Group actions on pro-$C^{*}$-algebras and their pro-$C^{*}$-crossed products

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by  Maria Joiţa

GROUP ACTIONS ON PRO-C∗-ALGEBRAS AND THEIR
PRO-C∗-CROSSED PRODUCTS
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1 MARIAJOIT¸A
0
2
Abstract. Inthispaper,wedefinethenotionsoffullpro-C∗-crossedproduct,
ct respectively reduced pro-C∗-crossedproduct, ofapro-C∗-algebraA[τΓ]bya
O strong bounded action α of a locally compact group G and investigate some
oftheirproperties.

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]
A 1. Introduction
O
Given a C∗-algebra A and a continuous action α of a locally compact group G
h. on A, we can construct a new C∗-algebra, called the crossed product of A by α,
t usuallydenotedbyG× A,andwhichcontains,insomesubtlesense,AandG. The
a α
m originof this constructiongoes back to Murray and von Neumann and their group
measure space construction by which they associated a von Neumann algebra to a
[
countablegroupactingonameasurespace. Theanalogofthisconstructionforthe

2 case of C∗-algebras is due to Gelfand with co-authors Naimark and Fomin. There
v is a vast literature on crossed products of C∗-algebras (see, for example, [W]), but
3
the corresponding theory in the context of non-normed topological ∗-algebras has
0
9 still a long way to go.
2 Crossed product of pro-C∗-algebras by inverse limit actions of locally compact
. groupswereconsideredbyPhillips[P2]andJoi¸ta[J2,J3,J4]. Ifαisaninverselimit
1
0 action of a locally compact group G on a pro-C∗-algebra A[τΓ] whose topology is
4 given by the family of C∗-seminorms Γ = {p } , then the covariance algebra
λ λ∈Λ
1 L1(G,α,A[τ ]) has a structure of locally m-convex ∗-algebra with topology given
Γ
:
v by the family of submultiplicative seminorms {Npλ}λ∈Λ, where
i
X N (f)= p (f(g))dg
pλ R λ
r G
a and the full crossed product of A[τ ] by α is defined as the enveloping pro-C∗-
Γ
algebraof L1(G,α,A[τ ]). In particular,for a giveninverselimit automorphismα
Γ
ofapro-C∗-algebraA[τ ],wecanassociatetothepair(A[τ ],α)apro-C∗-algebra
Γ Γ
bythecrossedproductconstruction,butifαisnotaninverselimitautomorphism,
this constructionis notpossible because the covariancealgebrahas nota structure
of locally m-convex ∗-algebra (N is not a submultiplicative ∗-seminorm). On
pλ
the other hand, a transformation group (X,G), with X a countably compactly
generated Hausdorff topological space (X is a direct limit of a countable family
of compact spaces {K } ), induces an action α of G on the pro-C∗-algebra C(X),
n n
2010 Mathematics Subject Classification. Primary46L05;Secondary 46H99.
Key words and phrases. pro-C∗-algebras;crossedproducts;covariantrepresentations .
Supported by a grant of the Romanian National Authority for Scientific Research, CNCS -
UEFISCDI,projectnumberPN-II-ID-PCE-2012-4-0201.
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whichisnotingeneralaninverselimitactionandso,wecannotassociateto(X,G)
a pro-C∗-algebra by the above construction.
Itis wellknownthatthe crossedproductofC∗-algebrasis auniversalobjectfor
nondegenerate covariant representations (see, for example, [R]). In [J3], we show
that the crossed product of pro-C∗-algebras by inverse limit actions has also the
universalproperty with respect to the nondegeneratecovariantrepresentations. In
thispaper,wedefinethefullcrossedproductofapro-C∗-algebraA[τ ]byanaction
Γ
α of a locally compact group G as a universal object for nondegenerate covariant
representationsandweshowthatthefullcrossedproductofpro-C∗-algebrasexists
for strong bounded actions. Strong boundless of the action α is essential to prove
the existence of a covariant representation. Unfortunately, if the action α of G
on A[τ ] is strongly bounded, then there is another family of C∗-seminorms on
Γ
A[τ ]which induces the same topologyon A, andα is aninverselimit actionwith
Γ
respect to this family of C∗-seminorms.
The organizationofthis paper is as follows. After preliminariesin Section 2, we
introduce the notion of strong bounded action of a locally compact group G on a
pro-C∗-algebraA[τ ]andpresentsomeexamplesinSection3. Weshowthatthere
Γ
isanondegeneratecovariantrepresentationofapro-C∗-algebraA[τ ]withrespect
Γ
to a strong bounded action α of a locally compact group G on A[τ ] in Section
Γ
4. In Section 5, we show that if α is strongly bounded, then there exists the full
pro-C∗-crossed product of A[τ ] by α. Also, we show that the full pro-C∗-crossed
Γ
productisinvariantundertheconjugacyoftheactions,andthefullpro-C∗-crossed
product of the maximal tensor product A[τΓ]⊗maxB[τΓ′] by the action α⊗id is
isomorphic to the maximal tensor product of the full pro-C∗-crossed product of
A[τΓ] by α and B[τΓ′]. In Section 6, we define the notion of reduced pro-C∗-
crossedproductofapro-C∗-algebraA[τ ]byastrongboundedactionα,andshow
Γ
that the reduced pro-C∗-crossed product is invariant under the conjugacy of the
actions. Also, we show that the reduced pro-C∗-crossed product of the minimal
tensorproductA[τΓ]⊗minB[τΓ′]by the actionα⊗idis isomorphicto the minimal
tensor product of the reduced pro-C∗-crossed product of A[τΓ] by α and B[τΓ′].
