ U ( A αε,K ) | u : [0 , → U ( A αε,K ) continuous, u (0) = o . Definition 1.7 (cf. [25, 2.1]) . Let G be a second-countable, locally compactgroup, A and B two C ∗ -algebras and α : G y A and β : G y B two actions.Let ϕ , ϕ : ( A, α ) → ( B, β ) be two equivariant ∗ -homomorphisms. TRONGLY SELF-ABSORBING C ∗ -DYNAMICAL SYSTEMS, III 7 (i) We say that ϕ and ϕ are approximately G -unitarily equivalent, if forevery ε >
0, every finite set F ⊂⊂ A and compact set K ⊂ G , there is aunitary v ∈ U (cid:0) ˜ B βε,K (cid:1) such that k ϕ ( x ) − vϕ ( x ) v ∗ k ≤ ε for all x ∈ F .We write ϕ ≈ u ,G ϕ .(ii) Assume that A is separable. We say that ϕ and ϕ are stronglyasymptotically G -unitarily equivalent, if for every ε > K ⊂ G , there is a continuous path of unitaries w : [1 , ∞ ) →U (cid:0) ˜ B βε ,K (cid:1) satisfying w (1) = B , ϕ ( x ) = lim t →∞ w ( t ) ϕ ( x ) w ( t ) ∗ for all x ∈ A, andlim t →∞ max g ∈ K k β g ( w ( t )) − w ( t ) k = 0 for every compact set K ⊂ G. (iii) Suppose that G is compact. Then by averaging, we may assume in (i)that v is in the fixed point algebra ˜ B β . We may also assume in (ii)that the path w takes values in the fixed point algebra ˜ B β .(iv) If we disregard equivariance and consider G = { } in (i) and (ii), thenwe say that the maps ϕ and ϕ are approximately unitarily equivalentor strongly asymptotically unitarily equivalent, respectively. Definition 1.8 (cf. [26, 3.1, 4.1]) . Let D be a separable, unital C ∗ -algebraand G a second-countable, locally compact group. Let γ : G y D be anaction. We say that γ is(i) strongly self-absorbing, if the equivariant first-factor embeddingid D ⊗ D : ( D , γ ) → ( D ⊗ D , γ ⊗ γ )is approximately G -unitarily equivalent to an isomorphism.(ii) semi-strongly self-absorbing, if it is strongly cocycle conjugate to astrongly self-absorbing action.Let us recall some results from [26]. Theorem 1.9 (see [26, 4.6]) . Let D be a separable, unital C ∗ -algebra and G a second-countable, locally compact group. Let γ : G y D be an action.The following are equivalent: (i) γ is semi-strongly self-absorbing; (ii) γ has approximately G -inner half-flip and there exists a unital andequivariant ∗ -homomorphism from ( D , γ ) to ( D ∞ ,γ ∩ D ′ , γ ∞ ) ; (iii) γ has approximately G -inner half-flip and γ ≃ scc γ ⊗∞ . Theorem 1.10 (see [26, 3.7, 4.7]) . Let G be a second-countable, locallycompact group. Let A be a separable C ∗ -algebra and ( α, u ) : G y A acocycle action. Let D be a separable, unital C ∗ -algebra and γ : G y D asemi-strongly self-absorbing action. The following are equivalent: (i) ( α, u ) ≃ scc ( α ⊗ γ, u ⊗ ) . (ii) ( α, u ) ≃ cc ( α ⊗ γ, u ⊗ ) . (iii) There exists a unital and equivariant ∗ -homomorphism from ( D , γ ) to (cid:0) F ∞ ,α ( A ) , ˜ α ∞ (cid:1) . Remark.
An action α satisfying condition 1.10(i) is called γ -absorbing. GÁBOR SZABÓ
We shall also recall the notion of a unitarily regular action.
Definition 1.11 (see [25, 1.17]) . Let G be a second-countable, locally com-pact group. Let A be a unital C ∗ -algebra and α : G y A an action. We saythat α is unitarily regular, if for every compact set K ⊂ G and ε >
0, thereexists δ > uvu ∗ v ∗ ∈ U ( A αε,K ) for every u, v ∈ U ( A αδ,K ). Theorem 1.12 (see [26, 2.2]) . If a semi-strongly self-absorbing action γ : G y D is unitarily regular, then γ has strongly asymptotically G -innerhalf-flip. In particular, the half-flip can be approximately implemented byunitaries in U (cid:0) ( D ⊗ D ) γ ⊗ γε,K (cid:1) for arbitrarily small ε > and large compactsets K ⊂ G . Strengthened McDuff-type theorem
The following is a continuous generalization of a key technical Lemmafrom [26]. Its proof is fairly analogous and employs a few straightforwardmodifications.
Lemma 2.1 (cf. [26, 2.1]) . Let G be a second-countable, locally compactgroup. Let ( α, u ) : G y A and ( β, w ) : G y B be two cocycle actions onseparable C ∗ -algebras. Let ϕ : ( A, α, u ) → ( B, β, w ) be an injective, non-degenerate and equivariant ∗ -homomorphism. Assume the following:For every ε > , compact subset K ⊂ G and finite subsets F A ⊂⊂ A, F B ⊂⊂ B ,there exists a unitary path z : [0 , → U ( ˜ B ) with z = satisfying (2.1a) k [ z t , ϕ ( a )] k ≤ ε for every a ∈ F A and ≤ t ≤ . (2.1b) dist( z ∗ bz , ϕ ( A )) ≤ ε for every b ∈ F B . (2.1c) ϕ ( a ) β g ( z t ) = ε ϕ ( a ) z t for every g ∈ K , a ∈ F A and ≤ t ≤ .Then ϕ is strongly asymptotically unitarily equivalent to an isomorphism ψ : A → B inducing very strong cocycle conjugacy between ( α, u ) and ( β, w ) .Proof. We first comment that by the non-degeneracy of ϕ , one can replacethe elements ϕ ( a ) in (2.1c), for a ∈ F A , by any element b ∈ F B .Let { a n } n ∈ N ⊂ A and { b n } n ∈ N ⊂ B be dense sequences. Since G is σ -compact, write G = S n ∈ N K n for an increasing union of compact subsets1 G ∈ K n . We are going to add paths of unitaries to ϕ step by step:In the first step, choose some a , ∈ A and z (1) : [0 , → U ( ˜ B ) with z (1)0 = such that for all 0 ≤ t ≤
