Crossed products of locally C-algebras

a rX iv:mat h /512128v1[mat h.OA ]6Dec25CROSSED PRODUCTS OF LOCALLY C ∗-ALGEBRAS MARIA JOIT ¸A February 2,2008ABSTRACT.The crossed products of locally C ∗-algebras are defined and a Takai duality theorem for inverse limit actions of a locally compact group on a locally C ∗-algebra is proved.2000AMS Mathematics subject classification.Primary 46L05,46L55.1Introduction Locally C ∗-algebras are generalizations of C ∗-algebras.Instead of being given by a single C ∗-norm,the topology on a locally C ∗-algebra is defined by a directed family of C ∗-semi-norms.In [9],Phillips defines the notion of action of a locally compact group G on a locally C ∗-algebra A whosetopology is determined by a countable family of C ∗-semi-norms,and also defines the crossed product of A by an inverse limit action α=lim ←nα(n )asbeing the inverse limit of crossed products of A n by α(n ).In this paper,by analogy with the case of C ∗-algebras,we define the concept of crossed product,respectively reduced crossed product of locally C ∗-algebras.1The Takai duality theorem says that ifαis a continuous action of an abelian locally compact group G on a C∗-algebra A,then we can recover the system(G,A,α)up to stable isomorphism from the double dual system in which G= G acts on the crossed product(A×αG)× α G by the dual action of the dual group.In[3],Imai and Takai prove a duality theorem for C∗-crossed products by a locally compact group that generalizes the Takai duality theorem[12].For a given C∗-dynamical system(G,A,α), they construct a”dual”C∗-crossed product of the reduced crossed product A×α,r G by an isomorphismβfrom A×α,r G into L(H),the C∗-algebra of all bounded linear operators on some Hilbert space H,and show that this is isomorphic to the tensor product A⊗K(L2(G))of A and K(L2(G)),the C∗-algebra of all compact operators on L2(G).If G is commutative,the ”dual”C∗-crossed product constructed by Imai and Takai is isomorphic to the double crossed product(A×αG)× α G.Katayama[6]shows that a non-degenerate coactionβof a locally compact group on a C∗-algebra A induces an action βof G on the crossed product A×βG and proves that the C∗-algebras(A×βG)× β,r G and A⊗K(L2(G))are isomorphic.In[13], Vallin shows that there is a bijective correspondence between the set of all actions of a locally compact group G on a C∗-algebra A and the set of all actions of the commutative Kac C∗-algebra C∗K a G associated with G on A.A coaction of G on A is an action of the symmetric Kac C∗-algebra C∗K s G associated with G.If G is commutative,we can identified C∗r(G)with C0( G) via the Fourier transform,whence becomes clear that a coaction of G is the same thing as an action of G.Thus we can regard the coactions of a locally compact group G as”actions of the dual group even there isn’t any dual group”.Also,Vallin shows that an actionα(coactionβ)of G on A induces2a coaction α(action β)of G on the crossed product A×α,r G(respectively A×βG)and proves a version of the Takai duality theorem