It is known that the full crossed product of a C∗-algebra by an action α of an
amenablelocallycompactgroupGcoincideswiththereducedcrossedproduct. We
show that this result is still valid for pro-C∗-crossed products.
2. Preliminaries
A seminorm p on a topological ∗-algebra A satisfies the C∗-condition (or is a
C∗-seminorm) if p(a∗a) = p(a)2 for all a ∈ A. It is known that such a seminorm
must be submultiplicative (p(ab) ≤ p(a)p(b) for all a,b ∈ A) and ∗-preserving
(p(a∗)=p(a) for all a∈A).
Apro-C∗-algebraisacompleteHausdorfftopological∗-algebraAwhosetopology
is given by a directed family of C∗-seminorms {p ;λ ∈ Λ}. Other terms used for
λ
pro-C∗-algebrasare: locallyC∗-algebras(A.Inoue,M.Fragoulopoulou,A.Mallios,
etc.), LMC∗-algebras (G. Lassner, K. Schmu¨dgen), b∗-algebras (C. Apostol).
Let A[τ ] be a pro-C∗-algebra with topology given by Γ = {p ;λ ∈ Λ} and let
Γ λ
B[τΓ′]be a pro-C∗-algebrawith topologygivenby Γ′ ={qδ;δ ∈∆}. Acontinuous
∗-morphismϕ:A[τΓ]→B[τΓ′](thatis,ϕislinear,ϕ(ab)=ϕ(a)ϕ(b)andϕ(a∗)=
ϕ(a)∗ for all a,b ∈ A and for each q ∈ Γ′, there is p ∈ Γ such that q (ϕ(a)) ≤
δ λ δ
p (a) for all a∈A) is called a pro-C∗-morphism. The pro-C∗-algebras A[τ ] and
λ Γ
GROUP ACTIONS AND THEIR PRO-C∗-CROSSED PRODUCTS 3
B[τΓ′] are isomorphic if there is a pro-C∗-isomorphism ϕ : A[τΓ] → B[τΓ′] (that
is, ϕ is invertible, ϕ and ϕ−1 are pro-C∗-morphisms).
If {A ;π } is an inverse system of C∗-algebras, then limA with
λ λµ λ≥µ,λ,µ∈Λ λ
←λ
topology given by the family of C∗-seminorms {p } , with p (a ) =
λ λ∈Λ λ(cid:16) µ µ∈Λ(cid:17)
ka k for all λ∈Λ, is a pro-C∗-algebra.
λ Aλ
Let A[τ ] be a pro-C∗-algebra with topology given by Γ = {p ;λ ∈ Λ}. For
Γ λ
λ ∈ Λ, kerp is a closed ∗-bilateral ideal and A = A/kerp is a C∗-algebra in
λ λ λ
the C∗-norm k·k induced by p (that is, kak = p (a), for all a ∈ A). The
pλ λ pλ λ
canonical map from A to A is denoted by πA, πA(a) = a+kerp for all a ∈ A.
λ λ λ λ
For λ,µ ∈ Λ with µ ≤ λ there is a surjective C∗-morphism πA : A → A such
λµ λ µ
that πA (a+kerp )=a+kerp , andthen {A ;πA } is aninversesystem of
λµ λ µ λ λµ λ,µ∈Λ
C∗-algebras. Moreover,pro-C∗-algebrasA[τ ] and limA are isomorphic (Arens-
Γ λ
←λ
Michael decomposition).
Let {(H ,h·,·i )} be a family of Hilbert spaces such that H ⊆ H and
λ λ λ∈Λ µ λ
h·,·i | = h·,·i for all λ,µ ∈ Λ with µ ≤ λ. H = limH with inductive limit
λ Hµ µ λ→ λ
topology is called a locally Hilbert space.
Let L(H) = {T : H → H;T = T| ∈ L(H ) and P T = T P for all
λ Hλ λ λµ λ λ λµ
λ,µ ∈ Λ with µ ≤ λ}, where P is the projection of H on H . Clearly, L(H) is
λµ λ µ
an algebra in an obvious way, and T →T∗ with T∗| =(T )∗ for all λ∈Λ is an
Hλ λ
involution.
For each λ ∈ Λ, the map p : L(H) → [0,∞) given by p (T) =
λ,L(H) λ,L(H)
kT| k is a C∗-seminormon L(H), and with topology given by the family of
Hλ L(Hλ)
C∗-seminorms {p } , L(H) becomes a pro-C∗-algebra.
λ,L(H) λ∈Λ
L(H) as a pro-C∗-algebra has an Arens-Michael decomposition, given by the
C∗-algebras L(H) = L(H)/kerp , λ ∈ Λ. Moreover, for each λ ∈ Λ, the map
λ λ
ϕ :L(H) →L(H ) givenby ϕ (T +kerp )=T| is anisometric ∗-morphism.