1, we have • z (1) ∗ b z (1)1 = / ϕ ( a , ); • k [ z (1) t , ϕ ( a )] k ≤ / • b β g ( z (1) t ) = / b z (1) t for all g ∈ K .In the second step, choose a , , a , ∈ A and z (2) : [0 , → U ( ˜ B ) with z (2)0 = such that for every 0 ≤ t ≤ • z (2) ∗ ( z (1) ∗ b j z (1)1 ) z (2)1 = / ϕ ( a ,j ) for j = 1 , • k [ z (2) t , ϕ ( a j )] k ≤ / j = 1 , • k [ z (2) t , ϕ ( a , )] k ≤ / • ( b j z (1)1 ) β g ( z (2) t ) = / ( b j z (2)1 ) z (2) t for all g ∈ K and j = 1 , TRONGLY SELF-ABSORBING C ∗ -DYNAMICAL SYSTEMS, III 9 Now assume that for some n ∈ N , we have found z (1) , . . . , z ( n ) : [0 , → U ( ˜ B )and { a m,j } m,j ≤ n ⊂ A satisfying for every 0 ≤ t ≤ z ( n ) ∗ ( z ( n − ∗ · · · z (1) ∗ b j z (1)1 · · · z ( n − ) z ( n )1 = − n ϕ ( a n,j ) for j ≤ n ;(e2.2) k [ z ( n ) t , ϕ ( a j )] k ≤ − n for j ≤ n ;(e2.3) k [ z ( n ) t , ϕ ( a m,j )] k ≤ − n for m < n and j < m ;(e2.4)( b j z (1)1 · · · z ( n − ) β g ( z ( n ) t ) = − n ( b j z (1)1 · · · z ( n − ) z ( n ) t for g ∈ K n and j ≤ n. Then we can again apply our assumptions to find z ( n +1) : [0 , → U ( ˜ B )with z ( n +1)0 = and { a n +1 ,j } j ≤ n +1 ⊂ A so that for every 0 ≤ t ≤ • z ( n +1) ∗ ( z ( n ) ∗ · · · z (1) ∗ b j z (1)1 · · · z ( n )1 ) z ( n +1)1 = − ( n +1) ϕ ( a n +1 ,j ) for j ≤ n + 1; • k [ z ( n +1) t , ϕ ( a j )] k ≤ − ( n +1) for j ≤ n + 1; • k [ z ( n +1) t , ϕ ( a m,j )] k ≤ − ( n +1) for m < n + 1 and j < n + 1; • ( b j z (1)1 · · · z ( n )1 ) β g ( z ( n +1) t ) = − ( n +1) ( b j z (1)1 · · · z ( n )1 ) z ( n +1) t for all g ∈ K n +1 and j ≤ n + 1.Carry on inductively. We define a norm-continuous path of unitaries x :[0 , ∞ ) → U ( ˜ B ) via x t = z (1)1 · · · z ( n )1 z ( n +1) t − n for n ≥ n ≤ t ≤ n +1. Notethat x = , and this map is well-defined since every path z ( n ) starts at theunit. Similarly we define a point-norm continuous path of ∗ -homomorphisms ψ t : A → B for t ≥ ψ t = Ad( x t ) ◦ ϕ .Now let us observe a number of facts: By condition (e2.2), the net ( ψ t ( a j )) t ≥ is Cauchy for all j ∈ N . Since the set { a j } j ∈ N ⊂ A is dense, this impliesthat the net ( ψ t ) t ≥ converges to some ∗ -homomorphism ψ : A → B in thepoint-norm topology. Then ψ is clearly strongly asymptotically unitarilyequivalent to ϕ .Exactly as in the proof of [26, 2.1], one deduces from (e2.1) and (e2.3)that ψ is surjective, and hence an isomorphism.By condition (e2.4), we have that the assignments t b j · x t β g ( x ∗ t ) yieldCauchy nets for every j ∈ N and g ∈ G , with uniformity on compact subsetsof G . Since { b j } j ∈ N ⊂ B is dense, it follows that every path of functions ofthe form [ g b · x t β g ( x ∗ t )] (for b ∈ B ) converges uniformly on compact sets.Since β is point-norm continuous, it follows that the functions g β g (cid:16) β g − ( b ) ∗ · x t β g ( x ∗ t ) (cid:17) ∗ = x t β g ( x ∗ t ) · b must also converge uniformly on compact sets of G as t → ∞ , for every b ∈ B .It follows that the strict limit v g = lim t →∞ x t β g ( x ∗ t ) ∈ U ( M ( B )) existsfor every g ∈ G , and that this convergence is uniform on compact subsets of G . In particular, the assignment g v g ∈ U ( M ( B )) is strictly continuous.Exactly as in the proof of [26, 2.1], it follows that ψ ◦ α g = Ad( v g ) ◦ β g ◦ ψ and v g β g ( v h ) w ( g, h ) v ∗ gh = ψ ( u ( g, h )) for all g, h ∈ G . This finishes theproof. (cid:3) Here comes the main result of this section, which is an improved versionof the equivariant McDuff theorem [26, 3.7, 4.7] for unitarily regular actions:
Theorem 2.2.
Let G be a second-countable, locally compact group. Let γ : G y D be a unitarily regular and semi-strongly self-absorbing action ona separable, unital C ∗ -algebra. Let ( α, u ) : G y A be a cocycle action on aseparable C ∗ -algebra. Suppose that there exists a unital and equivariant ∗ -homomorphism from ( D , γ ) to (cid:0) F ∞ ,α ( A ) , ˜ α ∞ (cid:1) . Then the equivariant second-factor embedding D ⊗ id A : ( A, α, u ) → ( D ⊗
A, γ ⊗ α, D ⊗ u ) is strongly asymptotically unitarily equivalent to an isomorphism that in-duces very strong cocycle conjugacy. In particular, we have ( α, u ) ≃ vscc ( α ⊗ γ, u ⊗ D ) .Proof. The proof is very similar to [26], apart from a small modification.Keeping in mind [26, 1.11], we have a natural isomorphism F ( D ⊗ A, ( D ⊗ A ) ∞ ) ∼ = F ( D ⊗ A, (cid:0) ( D ⊗ A ) ∼ (cid:1) ∞ ) . Denote by π : (cid:0) ( D ⊗ A ) ∼ (cid:1) ∞ ∩ ( D ⊗ A ) ′ → F ( D ⊗ A, ( D ⊗ A ) ∞ ) the canon-ical surjection. Note that by assumption, we have an equivariant, unital ∗ -homomorphism from ( D , γ ) to (cid:0) F ∞ ,α ( A ) , ˜ α ∞ (cid:1) . Consider the canonicalinclusions F ∞ ( A ) , D ⊂ F ( D ⊗ A, ( D ⊗ A ) ∞ ) , which define commuting C ∗ -subalgebras. Since these inclusions are natu-ral, they are equivariant with respect to the induced actions of α , γ and γ ⊗ α . By assumption, it follows that we have a unital and equivariant ∗ -homomorphism ϕ : ( D ⊗ D , γ ⊗ γ ) → (cid:0) F γ ⊗ α ( D ⊗ A, ( D ⊗ A ) ∞ ) , ( γ ⊗ α ) ∼∞ (cid:1) satisfying ϕ ( d ⊗ D ) · ( D ⊗ a ) = d ⊗ a and ϕ ( D ⊗ d ) · ( D ⊗ a ) ∈ D ⊗ A ∞ for all a ∈ A and d ∈ D .Now let ε > F D ⊂⊂D , F A ⊂⊂ A and K ⊂ G be a compact set. Withoutloss of generality, assume that F D and F A consist of contractions. Since γ isunitarily regular, we can apply 1.12 and choose a unitary path v : [0 , →U ( D ⊗ D ) with v = andmax ≤ t ≤ max g ∈ K k v t − ( γ ⊗ γ ) g ( v t ) k ≤ ε and v ∗ ( d ⊗ D ) v = ε D ⊗ d for all d ∈ F D . The unitary path u : [0 , → U (cid:16) F γ ⊗ α ( D ⊗ A, ( D ⊗ A ) ∞ ) (cid:17) , u t = ϕ ( v t )then satisfies u ∗ ( d ⊗ a ) u = ϕ ( v ( d ⊗ D ) v ∗ ) · ( D ⊗ a ) = ε ϕ ( D ⊗ d ) · ( D ⊗ a ) ∈ D ⊗ A ∞ for all a ∈ A with k a k ≤ d ∈ F D , and moreover(e2.5) k u t − ( γ ⊗ α ) ∼∞ ,g ( u t ) k ≤ k v t − ( γ ⊗ γ ) g ( v t ) k ≤ ε for all g ∈ K and 0 ≤ t ≤ TRONGLY SELF-ABSORBING C ∗ -DYNAMICAL SYSTEMS, III 11 Applying the unitary lifting theorem [2, 5.1], we can find paths of unitaries z ( n ) : [0 , → U (cid:0) ( D ⊗ A ) ∼ (cid:1) with z ( n )0 = and such that z = [( z ( n ) )] : [0 , → (cid:0) ( D ⊗ A ) ∼ (cid:1) ∞ ∩ ( D ⊗ A ) ′ satisfies u t = π ( z t ) for all 0 ≤ t ≤
1. Note that each u t is a continuouselement with respect to ( γ ⊗ α ) ∼∞ , so it follows that[ g ( D ⊗ a ) · ( γ ⊗ α ) ∞ ,g ( z t )]is a norm-continuous map on G for every a ∈ A . Using [26, 2.2], we thussee that also g (cid:16) ( D ⊗ a ) · ( γ ⊗ α ) g ( z ( n ) t ) (cid:17) n ∈ N ∈ ℓ ∞ ( N , D ⊗ A )is continuous. In particular, we obtain a uniformly continuous map[0 , × K → ℓ ∞ ( N , D ⊗ A ) , ( t, g ) (cid:16) ( D ⊗ a ) · ( γ ⊗ α ) g ( z ( n ) t ) (cid:17) n ∈ N . From (e2.5) it thus follows that(e2.6) lim sup n →∞ max g ∈ K max ≤ t ≤ (cid:13)(cid:13) ( D ⊗ a ) · (cid:0) ( γ ⊗ α ) g ( z ( n ) t ) − z ( n ) t (cid:1)(cid:13)(cid:13) ≤ ε for all a ∈ A with k a k ≤ z is a lift for u by choice, we havedist( z ∗ ( d ⊗ a ) z , D ⊗ A ∞ ) ≤ ε for all a ∈ A with k a k ≤ d ∈ F D .It follows that there exists some n with • max ≤ t ≤ k [ z ( n ) t , D ⊗ a ] k ≤ ε for all a ∈ F A ; • dist( z ( n ) ∗ ( d ⊗ a ) z ( n )1 , D ⊗ A ) ≤ ε for all d ∈ F D and a ∈ F A ; • max g ∈ K max ≤ t ≤ k ( D ⊗ a ) · (cid:0) z ( n ) t − ( γ ⊗ α ) g ( z ( n ) t ) (cid:1) k ≤ ε .So we have met the conditions of (2.1a), (2.1b) and (2.1c) for the equivariantembedding D ⊗ id A : ( A, α, u ) → ( D ⊗
A, γ ⊗ α, D ⊗ u ). The claim follows. (cid:3) Optimal McDuff-type theorem for compact groups
In this section, we turn to the case of compact group actions. Our mainobservation here is that, upon a close inspection of the proofs of 2.1 and2.2, one can further improve the absorption to conjugacy for compact groupactions.
Remark 3.1.
Let G be a compact group and γ : G y D a strongly self-absorbing action. It was observed in [26, 4.10] that an action α : G y A ona separable, unital C ∗ -algebra is γ -absorbing if and only if α is conjugateto α ⊗ γ . This relies on the more general observation that on unital C ∗ -algebras, two compact group actions are conjugate if and only if they arestrongly cocycle conjugate. This, in turn, relied on an observation [7, 2.4]of Izumi stating that cocycles close to the unit are coboundaries. It is notknown whether this can be generalized to the non-unital case.For unitarily regular actions γ , however, it turns out that we can alwaysobtain conjugacy between α and α ⊗ γ upon a closer inspection of the proofs in the previous section. The crucial part about unitary regularity here isthat, within the proof of the equivariant McDuff theorem, one needs somekind of gadget to lift a G -invariant unitary to a G -invariant unitary undera certain quotient map; it also gives us the strong asymptotic G -unitaryequivalence in the statement of 3.3. It would be natural to expect thatconjugacy can be arranged without assuming unitary regularity, howeverthis would for example presuppose a new way of proving the non-equivariant,non-unital McDuff theorem not relying on unitary regularity, which to thebest of my knowledge (and despite some effort) does not yet exist. Lemma 3.2.
Let G be a second-countable, compact group. Let α : G y A and β : G y B be two actions on separable C ∗ -algebras. Let ϕ : ( A, α ) → ( B, β ) be an injective, non-degenerate and equivariant ∗ -homomorphism.Suppose that for every ε > and every pair of finite subsets F A ⊂⊂ A, F B ⊂⊂ B ,there exists a unitary path z : [0 , → U ( ˜ B β ) with z = satisfying k [ z t , ϕ ( a )] k ≤ ε for every a ∈ F A and ≤ t ≤ , and dist( z ∗ bz , ϕ ( A )) ≤ ε for every b ∈ F B . Then ϕ is strongly asymptotically G -unitarily equivalent to an isomorphism.In particular, α and β are conjugate.Proof. Proving this is completely identical to the classical one-sided inter-twining result [20, 2.3.5]. Proceed as in the proof of 2.1; since the paths z ( n ) take values in the fixed point algebra, one may simply omit everythingrelated to the cocycles. (cid:3) Theorem 3.3.
