EQUIVARIANT *-HOMOMORPHISMS, ROKHLIN CONSTRAINTS AND EQUIVARIANT UHF-ABSORPTION 5 1 EUSEBIOGARDELLAANDLUISSANTIAGO 0 2 b e F Abstract. Weclassifyequivariant*-homomorphismsbetweenC*-dynamical systems associated to actions of finite groups with the Rokhlin property. In 5 addition, the given actions are classified. An obstruction is obtained for the Cuntz semigroupof aC*-algebraallowingsuchan action. Wealsoobtain an ] A equivariantUHF-absorptionresult. O Contents . h t 1. Introduction 1 a 2. Preliminary definitions and results 2 m 3. Classification of actions and equivariant *-homomorphisms 9 [ 4. Cuntz semigroup and K-theoretical constraints 22 2 5. Equivariant UHF-absorption 32 v References 38 0 6 5 5 0 1. Introduction . 1 Classification is a major subject in all areas of mathematics and has attracted 0 theattentionofmanytalentedmathematicians. InthecategoryofC*-algebras,the 5 program of classifying all amenable C*-algebras was initiated by Elliott, first with 1 the classification of AF-algebras, and later with the classification of certain simple : v C*-algebras of real rank zero. His work was followed by many other classification i X resultsfornuclearC*-algebras,bothinthestablyfiniteandthepurelyinfinitecase. The classification theory for von Neumann algebras precedes the classification r a program initiated by Elliott. In fact, the classification of amenable von Neumann algebras with separable pre-dual, which is due to Connes, Haagerup, Krieger and Takesaki, was completed more than 30 years ago. Connes moreover classified au- tomorphisms of the type II factor up to cocycle conjugacy in [7]. This can be 1 regardedas the first classificationresultfor actionson vonNeumann algebras,and it was followed by his own work on the classification of pointwise outer actions of amenable groups on von Neumann algebras in [8]. Several people have since then tried to obtain similar classification results for actions on C*-algebras. Early results in this direction include the work of Herman andOcneanuin[18]onintegeractionswiththeRokhlinpropertyonUHF-algebras, the work of Fack and Mar´echal in [14] and [15] for cyclic groups actions on UHF- algebras, and the work of Handelman and Rossmann [17] for locally representable Date:February6,2015. 1 2 EUSEBIOGARDELLAANDLUISSANTIAGO compactgroupactionsonAF-algebras. OtherresultshavebeenobtainedbyElliott andSuin[13]fordirectlimitactionsofZ onAF-algebras,andbyIzumiin[20]and 2 [21], where he proved a number of classification results for actions of finite groups on arbitrary unital separable C*-algebraswith the Rokhlin property, as well as for approximately representable actions. The classification result of Izumi regarding actionswiththe Rokhlinpropertyhasbeen extendedrecently byNawatain[24] to coveractionsoncertainnot-necessarilyunitalseparableC*-algebras(specificallyfor algebrasAsuchthatA⊆GL(A)). Itshouldbeemphasizedthattheclassificationof groupactionsonC*-algebrasisafarlessdevelopedsubjectthantheclassificationof C*-algebrasandevenfartherleess developedthanthe classificationofgroupactions on von Neumann algebras. In this paper we extend the classification results of Izumi and Nawata of finite groupactions on C*-algebraswith the Rokhlin property to actions of finite groups with the Rokhlin property on arbitrary separable C*-algebras. This is done by first obtaining a classification result for equivariant *-homomorphism between C*- dynamicalsystemsassociatedtoactionsoffinitegroupswiththeRokhlinproperty, and then applying Elliott’s intertwining argument. In this paper we also obtain