λ λ λ λ λ Hλ
The canonical maps from L(H) to L(H) ,λ ∈ Λ are denoted by πH,λ ∈ Λ, and
λ λ
πH(T)= T| . For a given pro-C∗-algebra A[τ ] there is a locally Hilbert space
λ Hλ Γ
H such that A[τ ] is isomorphic to a pro-C∗-subalgebra of L(H) (see [I, Theorem
Γ
5.1]).
A multiplier of A[τ ] is a pair (l,r) of linear maps l,r : A[τ ] → A[τ ] such
Γ Γ Γ
thatarerespectivelyleftandrightA-modulehomomorphismsandr(a)b=al(b)for
all a,b ∈ A. The set M(A[τ ]) of all multipliers of A[τ ] is a pro-C∗-algebra
Γ Γ
with multiplication given by (l ,r )(l ,r ) = (l l ,r r ), the involution given
1 1 2 2 1 2 2 1
by (l,r)∗ = (r∗,l∗), where r∗(a) = r(a∗)∗ and l∗(a) = l(a∗)∗ for all a ∈ A,
and the topology given by the family of C∗-seminorms {p } , where
λ,M(A[τΓ]) λ∈Λ
p (l,r) = sup{p (l(a));p (a) ≤ 1}. Moreover, for each p ∈ Γ, the
λ,M(A[τΓ]) λ λ λ
C∗-algebras (M(A[τ ])) and M(A ) are isomorphic. The strict topology on
Γ λ λ
M(A[τ ]) is given by the family of seminorms {p } , where p (l,r)=
Γ λ,a (λ,a)∈Λ×A λ,a
p (l(a))+p (r(a)), M(A[τ ])iscomplete withrespectto the stricttopologyand
λ λ Γ
A[τ ] is dense in M(A[τ ]) (see [P1] and [J1, Proposition 3.4]).
Γ Γ
A pro-C∗-morphism ϕ : A[τΓ] → M(B[τΓ′]) is nondegenerate if [ϕ(A)B] =
B[τΓ′],where[ϕ(A)B]denotestheclosedsubspaceofB[τΓ′]generatedby{ϕ(a)b;
a∈A,b∈B}. Anondegeneratepro-C∗-morphismϕ:A[τΓ]→M(B[τΓ′])extends
to a unique pro-C∗-morphism ϕ:M(A[τΓ])→M(B[τΓ′]).
4 MARIAJOIT¸A
3. Group actions on pro-C∗-algebras
Throughout this paper, A[τ ] is a pro-C∗-algebra with topology given by the
Γ
family of C∗-seminorms Γ={p } and G is a locally compact group.
λ λ∈Λ
Definition 3.1. (1) An action of G on A[τ ] is a group morphism α from G
Γ
to Aut(A[τ ]) such that the map t7→α (a) from G to A[τ ] is continuous
Γ t Γ
for each a∈A.
(2) An action α of G on A[τ ] is strongly bounded, if for each λ ∈ Λ there is
Γ
µ∈Λ such that
p (α (a))≤p (a)
λ t µ
for all t∈G and for all a∈A.
(3) An action α is an inverse limit action, if p (α (a))=p (a) for all a∈A,
λ t λ
for all t∈G and for all λ∈Λ.
Remark 3.2. (1) If α is an inverse limit action of G on A[τ ], then for each
Γ
λ ∈ Λ, there is an action αλ of G on A such that αλ◦πA = πA◦α for
λ t λ λ t
all t∈G, and then α =limαλ for all t∈G.
t t
↼λ
(2) Any inverse limit action of G on A[τ ] is strongly bounded.
Γ
(3) If A is a C∗-algebra, then any action of G on A is strongly bounded.
(4) IfGisacompact group, thenanyaction ofGon A[τ ]is stronglybounded.
Γ
Let X be a compactly countably generated Hausdorff topological space (that
is, X is a direct limit of a countable family {K } of compact spaces). The ∗-
n n
algebraC(X) of allcontinuous complex valued functions on X is a pro-C∗-algebra
with topology given by the family of C∗-seminorms {p } , where p (f) =
Kn n Kn
sup{|f(x)|;x∈K }.
n
Example 3.3. Let (G,X)be a transformation group (that is, there is a continuous
map (t,x) 7→t·x from G×X to X such that e·x=x and s·(t·x)=(st)·x for
all s,t ∈ G and for all x ∈ X) with X = limK a compactly countably generated
n
n→
Hausdorff topological space. Then there is an action α of G on the pro-C∗-algebra
C(X), given by
α (f)(x)=f t−1·x .
t
(cid:0) (cid:1)
If for any positive integer n, there is a positive integer m such that G·K ⊆K ,
n m
the action α is strongly bounded, since for each n, there is m such that
p (α (f))=sup{ f t−1·x ;x∈K }≤sup{|f(y)|;y ∈K }=p (f)
Kn t (cid:12) (cid:0) (cid:1)(cid:12) n m Km
(cid:12) (cid:12)
for all f ∈C(X) and for all t∈G. If G·K =K for all n, then α is an inverse
n n
limit action. Take, for instance, R=lim[−n,n]. Suppose that Z actions on R by
2
n→
0·x = x and 1·x = 2−x for all x ∈ R. Then (Z ,R) is a transformation group
2
bsuch that for ebach positive integer n, Z2·[−n,n]⊆[−n−2,n+2].