Let G be a second-countable, compact group. Let γ : G y D be a unitarily regular and strongly self-absorbing action on a separable, unital C ∗ -algebra. Let α : G y A be an action on a separable C ∗ -algebra. Supposethat there exists a unital and equivariant ∗ -homomorphism from ( D , γ ) to (cid:0) F ∞ ,α ( A ) , ˜ α ∞ (cid:1) . Then the equivariant second-factor embedding D ⊗ id A : ( A, α ) → ( D ⊗
A, γ ⊗ α ) is strongly asymptotically G -unitarily equivalent to an isomorphism. In par-ticular, α is conjugate to α ⊗ γ .Proof. Proceed exactly as in the proof of 2.2 until choosing the ∗ -homomor-phism ϕ . We make the additional observation that by [25, 3.7], the canonicalprojection π : (cid:0) ( D ⊗ A ) ∼ (cid:1) ∞ ∩ ( D ⊗ A ) ′ → F ( D ⊗ A, ( D ⊗ A ) ∞ )restricts to an equivariant, surjective ∗ -homomorphism on the continuousparts π : (cid:0) ( D ⊗ A ) ∼ (cid:1) ∞ ,γ ⊗ α ∩ ( D ⊗ A ) ′ → F γ ⊗ α ( D ⊗ A, ( D ⊗ A ) ∞ ) . As G is compact, this implies that π also becomes surjective after restrictingit to the fixed-point algebras (see [1, 3.9])(e3.1) π : (cid:0) ( D ⊗ A ) γ ⊗ α, ∼ (cid:1) ∞ ∩ ( D ⊗ A ) ′ → F ( D ⊗ A, ( D ⊗ A ) ∞ ) ( γ ⊗ α ) ∼ Now let ε > F D ⊂⊂D and F A ⊂⊂ A be given. As γ is strongly self-absorbingand unitarily regular, it has strongly asymptotically G -inner half-flip. As TRONGLY SELF-ABSORBING C ∗ -DYNAMICAL SYSTEMS, III 13 G is additionally compact, we can find a path of unitaries v : [0 , →U (cid:0) ( D ⊗ D ) γ ⊗ γ (cid:1) with v = and v ( d ⊗ D ) v ∗ = ε D ⊗ d for all d ∈ F D . Then u t = ϕ ( v t ) yields a unitary path u : [0 , → U (cid:0) F ( D ⊗ A, ( D ⊗ A ) ∞ ) ( γ ⊗ α ) ∼ (cid:1) such that u ∗ ( d ⊗ a ) u has distance at most ε from D ⊗ A ∞ for all d ∈ F D and all a ∈ A with k a k ≤ π by the unitarylifting theorem [2, 5.1]. That is, there exists a sequence of unitary paths z ( n ) : [0 , → U (cid:0) ( D ⊗ A ) γ ⊗ α, ∼ (cid:1) representing u . But then for sufficientlylarge n , we necessarily havemax ≤ t ≤ k [ z ( n ) t , D ⊗ a ] k ≤ ε and dist( z ( n ) ∗ ( d ⊗ a ) z ( n )1 , D ⊗ A ) ≤ ε for all d ∈ F D and a ∈ F A .As ε, F D , F A were arbitrary, the assertion follows from 3.2. (cid:3) Reduction to subgroups
In this section, we will study certain behavior of group actions in the casewhere the acting group arises as a union of open subgroups.
Notation 4.1.
Let G be a topological group with a distinguished sub-group H ⊂ G . Let A be a C ∗ -algebra and ( α, u ) : G y A a cocycleaction. Then we write ( α, u ) | H : H y A for the cocycle H -action on A thatarises by restriction. Let B be another C ∗ -algebra with a cocycle action( β, w ) : G y B , and let ϕ : ( A, α, u ) → ( B, β, w ) be a nondegenerate, equi-variant ∗ -homomorphism. Then we write ϕ | H : ( A, α, u ) | H → ( B, β, w ) | H for the equivariant ∗ -homomorphism between the restricted dynamical sys-tems. (This is equal to ϕ as a map, but is viewed as an arrow in a differentcategory.) For genuine actions, we will omit the 2-cocycles in this notation. Lemma 4.2.
Let G be a second-countable, locally compact group, A and B two C ∗ -algebras and α : G y A and β : G y B two actions. Let ϕ , ϕ : ( A, α ) → ( B, β ) be two equivariant ∗ -homomorphisms. Let { G n } n ∈ N be an increasing family of open subgroups of G such that G = S n ∈ N G n .Suppose that ϕ | G n ≈ u ,G n ϕ | G n for all n ∈ N . Then ϕ ≈ u ,G ϕ .Proof. Let ε > F ⊂⊂ A a finite subset and K ⊂ G a compact subset. Sincethe increasing subgroups G n are open, it follows by compactness that thereexists some N ∈ N with K ⊂ G N . As ϕ | G N ≈ u ,G N ϕ | G N by assumption,we find some v ∈ U (cid:0) ˜ B β | GN ε,K (cid:1) = U (cid:0) ˜ B βε,K (cid:1) with k ϕ ( x ) − vϕ ( x ) v ∗ k ≤ ε for all x ∈ F . This finishes the proof. (cid:3) Corollary 4.3.
Let G be a second-countable, locally compact group. Let D be a separable, unital C ∗ -algebra and γ : G y D an action. Let { G n } n ∈ N bean increasing family of open subgroups of G such that G = S n ∈ N G n . Then γ has approximately G -inner half-flip if and only if γ | G n has approximately G -inner half-flip for every n ∈ N . Remark 4.4.
Let G be a second-countable, locally compact group. Let A be a separable C ∗ -algebra and α : G y A an action. By the results in [25,Section 3], we have a natural identification F ∞ ,α ( A ) = ( A ∞ ,α ∩ A ′ ) / Ann(
A, A ∞ ,α ) . Moreover, the ideal Ann(
A, A ∞ ,α ) is a G - σ -ideal in A ∞ ,α ∩ A ′ , which impliesthat the quotient map from A ∞ ,α ∩ A ′ to F ∞ ,α ( A ) is strongly locally semi-split; see [25, 3.5, 3.6]. Denote by ℓ ∞ α ( N , A ) ⊂ ℓ ∞ ( N , A ) the C ∗ -algebra con-taining those bounded sequences ( x n ) n for which the map [ g ( α g ( x n )) n ]is norm-continuous. By a result of Brown [3, 2.1], we have A ∞ ,α = ℓ ∞ α ( N , A ) /c ( N , A ) . The following is nothing more than a fairly routine reindexation argument.
Lemma 4.5.