obstructions on the Cuntz semigroup, the Murray-von Neumann semigroup, and the K-groups of a C*-algebraallowing an action of a finite group with the Rokhlin property. These resultsareusedtogetherwiththe classificationresultofactionsto obtain an equivariant UHF-absorption result. This paper is organized as follows. In Section 2, we collect a number of defi- nitions and results that will be used throughout the paper. In Section 3, we give an abstract classification for equivariant *-homomorphism between C*-dynamical systemsassociatedtoactionsoffinite groupswiththe Rokhlinproperty,aswellas, a classification for the given actions. These abstract classification results are used together with known classification results of C*-algebras to obtain specific classifi- cationofequivariant*-homomorphismsandactionsoffinitegroupsonC*-algebras that can be written as inductive limits of 1-dimensional NCCW-complexes with trivial K -groups and for unital simple AH-algebras of no dimension growth. 1 In Section 4, we obtain obstructions on the Cuntz semigroup, the Murray-von Neumann semigroup,and the K -groupsof a C*-algebraallowingan actionof a fi- ∗ nitegroupwiththeRokhlinproperty. ThenusingtheCuntzsemigroupobstruction we show that the Cuntz semigroup of a C*-algebra that admits an action of finite group with the Rokhlin property has certain divisibility property. In this section we also compute the Cuntz semigroup, the Murray-von Neumann semigroup, and the K -groups of the fixed-point and crossed product C*-algebrasassociated to an ∗ action of a finite group with the Rokhlin property. In Section 5, we obtain divisibility results for the Cuntz semigroup of certain classesofC*-algebrasandusethistogetherwiththeclassificationresultsforactions obtained in Section 3 to prove an equivariant UHF-absorbing result. 2. Preliminary definitions and results Let A be a C*-algebra. We denote by M(A) its multiplier algebra, by A its unitization (that is, the C*-algebra obtained by adjoining a unit to A, even if A is unital). If A is unital, we denote by U(A) its unitary group. We denote by Auet(A) the automorphism group of A. The identity map of A is denoted id . A 3 TopologicalgroupsarealwaysassumedtobeHausdorff. IfGisalocallycompact group and A is a C*-algebra, then an action of G on A is a strongly continuous group homomorphism α: G → Aut(A). Strong continuity for α means that for each a in A, the map from G to A given by g 7→ α (a) is continuous with respect g to the norm topology on A. We denote by K the C*-algebra of compact operators on a separable Hilbert space. We take N={1,2,...}, Z ={0,1,2,...}, and Z =Z ∪{∞}. + + + 2.1. The Rokhlin property for finite group actions. Let us briefly recall the definition of the Rokhlin property, in the sense of [32, Definition 2], for actions of finite groups on (not necessarily unital) C*-algebras. Actions with the Rokhlin property are the main object of study of this work. Definition 2.1. Let A be a C*-algebra and let α: G→Aut(A) be an action of a finite group G on A. We say that α has the Rokhlin property if for any ε >0 and anyfinite subset F ⊆A there existmutually orthogonalpositive contractionsr in g A, for g ∈G, such that (i) kα (r )−r k<ε for all g,h∈G; g h gh (ii) kr a−ar k<ε for all a∈F and all g ∈G; g g (iii) k( r )a−ak<ε for all a∈F. g∈G g The elements r , for g ∈G, will be called Rokhlin elements for α for the choices of P g ε and F. It was shown in [32, Corollary 1] that Definition 2.1 agrees with [20, Definition 3.1] whenever the C*-algebra A is unital. It is also shown in [32, Corollary 