Example 3.4. Let X = limK be a compactly countably generated Hausdorff
n
n→
topological space and h:X →X a homeomorphism with the property that for each
positive integer n, there is a positive integer m such that hk(K ) ⊆ K for all
n m
integers k. Then the map n 7→ α from Z to Aut(C(X)), where α (f) = f ◦hn,
n n
is a strong bounded action of Z to C(X). If h(K ) = K for all n, then α is
n n
an inverse limit action. Take, for instance, R=lim[−n,n], the map h : R→R
n→
defined by h(x)=1−x is a homeomarphism such that for each positive integer n,
hk([−n,n])⊆[−n−1,n+1] for all integers k.
GROUP ACTIONS AND THEIR PRO-C∗-CROSSED PRODUCTS 5
Example 3.5. The ∗-algebra C[0,1] equipped with the topology ’cc’ of unifom con-
vergence on countable compact subsets is a pro-C∗-algebra denoted by C [0,1] (see,
cc
for example, [F,p. 104]). The action of Z2 on Ccc[0,1] given by αb0 =idCcc[0,1] and
αb1(f)(x)=f(1−x) for all f ∈Ccc[0,1] and for all x∈[0,1] is strongly bounded.
Remark 3.6. (1) Let α be a strong bounded action of G on A[τ ]. Then, for
Γ
each λ∈Λ, the map pλ :A→[0,∞) given by
pλ(a)=sup{p (α (a));t∈G}
λ t
is a continuous C∗-seminorm on A[τ ]. Let ΓG = {pλ} . Since, for
Γ λ∈Λ
each λ∈Λ, there is µ∈Λ such that
p ≤pλ ≤p ,
λ µ
ΓG defines on A a structure of pro-C∗-algebra, and moreover, the pro-C∗-
algebras A[τ ] and A τG are isomorphic.
Γ (cid:2) Γ(cid:3)
(2) If the action α of G on A[τ ] is strongly bounded, then α is an inverse
Γ
limit action of G on A τG .
(cid:2) Γ(cid:3)
4. Covariant representations
Definition 4.1. A pro-C∗-dynamical system is a triple (G,α,A[τ ]), where G is a
Γ
locally compact group, A[τ ] is a pro-C∗-algebra and α is an action of G on A[τ ].
Γ Γ
A representation ofapro-C∗-algebraA[τ ]onaHilbertspaceHisacontinuous
Γ
∗-morphismϕ:A[τ ]→L(H). A representation(ϕ,H) of A[τ ] is nondegenerate
Γ Γ
if [ϕ(A)H]=H.
Definition4.2. Acovariant representationof(G,α,A[τ ])onaHilbertspaceHis
Γ
a triple (ϕ,u,H)consisting of a representation (ϕ,H) of A[τ ] on H and a unitary
Γ
∗-representation (u,H) of G on H such that
ϕ(α (a))=u ϕ(a)u∗
t t t
for all a ∈ A and for all t ∈ G. A covariant representation (ϕ,u,H) is nondegen-
erate if (ϕ,H) is nondegenerate.
Two representations (ϕ,u,H) and (ψ,v,K) of (G,α,A[τ ]) are unitarily equiv-
Γ
alent if there is a unitary operator U : H → K such that Uϕ(a) = ψ(a)U for all
a∈A and Uu =v U for all t∈G.
t t
For each p ∈Γ, we denote by R (G,α,A[τ ]) the collection of all equivalence
λ λ Γ
classes of nondegenerate covariant representations (ϕ,u,H) of (G,α,A[τ ]) with
Γ
the property that kϕ(a)k≤p (a) for all a∈A. Clearly,
λ
R (G,α,A[τ ])=R(G,α,A[τ ]),
λ Γ Γ
S
λ
where R(G,α,A[τ ])denotes the collectionof allequivalence classesofnondegen-
Γ
erate covariant representations of (G,α,A[τ ]).
Γ
Remark 4.3. If α is an inverse limit action, then the map
(ϕ ,u,H)→ ϕ ◦πA,u,H
λ (cid:0) λ λ (cid:1)
is a bijection between R G,αλ,A and R (G,α,A[τ ]) (see, [J2]).
λ λ Γ
(cid:0) (cid:1)
6 MARIAJOIT¸A
By [J2], if α is an inverse limit action, then R(G,α,A[τ ]) is non empty.
Γ
From this result and Remark 3.6, we conclude that if α is strongly bounded, then
R(G,α,A[τ ]) is non empty too. In the following proposition we give another
Γ
proof for this result.
Proposition 4.4. Let (G,α,A[τ ]) be a pro-C∗-dynamical system such that α is
Γ
strongly bounded. Then there is a covariant representation of (G,α,A[τ ]).