Let G be a second-countable, locally compact group. Let A bea separable C ∗ -algebra with a cocycle action ( α, u ) : G y A . Let D be aseparable, unital C ∗ -algebra and γ : G y D an action. Let { G n } n ∈ N be anincreasing family of open subgroups of G such that G = S n ∈ N G n . Supposethat for every n ∈ N , there exists a unital and equivariant ∗ -homomorphismfrom ( D , γ | G n ) to (cid:0) F ∞ ,α | Gn ( A ) , ˜ α ∞ | G n (cid:1) . Then there exists a unital and equi-variant ∗ -homomorphism from ( D , γ ) to (cid:0) F ∞ ,α ( A ) , ˜ α ∞ (cid:1) .Proof. First let us observe that it suffices to consider only the case of genuineactions. Consider the Hilbert space H = ℓ ( N ) ¯ ⊗ L ( G ) and let δ : G y K ( H ) be the unitarily implemented action that is induced by the left-regularrepresentation of G on L ( G ). By [26, 1.10] and [1, 1.5], the dynamicalsystem on the central sequence algebra ( F ∞ ,α ( A ) , ˜ α ∞ ) does invariant undercocycle conjugacy or stabilization with the compacts. Thus we may withoutloss of generality replace ( α, u ) by ( α ⊗ δ, u ⊗ ) and show the claim in thiscase. By the Packer-Raeburn stabilization trick from [19, 3.4], the 2-cocycle u ⊗ is a coboundary, and so we may assume that α is a genuine action.Let ε n > F An ⊂⊂ A be increasingfinite subsets with dense union. Let G ′ ⊂ G be a countable, dense subgroup.Let D ⊂ D be a countable, dense, γ | G ′ -invariant Q [ i ]- ∗ -subalgebra. Let F Dn ⊂⊂ D be increasing finite subsets with D = S n ∈ N F n , and let K n ⊂ G bean increasing sequence of compact sets with G = S n ∈ N K n .Fix some n ∈ N . Then, as the subgroups G k ⊂ G are open, there existssome N ∈ N with K n ⊂ G N . Using 4.4, we find a commutative diagram ℓ ∞ α | GN ( N , A ) (cid:15) (cid:15) (cid:0) A ∞ ,α | GN ∩ A ′ , α ∞ | G N (cid:1) (cid:15) (cid:15) ( D , γ | G N ) κ =( κ l ) l tttttttttttttttttttttttttt ψ ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ ψ / / (cid:0) F ∞ ,α | GN ( A ) , ˜ α ∞ | G N (cid:1) where ψ is a G N -equivariant ∗ -homomorphism, ψ is a G N -equivariant c.p.c.order zero map and κ is a (not necessarily equivariant) linear map. TRONGLY SELF-ABSORBING C ∗ -DYNAMICAL SYSTEMS, III 15 As κ is a lift for both ψ and ψ , we observe the following properties forall x, y ∈ D , a ∈ A and compact sets K ⊂ G N : • lim sup l →∞ k κ l ( x ) k ≤ k x k ; • lim l →∞ k [ κ l ( x ) , a ] k = 0; • lim l →∞ k κ l ( ) a − a k = 0; • lim l →∞ k κ l ( xy ) κ l ( ) − κ l ( x ) κ l ( y ) k = 0; • lim l →∞ max g ∈ K k ( α g ◦ κ l )( x ) − ( κ l ◦ γ g )( x ) k = 0.In particular, we find l ( n ) ∈ N such that the following are satisfied for all x, y ∈ F Dn and a ∈ F An : • k κ l ( n ) ( x ) k ≤ k x k + ε n ; • k [ κ l ( n ) ( x ) , a ] k ≤ ε n ; • k κ l ( n ) ( ) a − a k ≤ ε n ; • k κ l ( n ) ( xy ) κ l ( n ) ( ) − κ l ( n ) ( x ) κ l ( n ) ( y ) k ≤ ε n ; • max g ∈ K n k ( α g ◦ κ l ( n ) )( x ) − ( κ l ( n ) ◦ γ g )( x ) k ≤ ε n .By these properties, the Q [ i ]- ∗ -linear map ϕ = [( κ l ( n ) ) n ] : D → A ∞ ,α iswell-defined, contractive and satisfies • [ ϕ ( x ) , a ] = 0 for all x ∈ D and a ∈ A ; • ϕ ( ) a = a for all a ∈ A ; • ϕ ( xy ) ϕ ( ) = ϕ ( x ) ϕ ( y ) for all x, y ∈ D ; • α g ◦ ϕ = ϕ ◦ γ g for all g ∈ G ′ .Thus this map extends continuously to an equivariant c.p.c. order zero map ϕ : ( D , γ ) → ( A ∞ ,α ∩ A ′ , α ∞ ) with ϕ ( ) a = a for all a ∈ A . Then ϕ = ϕ +Ann( A, A ∞ ,α ) yields the desired equivariant and unital ∗ -homomorphismfrom ( D , γ ) to (cid:0) F ∞ ,α ( A ) , ˜ α ∞ (cid:1) . (cid:3) Theorem 4.6.
Let G be a second-countable, locally compact group. Let A be a separable C ∗ -algebra with a cocycle action ( α, u ) : G y A . Let D be aseparable, unital C ∗ -algebra and γ : G y D an action. Let { G n } n ∈ N be anincreasing family of open subgroups of G such that G = S n ∈ N G n . (i) The action γ is semi-strongly self-absorbing if and only if for every n ∈ N , the restriction γ | G n is semi-strongly self-absorbing. (ii) Suppose that γ is semi-strongly self-absorbing. Then ( α, u ) ≃ cc ( α ⊗ γ, u ⊗ D ) if and only if for every n ∈ N , one has ( α, u ) | G n ≃ cc ( α ⊗ γ, u ⊗ D ) | G n . (iii) Suppose that γ is semi-strongly self-absorbing, and that β : G y D isanother action. Then β ≃ scc γ if and only if for every n ∈ N , one has β | G n ≃ scc γ | G n .Proof. The implication “ ⇒ ” is clear in every statement, so let us show the“ ⇐ ” implication everywhere.(i): Suppose that γ | G n is semi-strongly self-absorbing for every n ∈ N .Then for every n ∈ N , we see by 1.9 that the action γ | G n has approximately G n -inner half-flip and there exists a unital and equivariant ∗ -homomorphismfrom ( D , γ | G n ) to ( D ∞ ,γ | Gn ∩ D ′ , γ ∞ | G n ). By 4.3, it follows that γ has ap-proximately G -inner half-flip. By 4.5, it follows that there exists a unital and equivariant ∗ -homomorphism from ( D , γ ) to ( D ∞ ,γ ∩ D ′ , γ ∞ ). Thus γ is semi-strongly self-absorbing by 1.9.(ii): Suppose that ( α, u ) | G n ≃ cc ( α ⊗ γ, u ⊗ ) | G n for every n ∈ N .By the equivariant McDuff Theorem 1.10, this means that for every n ∈ N , there exists a unital and equivariant ∗ -homomorphism from ( D , γ | G n )to (cid:0) F ∞ ,α | Gn ( A ) , ˜ α ∞ | G n (cid:1) . By 4.5, there exists a unital and equivariant ∗ -homomorphism from ( D , γ ) to (cid:0) F ∞ ,α ( A ) , ˜ α ∞ (cid:1) . Thus ( α, u ) ≃ cc ( α ⊗ γ, u ⊗ )by the equivariant McDuff theorem.(iii): Suppose that β | G n ≃ scc γ | G n for every n ∈ N . Then in particular, forevery n ∈ N , the action β | G n is semi-strongly self-absorbing, and the actions β | G n and γ | G n absorb each other. Thus the claim follows upon combining(i) and (ii). (cid:3) Reducing Z -stable absorption to UHF-stable absorption Remark 5.1.