2] that Definition 2.1 agrees with [24, Definition 3.1] whenever the C*-algebra A is separable. If A is a C*-algebra, we denote by ℓ∞(N,A) the set of all bounded sequences (an)n∈N in Awith the supremumnormk(an)n∈Nk=supkank, andpointwiseoper- n∈N ations. Then ℓ∞(N,A) is a C*-algebra, and it is unital when A is (the unit being the constant sequence 1 ). Let A c0(N,A)= (an)n∈N ∈ℓ∞(N,A): lim kank=0 . n→∞ Then c (N,A) is an ideal innℓ∞(N,A), and we denote the quotieont 0 ℓ∞(N,A)/c (N,A) 0 by A∞, which we call the sequence algebra of A. Write π : ℓ∞(N,A) → A∞ for the quotient map, and identify A with the sub- A algebra of ℓ∞(N,A) consisting of the constant sequences, and with the subalgebra of A∞ by taking its image under π . We write A = A∞ ∩A′ for the relative A ∞ commutant of A inside of A∞, and call it the central sequence algebra of A. Let G be a finite group and let α: G → Aut(A) be an action of G on A. Then there are actions of G on A∞ and A which, for simplicity and ease of notation, ∞ and unless confusion is likely to arise, we denote simply by α. The following is a characterizationof the Rokhlin property in terms of elements of the sequence algebra A∞ ([32, Proposition 1]): Lemma 2.1. Let A be a C*-algebra and let α: G → Aut(A) be an action of a finite group G on A. Then the following are equivalent: (i) α has the Rokhlin property. 4 EUSEBIOGARDELLAANDLUISSANTIAGO (ii) For any finite subset F ⊆A there exist mutually orthogonalpositive con- tractions r in A∞∩F′, for g ∈G, such that g (a) α (r )=r for all g,h∈G; g h gh (b) ( r )b=b for all b∈F. g∈G g (iii) For any separable C*-subalgebra B ⊆ A there are orthogonal positive P contractions r in A∞∩B′ for g ∈G such that g (a) α (r )=r for all g,h∈G; g h gh (b) ( r )b=b for all b∈B. g∈G g The first paPrt of the following proposition is [32, Theorem 2 (i)]. The second part follows trivially from the definition of the Rokhlin property. Proposition 2.1. Let G be a finite group,let A be a C*-algebra,and let α: G→ Aut(A) be an action with the Rokhlin property. (i) If B is any C*-algebraand β: G→Aut(B) is any action of G on B, then the action α⊗β: G→Aut(A⊗ B) min defined by (α⊗β) =α ⊗β for all g ∈G, has the Rokhlin property. g g g (ii) If B is a C*-algebra and ϕ: A → B is an isomorphism, then the action g 7→ϕ◦α ◦ϕ−1 of G on B has the Rokhlin property. g The following example may be regarded as the “generating” Rokhlin action for agivenfinite groupG. For some classesof C*-algebras,itcanbe shownthat every action of G with the Rokhlin property tensorially absorbs the action we construct below. See [21, Theorems 3.4 and 3.5] and Theorem 5.2 below. Example 2.1. Let G be a finite group. Let λ: G → U(ℓ2(G)) be the left reg- ular representation, and identify ℓ2(G) with C|G|. Define an action µG: G → Aut(M ) by |G|∞ ∞ µG = Ad(λ ) g g n=1 O for all g ∈G. It is easy to check that α has the Rokhlin property. Note that µG is g approximately inner for all g ∈G. Itfollowsfrompart(i)ofProposition2.1thatanyactionoftheformα⊗µG has the Rokhlin property. One of our main results, Theorem 5.2, states that in some circumstances, every action with the Rokhlin property has this form. 2.2. The category Cu, the Cuntz semigroup, and the Cu∼-semigroup. In this subsection, we will recall the definitions of the Cuntz and Cu∼ semigroups, as well as the category Cu, to which these semigroups naturally belong. 