Γ
Proof. Let (ϕ,H) be a representation of A[τ ]. Then there is λ ∈ Λ such that
Γ
kϕ(a)k≤p (a) for all a∈A. Let a∈A and ξ ∈L2(G,H). Since, there is p ∈Γ
λ µ
such that
kϕ(αs−1(a))(ξ(s))k2ds ≤ kϕ(αs−1(a))k2kξ(s)k2ds
R R
G G
≤ pλ(αs−1(a))2kξ(s)k2ds≤pµ(a)2kξk2,
R
G
the map s 7→ ϕ(αs−1(a))(ξ(s)) defines an element in L2(G,H). Therefore, there
is ϕ(a)∈L(L2(G,H)) such that
e
ϕ(a)(ξ)(s)=ϕ(αs−1(a))(ξ(s)).
e
In this way, we obtain a map ϕ : A → L(L2(G,H)). Moreover, ϕ is a continuous
∗-morphism, and then ϕ,L2(G,H) is a representation of A[τ ].
e Γe
(cid:0) (cid:1)
Let λH,L2(G,H) bee the unitary ∗-representationofG onL2(G,H)givenby
(cid:16) G (cid:17)
λH (ξ)(s)=ξ t−1s . It is easy to verify that ϕ,λH,L2(G,H) is a covariant
(cid:16) G(cid:17)t (cid:0) (cid:1) (cid:16) G (cid:17)
representationof (G,α,A[τ ]). e (cid:3)
Γ
Remark 4.5. Let (G,α,A[τ ]) be a pro-C∗-dynamical system. Suppose that α is
Γ
strongly bounded. Then, for each representation (ϕ,H) of A[τ ], kerϕ ⊆ kerϕ.
Γ
Indeed, if ϕ(a) = 0, then ϕ(α (a))(ξ(s)) = 0 for all s ∈ G and for all ξ ∈
s e
L2(G,H), whence ϕ(a)(ξ(e))=0 for all ξ ∈L2(G,H) and so ϕ(a)=0.
e
5. The full pro-C∗-crossed product
Throughout this paper, B[τΓ′] is a pro-C∗-algebra with topology given by the
family of C∗-seminorms Γ′ = {q } . Let (G,α,A[τ ]) be a pro-C∗-dynamical
δ δ∈∆ Γ
system.
Definition 5.1. A covariant pro-C∗-morphism from (G,α,A[τ ]) to a pro-C∗-
Γ
algebra B[τΓ′] is a pair (ϕ,u) consisting of a pro-C∗-morphism ϕ : A[τΓ] →
M(B[τΓ′]) and a strict continuous group morphism u : G → U(M(B[τΓ′])) such
that
ϕ(α (a))=u ϕ(a)u∗
t t t
for all t ∈ G and for all a ∈ A. A covariant pro-C∗-morphism (ϕ,u) from
(G,α,A[τΓ]) to B[τΓ′] is nondegenerate if [ϕ(A)B]=B[τΓ′].
Theorem 5.2. Let (G,α,A[τ ]) be a pro-C∗-dynamical system. If α is strongly
Γ
bounded, then there is a locally Hilbert space H and a covariant pro-C∗-morphism
(i ,i ) from (G,α,A[τ ]) to L(H). Moreover, i and i are injective.
A G Γ A G
GROUP ACTIONS AND THEIR PRO-C∗-CROSSED PRODUCTS 7
Proof. Letλ∈Λ. ByProposition4.4,R (G,α,A[τ ])isnonempty. Let ϕλ,uλ,H
λ Γ λ
(cid:0) (cid:1)
be the direct sum of all equivalence classes of nondegenerate covariant reprsenta-
tions(ϕ,u,H )of(G,α,A[τ ]),(ϕ,u,H )∈R (G,α,A[τ ]). Then ϕλ,uλ,H
ϕ,u Γ ϕ,u λ Γ λ
(cid:0) (cid:1)
is a nondegenerate covariant representation of (G,α,A[τ ]) such that ϕλ(a) ≤
Γ (cid:13) (cid:13)
pλ(a) for all a∈A. (cid:13) (cid:13)
Let H =⊕ H . Then H=limH is a locally Hilbert space. For a∈A, the
λ µ≤λ µ λ
λ→
map iλ (a):H →H defined by
A λ λ
iλ (a) ⊕ ξ =⊕ ϕµ(a)ξ
A (cid:0) µ≤λ µ(cid:1) µ≤λ µ
is an element in L(H ) and iλ (a) ≤ p (a). Moreover, iλ (a∗) = iλ (a)∗ and
λ (cid:13) A (cid:13) λ A A
iλ (ab) = iλ (a)iλ (b) for all(cid:13)a,b ∈(cid:13)A. Clearly, iλ (a) is a direct system of
A A A (cid:0) A (cid:1)λ
bounded linear operators and i (a) = limiλ (a) is an element L(H) such that
A A
λ→
i (a∗)=i (a)∗ and i (ab)=i (a)i (b) for all a,b∈A. Moreover,
A A A A A
p (i (a))= iλ (a) ≤p (a)
λ,L(H) A (cid:13) A (cid:13) λ
for all a∈A and for all λ∈Λ. Therefore,(cid:13)i is a(cid:13)pro-C∗-morphism.