Recall that for two mutually coprime supernatural numbers p and q , one writes Z p,q = (cid:8) f ∈ C (cid:0) [0 , , M p ⊗ M q (cid:1) | f (0) ∈ M p ⊗ , f (1) ∈ ⊗ M q (cid:9) . It has been shown in [21, Section 3] that, if p and q are of infinite type,then there exists a trace-collapsing unital ∗ -monomorphism ϕ : Z p,q → Z p,q such that the stationary inductive limit lim −→ { Z p,q , ϕ } is isomorphic to theJiang-Su algebra Z .For proving the main result of this section, we need to recall some previousresults from [25]. Proposition 5.2 (see [25, 2.6]) . Let G be a second-countable, locally com-pact group. Let A be a unital C ∗ -algebra and α : G y A an action. Let D bea separable, unital C ∗ -algebra and γ : G y D a semi-strongly self-absorbingaction. Assume α ≃ cc α ⊗ γ . Let ϕ , ϕ : ( D , γ ) → ( A, α ) be two unitaland equivariant ∗ -homomorphisms. Then there exist sequences of unitaries u n , v n ∈ U ( A ) satisfying max g ∈ K (cid:16) k u n − α g ( u n ) k + k v n − α g ( v n ) k (cid:17) n →∞ −→ for every compact set K ⊂ G and Ad( u n v n u ∗ n v ∗ n ) ◦ ϕ n →∞ −→ ϕ in point-norm. Proposition 5.3 (see [25, 1.19]) . Let G be a second-countable, locally com-pact group. Let A be a unital C ∗ -algebra and α : G y A an action. Assume α ≃ cc α ⊗ id Z . Then α is unitarily regular. Moreover, for every separable, α ∞ -invariant C ∗ -subalgebra B ⊂ A ∞ ,α , the fixed-point algebra of the relativecommutant ( A ∞ ,α ∩ B ′ ) α ∞ is K -injective. Theorem 5.4 (see [25, 4.9]) . Let G be a second-countable, locally compactgroup. Let γ : G y D be a semi-strongly self-absorbing action. If γ isunitarily regular, then the class of all separable, γ -absorbing G - C ∗ -dynamicalsystems is closed under equivariant extensions. TRONGLY SELF-ABSORBING C ∗ -DYNAMICAL SYSTEMS, III 17 Recall the following technical property of semi-strongly self-absorbing ac-tions, which arises as a consequence from a basic homotopy Lemma provedin [25]. See also [6, Section 4], where compelling Model theoretic evidence isgiven for the fact that dynamical systems induced on relative commutantslike below are virtually indistinguishable from the surrounding system.
Lemma 5.5 (see [25, 2.14]) . Let G be a second-countable, locally compactgroup. Let D be a separable, unital C ∗ -algebra and γ : G y D a semi-strongly self-absorbing action. Let A be a unital C ∗ -algebra and α : G y A an action with α ≃ scc α ⊗ γ . Let ψ : ( D , γ ) → ( A ∞ ,α , α ∞ ) be a unital andequivariant ∗ -homomorphism. Then U (cid:16)(cid:0) A ∞ ,α ∩ ψ ( D ) ′ (cid:1) α ∞ (cid:17) = U (cid:0) ( A ∞ ,α ) α ∞ (cid:1) ∩ ψ ( D ) ′ . In other words, a unitary in the fixed-point algebra (cid:0) A ∞ ,α ∩ ψ ( D ) ′ (cid:1) α ∞ ishomotopic to precisely when it is homotopic to inside the larger fixed-point algebra ( A ∞ ,α ) α ∞ . The following is the main result of this section:
Theorem 5.6.
Let G be a second-countable, locally compact group. Let D be a separable, unital C ∗ -algebra and γ : G y D an action. Let A be aseparable C ∗ -algebra and ( α, u ) : G y A a cocycle action. Let p and q betwo mutually coprime supernatural numbers of infinite type. (i) The action γ ⊗ id Z is semi-strongly self-absorbing if and only if γ ⊗ id U is semi-strongly self-absorbing for U ∈ { M p , M q } . (ii) Suppose that γ is semi-strongly self-absorbing. Then one has ( α ⊗ id Z , u ⊗ Z ) ≃ cc ( α ⊗ γ ⊗ id Z , u ⊗ D ⊗ Z ) if and only if one has ( α ⊗ id U , u ⊗ U ) ≃ cc ( α ⊗ γ ⊗ id U , u ⊗ D ⊗ U ) for U ∈ { M p , M q } .Proof. The implication “ ⇒ ” is clear in every statement because of U ∼ = U ⊗Z ,so let us show the “ ⇐ ” implication everywhere. We shall start with (ii) anduse it to prove (i).(ii): We may assume without loss of generality that ( α, u ) ≃ cc ( α ⊗ id Z , u ⊗ ) and γ ≃ cc γ ⊗ id Z . Note that by 5.3 and 5.4, this implies that separable, γ -absorbing G -C ∗ -dynamical systems are closed under equivariant extensions.Consider the Hilbert space H = ℓ ( N ) ¯ ⊗ L ( G ) and let δ : G y K ( H ) bethe unitarily implemented action that is induced by the left-regular repre-sentation of G on L ( G ). By the Packer-Raeburn stabilization trick from[19, 3.4], the 2-cocycle u ⊗ with respect to α ⊗ δ is a coboundary, and thus( α ⊗ δ, u ⊗ ) is exterior equivalent to a genuine action. By [1, 4.30], theproperty of γ -absorption is invariant under equivariant Morita equivalence.In particular, we may replace ( α, u ) by a genuine action on A ⊗ K ( H ), oralternatively just assume that u = .Denote I = C (0 , ⊗ M p ⊗ M q and Q = M p ⊕ M q . From the canonicalextension of C ∗ -algebras0 / / I / / Z p,q / / Q / / , we get the equivariant extension0 / / ( A ⊗ I, α ⊗ id I ) / / ( A ⊗ Z p,q , α ⊗ id Z p,q ) / / ( A ⊗ Q, α ⊗ id Q ) / / . Since α ⊗ id U ≃ cc α ⊗ γ ⊗ id U for U ∈ { M p , M q } by assumption, it is clearthat α ⊗ id I ≃ cc α ⊗ γ ⊗ id I and α ⊗ id Q ≃ cc α ⊗ γ ⊗ id Q . Hence also α ⊗ id Z p,q ≃ cc α ⊗ γ ⊗ id Z p,q by virtue of this extension. As the Jiang-Sualgebra Z arises as a stationary inductive limit of Z p,q (see 5.1), we have( A ⊗ Z , α ⊗ id Z ) ∼ = lim −→ ( A ⊗ Z p,q , α ⊗ id Z p,q ) . Since γ -absorption passes to equivariant inductive limits by [25, 1.10], thisshows our claim.(i): Set U = M p , U = M q and W = U ⊗ U . Suppose that γ ⊗ id U i is semi-strongly self-absorbing for i = 1 ,