showing that the double crossed product(A×α,r G)× αG is isomorphic to A⊗K(L2(G)).We propose to prove a version of the Takai duality theorem for crossed products of locally C∗-algebras.The paper is organized follows.In Section2we present some basic definitions and results about locally C∗-algebras and Kac C∗-algebras.In Section3we define the notion of crossed product(reduced crossed product) of a locally C∗-algebra A by an inverse limit actionαof a locally compact group G and prove some basic properties of these.Section4is devoted to actions of a Kac C∗-algebra on a locally C∗-algebra.We show that there is a bijective correspondence between the set of all inverse limit actions of a locally compact group G on a locally C∗-algebra A and the set of all inverse limit actions of the commutative Kac C∗-algebra C∗K a G on A,Proposition 4.4.As a consequence of this result we obtain:for a compact group G,any action of the Kac C∗-algebra C∗K a G on A is an inverse limit of actions of the Kac C∗-algebras C∗K a G on A p,p∈S(A).In Section5,using the same arguments as in[13],we show that any inverse limit actionα(coactionβ)of a locally compact group G on a locally C∗-algebra A induces an inverse limit coaction α(action β)of G on the crossed product A×α,r G(respectively A×βG),Proposition5.5.Finally,we prove that ifαis an inverse limit action of a locally compact group G on a locally C∗-algebra A,then there is an isomorphism of locally C∗-algebras from(A×α,r G)× αG onto A⊗K(L2(G)) and the inverse limit actions αandα⊗adρare equivalent,Theorem5.6.32PreliminariesA locally C∗-algebra is a complete complex Hausdorfftopological∗-algebraA whose topology is determined by a family of C∗-semi-norms,see[1],[2],[4],[9],[10].If S(A)is the set of all continuous C∗-semi-norms on A,then for each p∈S(A),A p=A/ker(p)is a C∗-algebra with respect to theA p.The canonical maps from A onto norm induced by p,and A=lim←p∈S(A)A p,p∈S(A)are denoted byπp,the image of a underπp by a p,and the connecting maps of the inverse system{A p}p∈S(A)byπpq,p,q∈S(A)with p≥q.A morphism of locally C∗-algebras is a continuous∗-morphismΦfrom a locally C∗-algebra A to a locally C∗-algebra B.An isomorphism of locally C∗-algebras is a morphism of locally C∗-algebras which is invertible and its inverse is a morphism of locally C∗-algebras.An S-morphism of locally C∗-algebras is a morphismΦ:A→M(B),where M(B)is the multiplier algebra of B,with the property that for any approximate unit{e i}i of A the net{Φ(e i)}i converges to1with respect to the strict topology on M(B).If Φ:A→M(B)is an S-morphism of locally C∗-algebras,then it extends to a unique morphismlocally compact group,then M(A,C0(G))may be identified with the locallyC∗-algebra C b(G,A)of all bounded continuous functions from G to A.Let G be a locally compact group.C∗K a G=(C0(G),d a G,j a G,ds)is the commutative Kac C∗-algebra associated with G and C∗K s G=(C∗r(G),d s G,j s G,ϕG) is the symmetric Kac C∗-algebra associated with G,see[13].An action of a Kac C∗-algebra K=(B,d,j,ϕ)on