A
For t∈G, the map iλ (t):H →H defined by
G λ λ
iλ (t) ⊕ ξ =⊕ uµ(t)ξ
G (cid:0) µ≤λ µ(cid:1) µ≤λ µ
is a unitary element in L(H ). Moreover, the map t 7→ iλ (t) is a unitary ∗-
λ G
representationofGonH . Clearly, iλ (t) isadirectsystemofunitaryoperators,
λ (cid:0) G (cid:1)λ
and then i (t) = limiλ (t) is a unitary element L(H). Moreover, t 7→ i (t) is a
G G G
λ→
groupmorphismfromGtothe groupofunitaryoperatorsonH,andsinceforeach
ξ ∈ H, the map t 7→ i (t)ξ from G to H is continuous, t 7→ i (t) is a unitary
G G
∗-representationof G on H. We have
i (α (a)) ⊕ ξ = iλ (α (a)) ⊕ ξ =⊕ ϕµ(α (a)) ξ
A t (cid:0) µ≤λ µ(cid:1) A t (cid:0) µ≤λ µ(cid:1) µ≤λ t (cid:0) µ(cid:1)
= ⊕ uµ(t)ϕµ(a)uµ(t)∗ ξ
µ≤λ (cid:0) µ(cid:1)
= i (t)i (a)i (t)∗ ⊕ ξ
G A G (cid:0) µ≤λ µ(cid:1)
for all a∈A, for all t∈G and for all ⊕ ξ ∈H , λ∈Λ, and so
µ≤λ µ λ
i (α (a))=i (t)i (a)i (t)∗
A t G A G
for all a∈A and for all t∈G.
Suppose that i (a) = 0. Then iλ (a) = 0 for all λ ∈ Λ and so ϕ(a) = 0
A A
for all nondegenerate covariant representation (ϕ,u,H ) of (G,α,A[τ ]). By
ϕ,u Γ
Proposition4.4andRemark4.5,ψ(a)=0forallrepresentationsψofA. Therefore,
p (a)=0 for all λ∈Λ, and then a=0.
λ
Suppose that i (t)=id . Then iλ (t)=id for all λ∈Λ, and so u(t)=id
G H G Hλ Hϕ,u
forallnondegeneratecovariantrepresentation(ϕ,u,H )of(G,α,A[τ ]),whence
ϕ,u Γ
we deduce that t=e. (cid:3)
Proposition 5.3. Let (G,α,A[τ ]) be a pro-C∗-dynamical system. Then the fol-
Γ
lowing statements are equivalent.
(1) α is an inverse limit action.
(2) ThereisalocallyHilbertspaceHandacovariantpro-C∗-morphism(i ,i )
A G
from (G,α,A[τ ]) to L(H) such that p (i (a))=p (a) for all λ∈Λ
Γ λ,L(H) A λ
and a∈A.
8 MARIAJOIT¸A
Proof. (1)⇒(2) See [J3, Proposition 3.1] and [I, Theorem 5.1].
(2)⇒(1) From
∗
i (α (a))=i (t)i (a)i (t)
A t G A G
for all t ∈ G and for all a ∈ A, and taking into account that i (t) is a unitary
G
element in L(H) for all t∈G, we deduce that
∗
p (α (a)) = p (i (α (a)))=p i (t)i (a)i (t)
λ t λ,L(H) A t λ,L(H)(cid:0) G A G (cid:1)
= p (i (a))=p (a)
λ,L(H) A λ
for all t ∈ G, for all a ∈ A and for all t ∈ G. Therefore, α is an inverse limit
action. (cid:3)
IfuisastrictcontinuousgroupmorphismfromGtoU(M(B[τΓ′])),thenthere
is a ∗-morphism u : Cc(G) → M(B[τΓ′]) given by u(f) = f(s)usds, where ds
R
G
denotes the Haar measure on G (sse [J2]).
Definition5.4. Let(G,α,A[τ ])beapro-C∗-dynamicalsystem. Apro-C∗-algebra,
Γ
denoted by G× A[τ ], together with a covariant pro-C∗-morphism (ι ,ι ) from
α Γ A G
(G,α,A[τ ]) to G× A[τ ] which verifies the following:
Γ α Γ
(1) for each nondegenerate covariant representation (ϕ,u,H) of (G,α,A[τ ]),
Γ
there is a nondegenerate representation (Φ,H) of G× A[τ ] such that
α Γ
Φ◦ι =ϕ and Φ◦ι =u;
A G
(2) span{ι (a)ι (f);a∈A,f ∈C (G)}=G× A[τ ];
A G c α Γ
is called the full pro-C∗-crossed product of A[τ ] by α.
Γ
Remark 5.5. The covariant morphism (ι ,ι ) from the above definition is non-
A G
degenerate.
Proposition 5.6. Let (G,α,A[τ ]) be a pro-C∗-dynamical system such that there
Γ
is a full pro-C∗-crossed product of A[τ ] by α and (ϕ,u) a nondegenerate covariant
Γ
morphism from (G,α,A[τΓ]) to a pro-C∗-algebra B[τΓ′]. Then there is a unique
nondegenerate pro-C∗-morphism ϕ×u:G×αA[τΓ]→M(B[τΓ′]) such that
ϕ×u◦ι =ϕ and ϕ×u◦ι =u.
A G
Moreover, the map (ϕ,u) → ϕ×u is a bijection between nondegenerate covariant
morphisms of (G,α,A[τ ]) onto nondegenerate morphisms of G× A[τ ].
Γ α Γ
Proof. Let q ∈ Γ′ and (ψ ,H) a faithful nondegenerate representation of B .