2. We will need to go through twosteps in order to prove that γ ⊗ id Z is semi-strongly self-absorbing. Step 1:
The first and most difficult step is to show that γ ⊗ id Z hasapproximately G -inner flip. For a unital C ∗ -algebra C , denote by σ C ∈ Aut( C ⊗ C ) the flip automorphism. Set B = D ⊗ D , β = γ ⊗ γ and considerthe β -equivariant automorphism σ = σ D ∈ Aut(
B, β ) given by the flip.Then β ⊗ id U i is semi-strongly self-absorbing for i = 1 ,
2. Hence by 5.2, wecan find u i , v i ∈ U (cid:16) ( B ⊗ U i ) ( β ⊗ id) ∞ ∞ ,β ⊗ id (cid:17) , i = 1 , u i v i u ∗ i v ∗ i )( b ⊗ c i ) = σ ( b ) ⊗ c i for b ∈ B and c i ∈ U i . We maynaturally view B ⊗ U i ⊂ B ⊗ W for i = 1 ,
2, and thus define z = ( u v u ∗ v ∗ ) ∗ ( u v u ∗ v ∗ ) ∈ U (cid:16)(cid:0) ( B ⊗ W ) ∞ ,β ⊗ id ∩ ( B ⊗ W ) ′ (cid:1) ( β ⊗ id) ∞ (cid:17) By 5.3, it follows that the unitary z is homotopic to the unit inside ( B ⊗ W ) ( β ⊗ id) ∞ ∞ ,β ⊗ id . By the basic homotopy Lemma 5.5, we thus get that z is homo-topic to the unit by some unitary path (note the slight abuse of notation) z : [0 , → U (cid:16)(cid:0) ( B ⊗ W ) ∞ ,β ⊗ id ∩ ( B ⊗ W ) ′ (cid:1) ( β ⊗ id) ∞ (cid:17) with z (0) = and z (1) = z . Let us consider the unitary path w : [0 , → U (cid:16) ( B ⊗ W ) ( β ⊗ id) ∞ ∞ ,β ⊗ id (cid:17) , w ( t ) = ( u v u ∗ v ∗ ) z ( t ) . We see that w (0) = u v u ∗ v ∗ ∈ U (cid:16) ( B ⊗ U ) ( β ⊗ id) ∞ ∞ ,β ⊗ id (cid:17) and w (1) = u v u ∗ v ∗ ∈ U (cid:16) ( B ⊗ U ) ( β ⊗ id) ∞ ∞ ,β ⊗ id (cid:17) . Moreover, we have w ( t )( b ⊗ c ⊗ c ) w ( t ) ∗ = σ ( b ) ⊗ c ⊗ c for all b ∈ B, c i ∈ U i . Thus we can view w as a unitary w ∈ U (cid:16) ( B ⊗ Z p,q ) ( β ⊗ id) ∞ ∞ ,β ⊗ id (cid:17) . ⊂ U (cid:16) ( B ⊗ Z ) ( β ⊗ id) ∞ ∞ ,β ⊗ id (cid:17) with the property w ( b ⊗ ) w ∗ = σ ( b ) ⊗ for all b ∈ B . Note that the imageof ( Z , id Z ) → ( B ⊗ Z ) ( β ⊗ id) ∞ ∞ ,β ⊗ id , x w ( ⊗ x ) w ∗ commutes with B ⊗ . Using the uniqueness result 5.2 and a reindexationtrick, this map is G -unitarily equivalent to the canonical map x ⊗ x ,where we view it as a map with codomain being the relative commutant of TRONGLY SELF-ABSORBING C ∗ -DYNAMICAL SYSTEMS, III 19 B ⊗ . By perturbing w with the resulting unitary in ( B ⊗Z ) ( β ⊗ id) ∞ ∞ ,β ⊗ id ∩ ( B ⊗ ) ′ ,if necessary, we may thus assume that w ( b ⊗ x ) w ∗ = σ ( b ) ⊗ x for all b ∈ B and x ∈ Z .To summarize, all of this shows that σ D ⊗ id Z ≈ u ,G id B ⊗ id Z = id D⊗D ⊗ id Z . Combining this with the fact that
Z ∼ = Z ⊗ Z has approximately inner flip,we see that σ D⊗Z = σ − ◦ (id D⊗D ⊗ σ Z ) ◦ ( σ D ⊗ id Z⊗Z ) ◦ σ ≈ u ,G id D⊗Z⊗D⊗Z , where σ denotes the isomorphism from D ⊗ Z ⊗ D ⊗ D to D ⊗ D ⊗ Z ⊗ Z flipping the second and third tensors.
Step 2:
Let us now show the claim. From the previous step, we knowthat γ ⊗ id Z has approximately G -inner flip. Thus the infinite tensor poweraction ( γ ⊗ id Z ) ⊗∞ : G y ( D ⊗ Z ) ⊗∞ is strongly self-absorbing by [26, 3.3]. As U i ∼ = U i ⊗ Z for i = 1 ,
2, ourassumptions imply γ ⊗ id U i ≃ scc ( γ ⊗ id U i ) ⊗∞ ∼ = ( γ ⊗ id U i ) ⊗∞ ⊗ ( γ ⊗ id Z ) ⊗∞ ≃ scc γ ⊗ id U i ⊗ ( γ ⊗ id Z ) ⊗∞ for i = 1 ,
2. By part (ii) applied to γ in place of α and ( γ ⊗ id Z ) ⊗∞ in placeof γ , it follows that γ ⊗ id Z ≃ scc γ ⊗ id Z ⊗ ( γ ⊗ id Z ) ⊗∞ ∼ = ( γ ⊗ id Z ) ⊗∞ . Using 1.9 this shows that γ ⊗ id Z is semi-strongly self-absorbing. (cid:3) Application to actions on strongly self-absorbing C ∗ -algebras In this section, we shall obtain our main application of the results from theprevious sections. First, we need to recall some results from the literature.
Remark.
An automorphism α ∈ Aut( A ) on a unital C ∗ -algebra A is calledstrongly outer, if it is outer and if for every α -invariant tracial state τ ∈ T ( A ), the induced automorphism of α on the weak closure π τ ( A ) ′′ is outer.(Unlike in other sources from the literature, we shall not assume T ( A ) = ∅ for this definition. If T ( A ) = ∅ , then strongly outer just means outer byconvention.) If G is a discrete group, then a cocycle action ( α, u ) : G y A is called pointwise strongly outer, if α g is a strongly outer automorphism forevery g ∈ G \ { G } .The following is a combination of results proved in [14, 15, 11] due toMatui and Izumi-Matui. Theorem 6.1.
Let D be a strongly self-absorbing C ∗ -algebra satisfying theUCT that is not isomorphic to the Jiang-Su algebra. Let d ≥ be a number.Then any two pointwise strongly outer Z d -actions on D are strongly cocycleconjugate. Moreover, any such action is semi-strongly self-absorbing. Proof.
Note that by [27, 6.7], D must be isomorphic to either a UHF algebraof infinite type, one of the Cuntz algebras O or O ∞ , or tensor productsbetween these. By applying either one of the classification results [15, 5.4] ofMatui, [14, 5.2] of Matui or [11, 6.18, 6.20] of Izumi-Matui, it follows that anytwo pointwise strongly outer Z d -actions on D are strongly cocycle conjugate.It also follows from [26, 5.9, 5.12] that such actions are automatically semi-strongly self-absorbing. (cid:3) Here comes the main result of the paper:
Theorem 6.2.