a C∗-algebra A isan injective S-morphismαfrom A to M(A,B)such that(id A⊗σB◦d)◦α,see[13].3Crossed productsLet A be a locally C∗-algebra and let G be a locally compact group.Definition3.1An action of G on A is a morphismαfrom G to Aut(A),the set of all isomorphisms of locally C∗-algebras from A to A.The actionαis continuous if the function(t,a)→αt(a)from G×A to A is jointly continuous.Definition3.2A locally C∗-dynamical system is a triple(G,A,α),where Gis a locally compact group,A is a locally C∗-algebra andαis a continuous action of G on A.Definition3.3We say that G,Aδ,α(δ) δ∈∆is an inverse system of C∗-dynamical systems if{Aδ}δ∈∆is an inverse system of C∗-algebras and for each t in G, α(δ)t δ∈∆is an inverse system of C∗-isomorphisms.5Let A=lim←δ∈∆Aδandαt=lim←δ∈∆α(δ)t for each t∈G.Then the mapα:G→Aut(A)defined byα(t)=αt is a continuous action of G on A and (G,A,α)is a locally C∗-dynamical system.We say that(G,A,α)is the in-verse limit of the inverse system of C∗-dynamical systems G,Aδ,α(δ) δ∈∆. Definition3.4A continuous actionαof G on A is an inverse limit action if we can write A as inverse limit lim←δ∈∆Aδof C∗-algebras in such a way thatthere are actionsα(δ)of G on Aδsuch thatαt=lim←δ∈∆α(δ)t for all t in G (Definition5.1,[9]).Remark3.5The actionαof G on A is an inverse limit action if there is a cofinal subset of G-invariant continuous C∗-semi-norms on A(a continuous C∗-semi-norm p on A is G-invariant if p(αt(a))=p(a)for all a in A and for all t in G).The following lemma is Lemma5.2of[9].Lemma3.6Any continuous action of a compact group G on a locally C∗-algebra A is an inverse limit action.Let(G,A,α)be a locally C∗-dynamical system such thatαis an inverse limit action.By Remark3.5,we can suppose that S(A)coincides with the set of all G-invariant continuous C∗-semi-norms on A.Let C c(G,A)be the vector space of all continuous functions from G to A with compact support.6Lemma3.7Let f∈C c(G,A).Then there is a unique element G f(s)ds in A such that for any non-degenerate∗-representation(ϕ,Hϕ)of Aϕ( G f(s)ds)ξ,η = G ϕ(f(s))ξ,η dsfor allξ,ηin Hϕ.Moreover,we have:(1)p( G f(s)ds)≤M sup{p(f(s));s∈supp(f)}for some positive number M and for all p∈S(A);(2)( G f(s)ds)a= G f(s)ads for all a∈A;(3)Φ( G f(s)ds)= GΦ(f(s))ds for any morphism of locally C∗-algebras Φ:A→B;(4)( G f(s)ds)∗= G f(s)∗ds.Proof.Let p∈S(A).Thenπp◦f∈C c(G,A p)and so there is a unique ele-ment G(πp◦f)(s)ds in A p such that for any non-degenerate∗-representation )of A p(ϕp,Hϕpϕp( G(πp◦f)(s)ds)ξ,η = G ϕp((πp◦f)(s))ξ,η ds,see,for instance,Lemma7of[11].for allξ,ηin HϕpTo show that( G(πp◦f)(s)ds)p is a coherent net in A,let p,q∈S(A) with p≥q.Then we haveπpq( G(πp◦f)(s)ds)= Gπpq((πp◦f)(s))dsussing Lemma7of[11]= G(πq◦f)(s)ds.7Therefore( G(πp◦f)(s)ds)p∈A,and we define G f(s)ds=( G(πp◦f)(s)ds)p.Suppose that there is another element b in A such that for any non-degenerate∗-representation(ϕ,Hϕ)of Aϕ(b)ξ,η = G ϕ(f(s))ξ,η dsfor allξ,ηin Hϕ.Then for any p∈S(A)and for any non-degenerate∗-)of A prepresentation(ϕp,Hϕpϕp(πp(b))ξ,η = G ϕp((πp◦f)(s))ξ,η ds.From these facts and Lemma7of[11],we conclude that for allξ,ηin Hϕpπp(b)= G(πp◦f)(s)dsfor all p∈S(A).Therefore b= G f(s)ds and the uniqueness is proved. Using Lemma7of[11]it is easy to check that G f(s)ds satisfies the conditions(1)−(4).For any p ∈S (A ),define N p from C c (G,A )to [0,∞)byN p (f )= Gp (f (s ))ds.Straightforward computations show that N p ,p ∈S (A )are submultiplicative ∗-semi-norms on C c (G,A ).Let L 1(G,A,α)be the Hausdorffcompletion of C c (G,A )with respect to the topology defined by the family of submultiplicative ∗-semi-norms {N p }p ∈S (A ).Then by Theorem III 3.1of [7]L 1(G,A,α)=lim ←p ∈S (A ) L 1(G,A,α) p where L 1(G,A,α)p is the completion of the ∗-algebra C c (G,A )/ker (N p )with respect to the norm · p induced by N p .Lemma 3.8Let (G,A,α)be a locally C ∗-dynamical system such that αis an inverse limit action.ThenL 1(G,A,α) p =L 1 G,A p ,α(p )for all p ∈S (A ),up to a topological algebraic ∗-isomorphism.Proof.Let p ∈S (A )and f in C c (G,A ).Thenf +ker(N p ) p = G p (f (s ))ds =G πp (f (s )) p ds = πp ◦f 1.Therefore we can define a linear map ψp from C c (G,A )/ker (N p )to C c (G,A p )byψp (f +ker(N p ))=πp ◦f.It is not hard to check that ψp is a ∗-morphism,and since ψp is an isometric ∗-morphism from C c (G,A )/ker(N p )to C c (G,A p ),it can be9uniquely extended to an isometric∗-morphismψp from L1(G,A,α) p to L1 G,A p,α(p) .To show thatψp is surjective,let a∈A and f∈C c(G).Define f from G to A by f(s)=f(s)a.Clearly f∈C c(G,A)andψp f+ker(N p) (s)=f(s)πp(a)for all s in G.This implies thatA p⊗alg C c(G)⊆ψp L1(G,A,α) p ⊆L1(G,A p,α(p)) whence,since A p⊗alg C c(G)is dense in L1(G,A p,α(p))and sinceψp is an isometric∗-morphism,we deduce thatψp is surjective and the proposition is proved.Definition3.11A covariant representation of(G,A,α)is a triple(ϕ,u,H), where(ϕ,H)is a∗-representation of A and(u,H)is a unitary representation of G such thatϕ(αt(a))=u tϕ(a)u∗tfor all t∈G and for all a∈A.We say that the covariant representation(ϕ,u,H)of(G,A,α)is non-degenerate if the∗-representation(ϕ,H)of A is non-degenerate.Remark3.12(1).If(ϕ,u,H)is a covariant representation of(G,A,α)such that ϕ(a) ≤p(a)for all a∈A,then there is a unique covariant rep-resentation(ϕp,u,H)of the C∗-dynamical system G,A p,α(p) such that ϕp◦πp=ϕ.(2).If(ϕp,u,H)is a covariant representation of the C∗-dynamical system G,A p,α(p) ,then(ϕp◦πp,u,H)is a covariant representation of the locallyC∗-dynamical system(G,A,α).If R(G,A,α)denotes the non-degenerate covariant representations of (G,A,α),then it is easy to check thatR(G,A,α)= p∈S(A)R p(G,A,α)where R p(G,A,α)={(ϕ,u,H)∈R(G,A,α); ϕ(a) ≤p(a)for all a∈A}. Also it is easy to check that the mapϕp→ϕp◦πp from R(G,A p,α(p))to R p(G,A,α)is bijective.Proposition3.13Let(G,A,α)be a locally C∗-dynamical system such thatαis an inverse limit action.Then there is a bijection between the co-variant non-degenerate representations of(G,A,α)and the non-degenerate ∗-representations of L1(G,A,α).11Proof.Let(ϕ,u,H)∈R(G,A,α).Then,there