δ δ δ
Then, (ψ ◦πB ◦ϕ, ψ ◦πB ◦u,H) is a nondegenerate covariant representation
δ δ δ δ
of (G,α,A[τ ]), and by Definition 5.4, there is a nondegenerate representation
Γ
(φ ,H) of G× A[τ ] such that
δ α Γ
φ ◦ι =ψ ◦πB◦ϕ and φ ◦ι =ψ ◦πB ◦u.
δ A δ δ δ G δ δ
LetΦ =ψ−1◦φ . ThenΦ isanondegeneratepro-C∗-morphismfromG× A[τ ]
δ δ δ δ α Γ
to M(B ). Moreover, for q ,q ∈ Γ′ with q ≥ q , we have πB ◦ Φ =
δ δ1 δ2 δ1 δ2 δ1δ2 δ1
Φ . Therefore, there is a nondegenerate pro-C∗-morphism ϕ×u: G× A[τ ]→
δ2 α Γ
M(B[τΓ′]) such that
πB ◦ϕ×u=Φ
δ δ
forallq ∈Γ′. Moreover,ϕ×u◦ι =ϕandϕ×u◦ι =u,andsince{ι (a)ι (f),
δ A G A G
a∈A,f ∈C (G)}generatesG× A[τ ],ϕ×uisuniquewiththeaboveproperties.
c α Γ
GROUP ACTIONS AND THEIR PRO-C∗-CROSSED PRODUCTS 9
Let Φ:G×αA[τΓ]→M( B[τΓ′]) be a nondegeneratepro-C∗-morphism. Then
ϕ = Φ◦ιA is a nondegenerate pro-C∗-morphism from A[τΓ] to M(B[τΓ′]) and
u= Φ◦ιG is a strict continuous morphism from G to U(M(B[τΓ′])), since ιG is a
strictcontinuousmorphismfromGtoM(G× A[τ ])andΦisstronglycontinuous
α Γ
on the bounded subsets of M(G× A[τ ]). Moreover, (ϕ,u) is a nondegenerate
α Γ
covariant morphism from A[τΓ] to B[τΓ′], and ϕ×u = Φ. If (ψ,v) is another
nondegenerate covariant morphism from A[τΓ] to B[τΓ′] such that ψ ×v = Φ,
then ψ = Φ◦ι =ϕ and v = Φ◦ι =u. (cid:3)
A G
The following corollary provides uniqueness of the full pro-C∗-crossed product
by strong bounded actions.
Corollary 5.7. Let (G,α,A[τ ]) be a pro-C∗-dynamical system such that there is
Γ
a full pro-C∗-crossed product of A[τ ] by α. Then the full pro-C∗-crossed product
Γ
of A[τ ] by α is unique up to a pro-C∗-isomorphism.
Γ
Proof. Let B[τΓ′] be a pro-C∗-algebra and (jA,jG) a covariant pro-C∗-morphism
from (G,α,A[τΓ]) to B[τΓ′] which satisfy the relations (1)−(2) from Definition
5.4. Then,byProposition5.6,thereisanondegeneratepro-C∗-morphismΦ:G×
α
A[τΓ] → M(B[τΓ′]) such that Φ◦ιA = jA and Φ◦ιG = jG. Since {ιA(a)ιG(f),
a ∈ A, f ∈ C (G)} generates G× A[τ ] and {j (a)j (f), a ∈ A, f ∈ C (G)}
c α Γ A G c
generates B[τΓ′], Φ(G×αA[τΓ])⊆B.
Inthesameway,thereisapro-C∗-morphismΨ:B[τΓ′]→G×αA[τΓ]suchthat
Ψ◦j = ι and Ψ◦j = ι . From these facts and Definition 5.4(2), we deduce
A A G G
that Φ◦Ψ=id and Ψ◦Φ=id , and so Φ is a pro-C∗-isomorphism. (cid:3)
B G×αA[τΓ]
The followingpropositionrelatesthe nondegeneratecovariantrepresentationsof
a pro-C∗-dynamical system (G,α,A[τ ]) with the nondegenerate representations
Γ
of the full pro-C∗-crossedproduct of A[τ ] by α.
Γ
Proposition 5.8. Let (G,α,A[τ ]) be a pro-C∗-dynamical system such that there
Γ
isthefullpro-C∗-crossedproductofA[τ ]byα. Thereisabijectivecorrespondence
Γ
between nondegenerate covariant representations of (G,α,A[τ ]) and nondegener-
Γ
ate representations of G× A[τ ].
α Γ
Proof. Let (ϕ,u,H) be a nondegenerate covariant representation of (G,α,A[τ ]).
Γ
Then,by Definition5.4,thereis a representation(ϕ×u,H)suchthatϕ×u◦ι =
A
ϕ and ϕ×u◦ι = u. Moreover, by Definition 5.4(2), (ϕ×u,H) is unique, and
G
since ϕ is nondegenerate, it is nondegenerate too.
Let (Φ,H) be a nondegenerate representation of G× A[τ ]. Then (Φ ◦ι ,
α Γ A
Φ◦ι ,H) is a covariant representation of (G,α,A[τ ]), and moreover, Φ◦ι ×
G Γ A
(cid:0) (cid:1)
Φ◦ι = Φ. Since ι and Φ are nondegenerate, the net {Φ(ι (e ))} , where
G A A i i
(cid:0) (cid:1)
{e } is an approximate unit of A[τ ], converges strictly to id , and so Φ◦ι is
i i Γ H A
nondegenerate.