2.2.1. The category Cu. Let S be an ordered semigroup and let s,t ∈ S. We say that s is compactly contained in t, and denote this by s ≪ t, if whenever (tn)n∈N is an increasing sequence in S such that t ≤ supt , there exists k ∈ N such that n n∈N s ≤ tk. A sequence (sn)n∈N is said to be rapidly increasing if sn ≪ sn+1 for all n∈N. Definition 2.2. An ordered abelian semigroup S is an object in the category Cu if it has a zero element and it satisfies the following properties: (O1) Every increasing sequence in S has a supremum; 5 (O2) For every s ∈ S there exists a rapidly increasing sequence (sn)n∈N in S such that s=sups . n n∈N (O3) If (sn)n∈N and (tn)n∈N are increasing sequences in S, then sups +supt =sup(s +t ); n n n n n∈N n∈N n∈N (O4) Ifs ,s ,t ,t ∈S aresuchthats ≪t ands ≪t ,thens +s ≪t +t . 1 2 1 2 1 1 2 2 1 2 1 2 Let S and T be semigroups in the category Cu. An order-preservingsemigroup map ϕ: S → T is a morphism in the category Cu if it preserves the zero element and it satisfies the following properties: (M1) If (sn)n∈N is an increasing sequence in S, then ϕ sups =supϕ(s ); n n (cid:18)n∈N (cid:19) n∈N (M2) If s,t∈S are such that s≪t, then ϕ(s)≪ϕ(t). It is shown in [9, Theorem 2] that the category Cu is closed under sequential inductive limits. The following description of inductive limits in the category Cu follows from the proof of this theorem. Proposition 2.2. Let (Sn,ϕn)n∈N, with ϕn: Sn →Sn+1, be an inductive system in the category Cu. For m,n ∈ N with m ≥ n, let ϕ : S → S denote the n,m n m+1 compositionϕn,m =ϕm◦···◦ϕn. Apair(S,(ϕn,∞)n∈N),consistingofasemigroup S andmorphismsϕ : S →S inthecategoryCusatisfyingϕ ◦ϕ =ϕ n,∞ n n+1,∞ n n,∞ for all n∈N, is the inductive limit of the system (Sn,ϕn)n∈N if and only if: (i) Foreverys∈Sthereexistelementss ∈S forn∈N,suchthatϕ (s )≪ n n n n s for all n∈N and n+1 s=supϕ (s ); n,∞ n n∈N (ii) Whenever s,s′,t ∈ S satisfy ϕ (s) ≤ ϕ (t) and s′ ≪s, there exists n n,∞ n,∞ m≥n such that ϕ (s′)≤ϕ (t). n,m n,m Lemma 2.2. LetS beasemigroupinCu,letsbeanelementinS andlet(sn)n∈N be a rapidly increasing sequence in S such that s=sups . Let T be a subset of S n n∈N such that every element of T is the supremum of a rapidly increasing sequence of elements in T. Suppose that for every n ∈ N there is t ∈ T such that s ≪ t ≤ s. n Then there exists an increasing sequence (tn)n∈N in T such that s=suptn. n∈N Proof. It is sufficient to construct an increasing sequence (nk)k∈N of natural num- bers and a sequence (tk)k∈N in T such that snk ≤ tk ≤ snk+1 for all k ∈ N, since this implies that s=supt . k k∈N Fork =1,setn =1ands =0. Assumeinductivelythatwehaveconstructed 1 n1 n and t for all j ≤ k and let us construct n and t . By the assumptions j j k+1 k+1 of the lemma, there exists t ∈ T such that s ≪ t ≤ s. Also by assumption, t is nk thesupremumofarapidlyincreasingsequenceofelementsofT. Hencethereexists t′ ∈T such that s ≤t′ ≪s. Use that s=sups and t′ ≪s, to choose n ∈N nk n∈N n k+1 with n >n such that t′ ≤s ≪s. Set t =t′. Then s ≤t ≤s . k+1 k nk+1 k+1 nk k+1 nk+1 This completes the proof of the lemma. (cid:3) 6 EUSEBIOGARDELLAANDLUISSANTIAGO Definition 2.3. Let S be a semigroup in the category Cu. Let I be a nonempty set and let γ : S →S for i∈I, be a family of endomorphisms of S in the category i Cu. We introduce the following notation: s ≪s if r <t, s =sups ∀ t∈(0,1], r t t r Sγ = s∈S: ∃ (s ) in S: r<t , t t∈(0,1] ( s =s, and γ (s )=s ∀ t∈(0,1] and ∀ i∈I) 1 i t t and s ≪s ∀ n∈N, s=sups , n n+1 n SNγ = s∈S: ∃ (sn)n∈N in S: n∈N . ( and γ (s )=s ∀ n∈N and ∀ i∈I) i n n Lemma 2.3. Let S be a semigroup in the category Cu. Let I be a nonempty set and let γ : S → S for i ∈ I, be a family of endomorphisms of S in the category i Cu. Then (i) Sγ is closed under suprema