is p∈S(A)and(ϕp,u,H)∈R(G,A p,α(p))such thatϕ=ϕp◦πp.Since(ϕp,u,H)∈R(G,A p,α(p))there is a unique non-degenerate∗-representation(ϕp×u,H)of L1(G,A p,α(p)) such that(ϕp×u)(f)= Gϕp(f(t))u t dtfor all f∈L1 G,A p,α(p) ,see,for instance,Proposition7.6.4of[8].Letϕ×u=(ϕp×u)◦ πp,where πp is the canonical map from L1(G,A,α) to L1(G,A p,α(p)), πp(f)=πp◦f for all f in L1(G,A,α).Then,clearly (ϕ×u,H)is a non-degenerate∗-representation of L1(G,A,α)and moreover, (ϕ×u)(f)=(ϕp×u)(πp◦f)= Gϕp((πp◦f)(t))u t dt= Gϕ(f(t))u t dt for all f∈L1(G,A,α).Thus we have obtained a map(ϕ,u,H)→(ϕ×u,H) from R(G,A,α)to R(L1(G,A,α)).To show that this map is bijective,let (Φ,H)be a non-degenerate∗-representation of L1(G,A,α).Then there is p∈S(A)and a non-degenerate∗-representation(Φp,H)of L1(G,A p,α(p)) such thatΦ=Φp◦πp.By Proposition7.6.4of[8]there is a unique non-degenerate covariant representation(ϕp,u,H)of(G,A p,α(p))such that (φp,H)=(ϕp×u,H).Therefore there is a non-degenerate covariant rep-resentation(ϕ,u,H)of(G,A,α),whereϕ=ϕp◦πp,such that(Φ,H)= (ϕ×u,H).To show that(ϕ,u,H)is unique,let(ψ,v,K)be another non-degenerate covariant representation of(G,α,A)such that(ψ×v,K)=(Φ,H).Then there is q∈S(A)with q≥p such that(ψ,v,K)∈R q(G,A,α)and(Φ,K)∈R q L1(G,A,α) .ThereforeΦ=Φq◦ πq with(Φq,H)∈R L1(G,A q,α(q)) andψ=ψq◦πq with(ψq,v,K)∈R G,A q,α(q) and moreover,(Φq,H)=12(ψq×v,K).On the other hand,(ϕp◦πpq,u,H)∈R G,A q,α(q) and ((ϕp◦πpq)×u)(f)= G(ϕp◦πpq)(f(t))u t dt= Gϕp( πpq(f)(t))u t dt=φp( πpq(f))=(φp◦ πpq)(f)=φq(f)for all f∈L1(G,A q,α(q)).From these facts and Proposition7.6.4of[8],we conclude that the covariant representations(ψq,v,K)and(ϕp◦πpq,u,H) of G,A q,α(q) coincide,and so the covariant representations(ψ,v,K)and (ϕ,u,H)of(G,A,α)coincide.Proof.Since A×αG is the enveloping locally C∗-algebra of the com-plete locally m-convex∗-algebra L1(G,A,α),there is a bijection between the non-degenerate representations of A×αG and the non-degenerate rep-resentations of L1(G,A,α),[2,pp.37].From this fact and Proposition 3.13we conclude that there is a bijection between the non-degenerate rep-resentations of A×αG and the non-degenerate covariant representations of (G,A,α).Let p∈S(A).The map r p:L1(G,A,α)→[0,∞)defined byr p(f)= ( ϕp×λ)(f)is a C∗-semi-norm on L1(G,A,α)with the property that r p(f)≤N p(f)for all f in L1(G,A,α).14Let I= p∈S(A)ker(r p).Clearly I is a closed two-sided ideal of L1(G,A,α)and L1(G,A,α)/I is a pre-locally C∗-algebra with respect to the topol-ogy determined by the family of C∗-semi-norms{ r p}p∈S(A), r p(f+I)=inf{r p(f+h);h∈I}.Definition3.18The reduced crossed product of A by the actionα,denotedby A×α,r G,is the Hausdorffcompletion of L1(G,A,α),{r p}p∈S(A) (that is,A×α,r G is the completion of the pre-locally C∗-algebra L1(G,A,α)/I,{ r p}p∈S(A) ).Lemma3.19Let(G,A,α)be a locally C∗-dynamical system such thatαis an inverse limit action.Then=A p×α(p),r G(A×α,r G)pfor all p∈S(A),up to an isomorphism of C∗-algebras.Proof.Let p∈S(A).If f∈L1(G,A,α),then we have(f+I)+ker( r p) r p= r p(f+I)=inf{ ( ϕp×λ)(f+h) ;h∈I}=inf{ ( ϕp×λ)(f) ;h∈I}=r p(f)= f+ker(r p).r pis isomorphic to the com-From this relation,we conclude that(A×α,r G)ppletion of L1(G,A,α)/ker(r p)with respect to the C∗-norm induced by r p. On the other hand,A p×α(p),r G is the completion of L1 G,A p,α(p) /I p, where I p={f∈L1 G,A p,α(p) /( ϕp,u×λ)(f)=0},with respect to theϕp,u×λ)(f) ≤ f 1.But the completionnorm · ′given by f+I p ′= (is isomorphic to theof L1(G,A,α)/ker(r p)with respect to the norm · rp15completion of L1 G,A,α(p) /I p with respect to the norm · ′,since=r p(f)= ( ϕp×λ)(f)f+ker(r p)r pϕp,u×λ)(πp◦f) = πp(f)+I p ′= (for all f∈L1(G,A,α).Therefore the C∗-algebras(A×α,r G)and A p×α(p),rpG are isomorphic.α⊗id B ◦α=Two actionsα1andα2of C∗K on the locally C∗-algebras A1respectively A2are said to be equivalent if there is an isomorphism of locally C∗-algebras Φ:A1→A2such thatα2◦Φ=2for all t in U0,and sinceαis a continuous∗-morphism,there is a neighbor-hood V0of a0such thatε α(a)−α(a0) p=sup{p(α(a)(t)−α(a0)(t));t∈G}<<ε217for all(t,a)∈U0×V0and the proposition is proved.β(δ)t for each t in G.By Proposition5.1.5of[13],for eachδ∈∆there is an actionα(δ)of C∗K a G on Aδsuch thatΣ(α(δ))=β(δ).It is not difficult to verify that{α(δ)}δ∈∆is an inverse system of injective S-morphisms ofα(δ).Thenαis an injective S-morphism of locally C∗-algebras.Letα=lim←δ∈∆C∗-algebras andα(δ)⊗id C0(G) ◦α(δ)=lim←δ∈∆ id A⊗σC0(G)◦d a G ◦α.Thereforeαis an inverse limit action of C∗K a G on A andΣ(α)=β.Thus we showed thatΣis bijective.5The Takai duality theoremLet G be a locally compact group and let A be a locally C∗-algebra. Lemma5.1Letαbe an inverse limit action of G on A.Then the re-duced crossed product of A by the actionαis isomorphic to the locally C∗-19subalgebra of M A⊗L L2(G) generated by{α(a) 1M(A)⊗λ(f) ;a∈A,f∈C c(G)},whereλis the left regular representation of L1(G).Proof.Let p∈S(A).By Remark5.2.1.1of[13],the mapΦp from the C∗-subalgebra of M A p⊗L L2(G) generated by{α(p)(a p) 1M(A p)⊗λ(f) ;a p∈A p,f∈C c(G)}to A p×α(p),r G,that appliesα(p)(a p)(1M(Ap)⊗λ(f))to f+I p, where f(t)=f(t)a p,t∈G,see the proof of Lemma3.19,is an isomorphism of C∗-algebras.Ifπ′pq,p,q∈S(A),p≥q are the connecting maps of the inverse system {M(A p⊗L L2(G) )}p∈S(A)and πpq,p,q∈S(A),p≥q are the connecting maps of the inverse system{A p×α(p),r G}p∈S(A),then we have(Φq◦π′pq)(α(p)(a p)(1M(Ap)⊗λ(f)))=Φq(α(q)(πpq(a p))(1M(Aq)⊗λ(f)))=πpq(a p)⊗f+I p= πpq(a p⊗f)+I p= πpq(a p⊗f+I p)=( πpq◦Φp)(α(p)(a p)(1M(Ap)⊗λ(f)))for all a p in A p,for all f in C c(G)and for all p,q∈S(A)with p≥q. Therefore{Φp}p∈S(A)is an inverse system of isomorphisms of C∗-algebras and the lemma is proved.