Let D be a strongly self-absorbing C ∗ -algebra satisfying theUCT. Let G be a countable, torsion-free abelian group. Then any two point-wise strongly outer G -actions on D are very strongly cocycle conjugate.Moreover, any such action is semi-strongly self-absorbing.Proof. Let γ , γ : G y D be two pointwise strongly outer actions. Wemay write G = S n ∈ N G n for an increasing sequence of finitely generatedsubgroups. As G is torsion-free and abelian, this implies in particular thatfor every n , the group G n is isomorphic to Z d n for some d n ∈ N . Let p and q be two mutually coprime supernatural numbers of infinite type. Set U = M p and U = M q .Then D ⊗ U j is a strongly self-absorbing C ∗ -algebras satisfying the UCTthat is not isomorphic to the Jiang-Su algebra for j = 1 ,
2. Thus 6.1 appliesand we see that for every n , the G n -action ( γ i ⊗ id U j ) | G n is a semi-stronglyself-absorbing action for i = 1 , j = 1 ,
2, with( γ ⊗ id U j ) | G n ≃ scc ( γ ⊗ id U j ) | G n ≃ scc ( γ ⊗ γ ⊗ id U j ) | G n , j = 1 , . Thus we can apply 5.6 to deduce that for every n , the G n -action ( γ i ⊗ id Z ) | G n is semi-strongly self-absorbing for i = 1 ,
2, with( γ ⊗ id Z ) | G n ≃ scc ( γ ⊗ id Z ) | G n ≃ scc ( γ ⊗ γ ⊗ id Z ) | G n . As n was arbitrary, it follows from 4.6 that the actions γ ⊗ id Z and γ ⊗ id Z are semi-strongly self-absorbing and are strongly cocycle conjugate. Nowone has γ ≃ scc γ ⊗ id Z and γ ≃ scc γ ⊗ id Z due to a result [17, 4.11]of Matui-Sato. So γ and γ are equivariantly Z -stable and in particularunitarily regular by 5.3. Since they absorb each other tensorially, it followsfrom 2.2 that in fact γ ≃ vscc γ ⊗ γ ≃ vscc γ . This finishes the proof. (cid:3) Remark 6.3.
The strategy of the proof of 6.2 in order to obtain uniquenessresults for actions on the Jiang-Su algebra, making crucial use of 5.6, shouldhave more applications in the future because it relies on a general principlenot depending on the acting group. Note that the results from Section 5 inparticular allow one to bypass having to solve some hard problems related tothe vanishing of general cocycles, which has been considered by Matui-Satoin [16, 17] to show uniqueness for Z -actions on Z , and to show uniquenessfor actions of the Klein bottle group Z ⋊ − Z on Z . What is however seem-ingly inaccessible with our approach at the moment is to determine underwhat conditions a cocycle action on a strongly self-absorbing C ∗ -algebra iscocycle conjugate to a genuine action; this could also be successfully tackledin Matui-Sato’s approach [16, 17]. TRONGLY SELF-ABSORBING C ∗ -DYNAMICAL SYSTEMS, III 21 References [1] S. Barlak, G. Szabó: Sequentially split ∗ -homomorphisms between C ∗ -algebras. Int.J. Math. 27 (2016), no. 12. 48 pages.[2] B. Blackadar: The homotopy lifting theorem for semiprojective C ∗ -algebras. Math.Scand. 118 (2016), no. 2, pp. 291–302.[3] L. G. Brown: Continuity of actions of groups and semigroups on Banach spaces. J.London Math. Soc. 62 (2000), no. 1, pp. 107–116.[4] M. Dadarlat, W. Winter: The KK -theory of strongly self-absorbing C ∗ -algebras.Math. Scand. 104 (2009), no. 1, pp. 95–107.[5] E. Gardella, M. Lupini: Cocycle superrigidity and group actions on stably finiteC ∗ -algebras (2016). URL https://arxiv.org/abs/1612.08217 .[6] E. Gardella, M. Lupini: Equivariant logic and applications to C ∗ -dynamics (2016).URL https://arxiv.org/abs/1608.05532 .[7] M. Izumi: Finite group actions on C ∗ -algebras with the Rohlin property I. DukeMath. J. 122 (2004), no. 2, pp. 233–280.[8] M. Izumi: Finite group actions on C ∗ -algebras with the Rohlin property II. Adv.Math. 184 (2004), no. 1, pp. 119–160.[9] M. Izumi: Group Actions on Operator Algebras. Proc. Intern. Congr. Math. (2010),pp. 1528–1548.[10] M. Izumi: Poly- Z group actions on Kirchberg algebras. Oberwolfach Rep. 9 (2012),pp. 3170–3173.[11] M. Izumi, H. Matui: Z -actions on Kirchberg algebras. Adv. Math. 224 (2010), pp.355–400.[12] V. F. R. Jones: A converse to Ocneanu’s theorem. J. Operator Theory 10 (1983), pp.61–63.[13] E. Kirchberg: Central sequences in C ∗ -algebras and strongly purely infinite algebras.Operator Algebras: The Abel Symposium 1 (2004), pp. 175–231.[14] H. Matui: Classification of outer actions of Z N on O . Adv. Math. 217 (2008), pp.2872–2896.[15] H. Matui: Z N -actions on UHF algebras of infinite type. J. reine angew. Math. 657(2011), pp. 225–244.[16] H. Matui, Y. Sato: Z -stability of crossed products by strongly outer actions. Comm.Math. Phys. 314 (2012), no. 1, pp. 193–228.[17] H. Matui, Y. Sato: Z -stability of crossed products by strongly outer actions II. Amer.J. Math. 136 (2014), pp. 1441–1497.[18] A. Ocneanu: Actions of discrete amenable groups on von Neumann algebras, Vol.1138. Springer-Verlag, Berlin (1985).[19] J. A. Packer, I. Raeburn: Twisted crossed products of C ∗ -algebras. Math. Proc.Cambridge Philos. Soc. 106 (1989), no. 2, pp. 293–311.[20] M. Rørdam: Classification of Nuclear C ∗ -Algebras. Encyclopaedia of MathematicalSciences. Springer (2001).[21] M. Rørdam, W. Winter: The Jiang-Su algeba revisited. J. reine angew. Math. 642(2010), pp. 129–155.[22] Y. Sato: The Rohlin property for automorphisms of the Jiang-Su algebra. J. Funct.Anal. 259 (2010), no. 2, pp. 453–476.[23] Y. Sato: Actions of amenable groups and crossed products of Z -absorbing C ∗ -algebras(2016). URL https://arxiv.org/abs/1612.08529 .[24] G. Szabó: Equivariant Kirchberg-Phillips-type absorption for amenable group actions(2016). URL https://arxiv.org/abs/1610.05939 .[25] G. Szabó: Strongly self-absorbing C ∗ -dynamical systems, II. J. Noncommut. Geom.,to appear (2016). URL http://arxiv.org/abs/1602.00266 .[26] G. Szabó: Strongly self-absorbing C ∗ -dynamical systems. Trans. Amer. Math. Soc.,to appear (2016). URL http://arxiv.org/abs/1509.08380 .[27] A. Tikuisis, S. White, W. Winter: Quasidiagonality of nuclear C ∗ -algebras. Ann. ofMath. 185 (2017), no. 1, pp. 229–284. [28] A. S. Toms, W. Winter: Strongly self-absorbing C ∗ -algebras. Trans. Amer. Math.Soc. 359 (2007), no. 8, pp. 3999–4029.[29] W. Winter: Strongly self-absorbing C ∗ -algebras are Z -stable. J. Noncomm. Geom. 5(2011), no. 2, pp. 253–264.[30] W. Winter: Localizing the Elliott conjecture at strongly self-absorbing C ∗ -algebras,with an appendix by H. Lin. J. reine angew. Math. 692 (2014), pp. 193–231. Fraser Noble Building, Institute of Mathematics, University of Aberdeen,Aberdeen AB24 3UE, Scotland, UK
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