of increasing sequences; N (ii) Sγ is an object in Cu. Proof. (i). Let (sn)n∈N be an increasing sequence in SNγ. For each n∈N, choose a rapidlyincreasingsequence(sn,m)m∈N inS suchthatsn = supsn,m andγi(sn,m)= m∈N s foralli∈I andm∈N. Bythedefinitionofthecompactcontainmentrelation, n,m there exist increasing sequences (nj)j∈N and (mj)j∈N in N such that sk,l ≤ snj,mj whenever 1 ≤ k,l ≤ j, and such that (snj,mj)j∈N is increasing. Let s be the supremum of (snj,mj)j∈N in S. Then s ∈ SNγ, and it is straightforward to check, using a diagonal argument, that s=sups , as desired. n n∈N (ii). It is clear that Sγ satisfies O2, O3 and O4. Now let us check that Sγ satisfies axiom O1. Let (s(n))n∈N be an increasing sequence in Sγ and let s be its supremum in S. It is sufficient to show that s∈Sγ. For each n ∈ N, choose a path (s(n)) as in the definition of Sγ for s(n). t t∈(0,1] Using that s(n) ≪ s(n+1) for all n ∈ N and all t ∈ (0,1), together with a diagonal t argument,chooseanincreasingsequence(tn)n∈N in(0,1]convergingto1,suchthat s(n) ≪s(n+1) ∀ n∈N, and s= sups(n). tn tn+1 n∈N tn Thisimplies,usingthedefinitionofthecompactcontainmentrelation,thatforeach n∈N there exists t′ such that t <t′ <t and n+1 n n+1 n+1 s(n) ≪s(n+1) ≤s(n+1) for all t∈(t′ ,t ]. tn t tn+1 n+1 n+1 Choose an increasing function f: (0,1]→(0,1] such that 1 1 f 1− ,1− =(t′ ,t ] n n+1 n+1 n+1 (cid:18)(cid:18) (cid:21)(cid:19) for all n∈N. Define a path (s ) in S by taking s =s and t t∈(0,1] 1 1 1 s =s(n+1) for t∈ 1− ,1− . t f(t) n n+1 (cid:18) (cid:21) Then γ (s )=s for all t∈(0,1] and all i∈I, so s∈Sγ. It is clear that this path i t t satisfies the conditions in the definition of Sγ for s. (cid:3) 7 2.2.2. The Cuntz semigroup. Let A be a C*-algebra and let a,b ∈ A be positive elements. We say that a is Cuntz subequivalent to b, and denote this by a - b, if there exists a sequence (dn)n∈N in A such that lim kd∗nbdn−ak=0. We say that n→∞ a is Cuntz equivalent to b, and denote this by a∼b, if a-b and b-a. It is clear that - is a preorder relation in the set of positive elements of A, and thus ∼ is an equivalence relation. We denote by [a] the Cuntz equivalence class of the element a∈A . + The first conclusion of the following lemma was proved in [30, Proposition 2.2] (see also [23, Lemma 2.2]). The second statement was shown in [29, Lemma 1]. Lemma 2.4. LetAbeaC*-algebraandletaandbbepositiveelementsinAsuch that ka−bk < ε. Then (a−ε) - b. More generally, if r is a non-negative real + number, then (a−r−ε) -(b−r) . + + The Cuntz semigroup of A, denoted by Cu(A), is defined as the set of Cuntz equivalence classes of positive elements of A⊗K. Addition in Cu(A) is given by [a]+[b]=[a′+b′], where a′,b′ ∈(A⊗K) are orthogonaland satisfy a′ ∼a and b′ ∼b. Furthermore, + Cu(A) becomes an ordered semigroup when equipped with the order [a] ≤ [b] if a - b. If φ: A → B is a *-homomorphism, then φ induces an order-preserving map Cu(φ): Cu(A) → Cu(B), given by Cu(φ)([a]) = [(φ⊗id )(a)] for every a ∈ K (A⊗K) . + Remark 2.1. Let A be a C*-algebra, let a ∈ A and let ε > 0. It can be checked that[(a−ε) ]≪[a]andthat[a]=sup[(a−ε) ],thusshowingthatCu(A)satisfies + + ε>0 Axiom O2. It is shown in [9, Theorem 1] that Cu is a functor from the category of C*- algebras to the category Cu. Lemma 2.5. Let A and B be C*-algebras and let ρ: Cu(A) → Cu(B) be an order-preservingsemigroup map. Suppose that for all a∈(A⊗K) one has + (i) ρ([a])=supρ([(a−ε) ]), + ε>0 (ii) ρ([(a−ε) ])≪ρ([a]) for all ε>0. + ThenρisamorphisminthecategoryCu;thatis,itpreservessupremaofincreasing sequences and the compact