β(δ)be an inverse limit coaction of G on A such Remark5.3Letβ=lim←δ∈∆that the connecting maps of the inverse system{Aδ}δ∈∆are all surjective. Then,by Theorem3.14of[10]M(A⊗L L2(G) )=lim←δ∈∆M(Aδ⊗L L2(G) )up to an isomorphism of locally C∗-algebras,and by Lemma III3.2of[7],A×βG=limAδ×β(δ)G←δ∈∆up to an isomorphism of locally C∗-algebras.Remark5.4Let G be a commutative locally compact group.Exactly as in the proof of Proposition5.1.6of[13]we show that ifβis an inverse limit coaction of G on A,thenβ′=(id A⊗ad F)◦β,where F is the Fourier-Plancherel isomorphism from L2(G)onto L2( G),is an inverse limit action of G on A and conversely,ifαis an inverse limit action of G on A then α′=(id A⊗ad F∗)◦αis an inverse limit coaction of G on A.Therefore an inverse limit coaction of G can be identified with an inverse limit action of G and id A⊗ad F is an isomorphism between A×βG and A×β′,r G.The following proposition is a generalization of Theorem5.2.6of[13]for inverse limit actions of a locally compact group on a locally C∗-algebra. Proposition5.5Let A be a locally C∗-algebra and let G be a locally com-pact group.(1).Ifαis an inverse limit action of G on A,then there is an inverse limit coaction αof G on A×α,r G,called the dual coaction associated toα,21such thatα(α(a)(1M(A)⊗λ(f)))=(α(a)⊗1G)(1M(A)⊗d s G(λ(f)))(*) for all a in A and for all f in C c(G).β(δ)is an inverse limit coaction of G on A such that the(2).Ifβ=lim←δ∈∆connecting maps of the inverse system{Aδ}δ∈∆are all surjective,then there is an inverse limit action βof G on A×βG,called the dual action associated toβ,such thatβ(β(a)(1M(A)⊗f))=(β(a)⊗1G)(1M(A)⊗(⊗j a G)d a G(f))id C0(G)for all aδin Aδand for all f in C c(G).Using this relation and Remark5.3 it is not difficult to check that{ β(δ)}δ∈∆is an inverse system of injective22S-morphisms.Let β=lim←δ∈∆ β(δ).Then βis a continuous action of G on A×βG and moreover,it verifies the condition(∗∗).)◦(α⊗adρ)Π⊗id C0(G)whereρis the right regular representation of L1(G).Proof.By Proposition3.2of[10],A p⊗K(L2(G))A⊗K(L2(G))=lim←p∈S(A)up to an isomorphism of locally C∗-algebrasSinceαis an inverse limit action,according to the proof of Proposition 5.5(1),α=lim←p∈S(A) α(p)where α(p)is the dual coaction associated toα(p)for each p∈S(A).Then, since the connecting maps of the inverse system{A p×α(p),r G}p∈S(A)are all surjective,by Proposition5.5(2),α=lim←p∈S(A) α(p)and by Remark5.3,(A×α,r G)× αG=lim←p∈S(A) A p×α(p),r G × α(p)G23up to an isomorphism of locally C∗-algebras.Let p∈S(A).According to Theorem5.2of[13],there is an isomorphism Π(p)from A p⊗K(L2(G))onto A p×α(p),r G × α(p)G such thatα(p)◦Π(p)=(Π⊗id C(G))◦(α⊗adρ)and the theorem is proved.from A⊗K(L2(G))onto(A×α,r G)× αG such thatα◦Π=([8]G.K.Pedersen,C∗-algebras and their automorphism groups,AcademicPress,London,New-York,San Francisco,1979.[9]N.C.Phillips,Representable K-theory forσ-C∗-algebras,K-Theory,3(1989),5,441-478.[10]N.C.Phillips,Inverse limits of C∗-algebras,J.Operator Theory,19(1988),159-195.[11]I.Raeburn,On crossed products and Takai duality,Proc.EdinburghMath.Soc.,31(1988),321-330.[12]H.Takai,On duality for crossed products of C∗-algebras,J.Func.Anal-ysis,19(1975),25-39.[13]J.M.Vallin,C∗-alg`e bres de Hopf et C∗-alg`e bres de Kac,Proc.LondonMath.Soc.(3),50(1985),131-174.Department of Mathematics,Faculty of Chemistry,Univer-sity of Bucharest,Bd.Regina Elisabeta nr.4-12,Bucharest, Romaniae-mail address:mjoita@fmi.unibuc.ro26。