Suppose that there is another nondegenerate covariant representation (ϕ,u,H)
of (G,α,A[τ ]) such that ϕ×u = Φ. Then ϕ = ϕ×u◦ι = Φ◦ι and u =
Γ A A
ϕ×u◦ι =Φ◦ι . Therefore, the map (ϕ,u,H)7→(ϕ×u,H) is bijective. (cid:3)
G G
Theorem 5.9. Let (G,α,A[τ ]) be a pro-C∗-dynamical system such that α is
Γ
strongly bounded. Then, there is the full pro-C∗-crossed product of A[τ ] by α.
Γ
10 MARIAJOIT¸A
Proof. By Theorem 5.2, there is a locally Hilbert space H and a covariantpro-C∗-
morphism (i ,i ) from A[τ ] to L(H).
A G Γ
LetB =span{i (a)i (f);a∈A,f ∈C (G)}⊆L(H). ToshowthatBisapro-
A G c
C∗-algebra,wemustshowthatB isclosedundertakingadjointsandmultiplication.
For this, since B = lim πH(B) ([M, Chapter III, Theorem 3.1]), it is sufficient to
λ
←−λ
show that for each p ∈Γ, πH(i (f)i (a)) and πH(i (b)i (f)i (a)i (h)) are
λ λ G A λ A G A G
elementsintheclosureofπH(B)inL(H )foralla,b∈Aandforallf,h∈C (G).
λ λ c
The map s → πA(f(s)α (a)) from G to A defines an element in C (G,A ),
λ s λ c λ
and so there is a net {πA(a )⊗f } in A ⊗ C (G) with suppf , suppf ⊆K
λ j j j∈J λ alg c j
for some compact subset K, which converges uniformly to this map.
By [J4, Lemma 3.7],
πH(i (f)i (a)) = f(s)iλ(s)dsπH(i (a))= f(s)πH(i (s)i (a))ds
λ G A R λ A R λ G A
G G
= f(s)πH(i (α (a))i (s))ds
R λ A s G
G
= f(s)iλ(α (a))iλ(s)ds
R A s G
G
and
πH(i (a )i (f ))=πH(i (a )) f (s)iλ(s)ds= iλ(a )f (s)iλ(s)ds
λ A j G j λ A j R j G R A j j G
G G
for each j ∈J. Then, we have
πH(i (f)i (a))−πH(i (a )i (f ))
(cid:13) λ G A λ A j G j (cid:13)L(Hλ)
(cid:13) (cid:13)
≤ f(s)iλ(α (a))iλ(s)−iλ(a )f (s)iλ(s) ds
RG(cid:13)(cid:13) A s G A j j G (cid:13)(cid:13)L(Hλ)
≤ Msup{ f(s)iλ(α (a))iλ(s)−iλ(a )f (s)iλ(s) ,s∈K}
(cid:13) A s G A j j G (cid:13)L(Hλ)
(cid:13) (cid:13)
= Msup{ iλ(f(s)α (a)−f (s)a ) iλ(s) ,s∈K}
(cid:13) A s j j (cid:13)L(Hλ)(cid:13) G (cid:13)L(Hλ)
(cid:13) (cid:13) (cid:13) (cid:13)
≤ Msup{p (f(s)α (a)−f (s)a ),s∈K}
λ s j j
= Msup{ πA(f(s)α (a))−f (s)πA(a ) ,s∈K}
(cid:13) λ s j λ j (cid:13)Aλ
(cid:13) (cid:13)
for all j ∈J, where M = dg, and so πH(i (f)i (a))∈πH(B).
R λ G A λ
K
On the other hand,
πH(i (b)i (f)i (a)i (h))−πH(i (ba )i (f ∗h))
(cid:13) λ A G A G λ A j G j (cid:13)L(Hλ)
(cid:13) (cid:13)
= πH(i (b)i (f)i (a)i (h)−i (b)i (a )i (f )i (h))
(cid:13) λ A G A G A A j G j G (cid:13)L(Hλ)
(cid:13) (cid:13)
≤ πH(i (b))πH(i (f)i (a)−i (a )i (f ))πH(i (h))
(cid:13) λ A λ G A A j G j λ G (cid:13)L(Hλ)
(cid:13) (cid:13)
≤ πH(i (b)) πH(i (h)) πH(i (f)i (a))−πH(i (a )i (f ))
(cid:13) λ A (cid:13)L(Hλ)(cid:13) λ G (cid:13)L(Hλ)(cid:13) λ G A λ A j G j (cid:13)L(Hλ)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
whence, we deduce that πH(i (b)i (f)i (a)i (h)) ∈ πH(B). Thus, we showed
λ A G A G λ
that πH(i (f)i (a)),πH(i (b)i (f)i (a)i (h)) ∈ πH(B) for each λ ∈Λ, and
λ G A λ A G A G λ
so i (f)i (a), i (b)i (f)i (a)i (h)∈B. Therefore, B is a pro-C∗-algebra.
G A A G A G
In the same manner, we show that for each a ∈ A,i (a)i (b)i (f) ∈ B and
A A G
i (b)i (f)i (a)∈B for all b∈A and for all f ∈C (G), and so i (a)∈M(B).
A G A c A