containment relation. Proof. We show first that ρ preserves suprema of increasing sequences. Let a be a positive element in A⊗K and let (an)n∈N be an increasing sequence of positive elements in A⊗K such that sup[a ] = [a]. Then ρ([a ]) ≤ ρ([a]) for all n ∈ N. n n n∈N Supposethatb∈(B⊗K) issuchthatρ([a ])≤[b]foralln∈Nandletε>0. By + n thedefinitionofthecompactcontainmentrelationandthefactthat[(a−ε) ]≪[a], + there exists n ∈ N such that [(a−ε) ] ≤ [a ]. By applying ρ to this inequality 0 + n0 we get ρ([(a−ε) ])≤ρ([a ])≤[b]. + n0 By taking supremum in ε>0 and applying (i) we get ρ([a])=supρ([(a−ε) ])≤[b]. + ε>0 This shows that ρ([a]) is the supremum of (ρ([an]))n∈N, as desired. 8 EUSEBIOGARDELLAANDLUISSANTIAGO We proceed to show that ρ preserves the compact containment relation. Let a and b be positive elements in A⊗K such that [a] ≪ [b]. Choose ε > 0 such that [a]≤[(b−ε) ]≤[b]. It follows that + ρ([a])≤ρ([(b−ε) ])≤ρ([b]). + By (ii) applied to [b] we get ρ([a])≪ρ([b]), which concludes the proof. (cid:3) The following lemma is a restatement of [29, Lemma 4]. Lemma 2.6. Let A be a C*-algebra, let (x )n be elements of Cu(A) such that i i=0 x ≪x for all i=0,...,n, and let ε>0. Then there exists a∈(A⊗K) such i+1 i + that x ≪[(a−(n−1)ε) ]≪x ≪[(a−(n−2)ε) ]≪··· n + n−1 + ···≪x ≪[(a−2ε) ]≪x ≪[(a−ε) ]≪x ≪[a]=x . 3 + 2 + 1 0 2.2.3. The Cu∼-semigroup. Here we define the Cu∼-semigroup of a C*-algebra. This semigroup was introduced in [28] in order to classify certain inductive limits of 1-dimensional NCCW-complexes. Definition2.4. LetAbeC*-algebraandletπ: A→A/A∼=Cdenotethequotient map. Then π induces a semigroup homomorphism e e Cu(π): Cu(A)→Cu(C)∼=Z . + We define the semigroup Cu∼(A) by e Cu∼(A)={([a],n)∈Cu(A)×Z |Cu(π)([a])=n}/∼, + where ∼ is the equivalence relation defined by e ([a],n)∼([b],m) if [a]+m[1]+k[1]=[b]+n[1]+k[1], for some k ∈ N. The image of the element ([a],n) under the canonical quotient map is denoted by [a]−n[1]. Addition in Cu∼(A) is induced by pointwise addition in Cu(A) × Z . The + semigroupCu∼(A)canbeendowedwithanorder: wesaythat[a]−n[1]≤[b]−m[1] in Cu∼(A) if there exists k in Z such that e + [a]+(m+k)[1]≤[b]+(n+k)[1] in Cu(A). The assignment A 7→ Cu∼(A) can be turned into a functor as follows. Let φ: A→eB be a *-homomorphismand let φ: A→B denote the unital extension of φ to the unitizations of A and B. Let us denote by Cu∼(φ): Cu∼(A) → Cu∼(B) the map defined by e e e Cu∼(φ)([a]−n[1])=Cu(φ)([a])−n[1]. It is clear that Cu∼(φ) is order-preserving, and thus Cu∼ becomes a functor from e the category of C*-algebrasto the category of ordered semigroups. It was shown in [28] that the Cu∼-semigroup of a C*-algebra with stable rank one belongs to the category Cu, that Cu∼ is a functor from the category of C*- algebras of stable rank one to the category Cu, and that it preserves inductive limits of sequences. 9 3. Classification of actions and equivariant *-homomorphisms In this section we classify equivariant *-homomorphisms whose codomain C*- dynamicalsystemhavetheRokhlinproperty. Weusethisresultstoclassifyactions of finite groups on separable C*-algebras with the Rokhlin property. Our results complement and extend those obtained by Izumi in [20] and [21] in the unital setting, and by Nawata in [24] for C*-algebras A that satisfy A⊆GL(A). 3.1. Equivariant *-homomorphisms. LetAandB beC*-algebrasandletGbe e a compact group. Let α: G → Aut(A) and β: G → Aut(B) be (strongly continu- ous) actions. Recall that a *-homomorphismφ: A→B is said to be equivariant if φ◦α =β ◦φ for all g ∈G. g g Definition 3.1. LetAandB be C*-algebrasandletα: G→Aut(A)andβ: G→ Aut(B) be actions of a compact group G. Let φ,ψ: A → B be equivariant *- homomorphisms. We say that φ and ψ are equivariantly approximately unitarily equivalent, and denote this by φ ∼ ψ, if for any finite subset F ⊆ A and for G−au any ε>0 there exists a unitary u∈Bβ such that kφ(a)−u∗ψ(a)uk<ε, f for all a∈F. Note that when G is the trivial group, this definition agrees with the standard definitionofapproximateunitaryequivalenceof*-homomorphisms. Inthiscasewe will omit the group G in the notation ∼ , and write simply ∼ . G−au au Thefollowinglemmacanbeprovedusingastandardsemiprojectivityargument. Its proof is left to the reader. Lemma 3.1. Let A be a unital C*-algebra and let u be a unitary in A . Given ∞ ε > 0 and given a finite subset F ⊆ A, there exists a unitary v ∈ A such that kva − avk < ε for all a ∈ F. If moreover A is separable, then there exists a sequence (un)n∈N of unitaries in A with lim ku a−au k=0 n n n→∞ for all a∈A, such that πA((un)n∈N)=u in A∞. Proposition3.1. LetAandBbeC*-algebrasandletα: G→Aut(A)andβ: G→ Aut(B) be actions of a finite group G such that β has the Rokhlin property. Let φ,ψ: (A,α) → (B,β) be equivariant *-homomorphisms such that φ ∼ ψ. Then au φ∼ ψ. G−au Proof. Let F be a finite subset of A and let ε > 0. We have to show that there exists a unitary w ∈Bβ such that kφ(a)−w∗ψ(a)wk<ε, f for all a ∈ F. Set F′ = α (F), which is again a finite subset of A. Since g g∈G φ∼ ψ, there exists a unitaSry u∈B such that au (3.1) kφ(b)−u∗ψ(b)uk<ε e forall b∈F′. Choose x∈B andλ∈C ofmodulus 1 suchthatu=x+λ1e. Then B equation (3.1) above is satisfied if one replaces u with λu. Thus, we may assume that the unitary u has the form u=x+1e for some x∈B. B 10 EUSEBIOGARDELLAANDLUISSANTIAGO Fix g ∈ G and a ∈ F. Then b = α (a) belongs to F′. Using equation (3.1) g−1 and the fact that φ and ψ are equivariant, we get kβ (φ(a))−u∗β (ψ(a))uk<ε. g−1 g−1 By applying β to the inequality above, we conclude that g kφ(a)−β (u)∗ψ(a)β (u)k<ε g g for all a∈F and g ∈G Choose positive orthogonal contractions (r ) ⊆ B as in the definition of g g∈G ∞ the Rokhlin property for β, and set v = βg(x)rg +1Be. Using that xg+1Be is a g∈G unitary in B, one checks that P e v∗v = βg(x∗x)rg2+βg(x)rg +βg(x)rg +1Be =1Be. g∈G X(cid:0) (cid:1) Analogously, we have vv∗ = 1e, and hence v is a unitary in B. For every b ∈ B, B we have v∗bv = r β (u)∗bβ (u). e g g g g∈G X Therefore, kφ(a)−v∗ψ(a)vk= r φ(a)− r β (u)∗ψ(a)β (u) <ε, g g g g (cid:13) (cid:13) (cid:13)g∈G g∈G (cid:13) (cid:13)X X (cid:13) (cid:13) (cid:13) for all a ∈ F (here we are c(cid:13)onsidering φ and ψ as maps from A(cid:13) to (B)∞, by (cid:13) (cid:13) composingthemwiththenaturalinclusionofB in(B)∞). Sincev = β (xr )+ g e g∈G e 1e, we have v ∈(Bβ)∞ ⊆(B)∞. By Lemma 3.1, weecan choose a uniPtary w ∈Bβ B such that f ekφ(a)−w∗ψ(a)wk<ε, f for all a∈F, and the proof is finished. (cid:3) Lemma3.2. LetAandBbeC*-algebrasandletψ: A→Bbea*-homomorphism. Suppose there exists a sequence (vn)n∈N of unitaries in B such that the sequence (vnφ(x)vn∗)n∈N convergesin B for all x in a dense subset of A. Then there exists a *-homomorphismψ: A→B such that e lim v φ(x)v∗ =ψ(x) n n n→∞ for all x∈A. Proof. Let S ={x∈A: (vnφ(x)vn∗)n∈N converges in B}⊆A. Then S is a dense *-subalgebra of A. For each x ∈ S, denote by ψ (x) the limit 0 of the sequence (vnφ(x)vn∗)n∈N. The map ψ0: S → B is linear, multiplicative, preservesthe adjointoperation,andisboundedby kφk,soitextends bycontinuity to a *-homomorphism ψ: A → B. Given a ∈ A and given ε > 0, use density