- Email: [email protected]
A note on crossed products
Journal Pre-proof A note on crossed products Wojciech Chojnacki
PII: DOI: Reference:
S0723-0869(19)30045-3 https://doi.org/10.1016/j.exmath.2019.08.001 EXMATH 25374
To appear in:
Expositiones Mathematicae
Received date : 29 March 2019 Revised date : 30 August 2019 Accepted date : 30 August 2019 Please cite this article as: W. Chojnacki, A note on crossed products, Expositiones Mathematicae (2019), doi: https://doi.org/10.1016/j.exmath.2019.08.001. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2019 Elsevier GmbH. All rights reserved. ⃝
Journal Pre-proof
A note on crossed products Wojciech Chojnackia,b a School b Faculty
of Computer Science, The University of Adelaide, SA 5005, Australia of Mathematics and Natural Sciences, College of Sciences, Cardinal Stefan Wyszy´ nski University, ul. Dewajtis 5, 01-815 Warszawa, Poland
Abstract
f
We give an exposition of two fundamental results of the theory of crossed products. One of these states that every regular representation of a reduced crossed product is faithful whenever the underlying Hilbert space representation of the C ∗ -algebra that together with an automorphism group gives rise to the crossed product is faithful. The other result states that a full and a reduced crossed products coincide whenever their common underlying automorphism group is amenable.
roo
Keywords: C ∗ -dynamical system, crossed product, representation, amenable group 2010 MSC: Primary 47L65, Secondary 43A07, 46L55
rep
1. Introduction
Jo u
rna
lP
The theory of crossed products of C ∗ -algebras by automorphism groups is a deep and interesting area of the modern theory of operator algebras, with links to such disciplines as non-commutative geometry and mathematical physics. The mathematical significance of the theory is attested, among others, by the fact that a substantial portion of the classic treatise by Pedersen [7] and the entire recent monograph by Williams [12] are devoted to crossed products. For physicists the crossed product construction is important as it allows exhibition of various simple and primitive (in algebraic sense) C ∗ -algebras that can serve as algebras of quantum observables. With applications in mind, including specifically those in mathematical physics, crossed products are treated in numerous introductory lecture notes and tutorials. However, as a rule, rudimentary texts on the subject stop short of presenting the proofs of two fundamental results on crossed products, namely that (1) every regular representation of a reduced crossed product is faithful whenever the underlying Hilbert space representation of the C ∗ -algebra that together with an automorphism group gives rise to the crossed product is faithful; and that (2) a full and a reduced crossed products coincide whenever their common underlying automorphism group is amenable. This state of affairs is reflective of the fact that the relevant proofs, as presented, say, in [7] and [12], require combining together a number of disparate elements that are somewhat hard to trace and follow. The purpose of this note is to provide a Email address: [email protected], [email protected] (Wojciech Chojnacki)
1
Journal Pre-proof
self-contained exposition of the above-mentioned fundamental results. It is hoped that our presentation will serve, among others, to alleviate the difficulties that writers of introductory texts on crossed products typically encounter when they attempt to strike a balance between conciseness and completeness. 2. Preliminaries We first recall some concepts and facts that will be of relevance throughout the paper. A basic familiarity with locally compact groups and C ∗ -algebras will be assumed. 2.1. Amenable groups Let G be a locally compact group, and let mG denote a left-invariant Haar measure on the group. Let f be a function on G. The left translate of f by t ∈ G is the function λt f defined by λt f (s) = f (t−1 s) (s ∈ G). The right translate of f by t ∈ G is the function ρt f defined by
f
(s ∈ G).
roo
ρt f (s) = f (st)
Let L∞ (G) denote the Banach space of equivalence classes of essentially bounded Haarmeasurable complex-valued functions on G which agree locally almost everywhere. A linear functional m : L∞ (G) → C is called a mean on L∞ (G) if
rep
(i) m(f ) = m(f ), (ii) m(1) = 1,
(iii) f ≥ 0 implies m(f ) ≥ 0.
A mean is called left-invariant if
lP
(iv) m(λt (f )) = m(f ) for all t ∈ G and f ∈ L∞ (G).
rna
A locally group G is called amenable if there exists a left-invariant mean on L∞ (G). As it turns out, L∞ (G) has a left-invariant mean if and only if it has a right-invariant mean and this is equivalent to L∞ (G) having a two-sided invariant mean. It is also equivalent to L∞ (G) having a topologically invariant mean, that is, a mean m such that Z m(g ∗ f ) = g dmG m(f ) (g ∈ L1 (G), f ∈ L∞ (G)). G
Jo u
Here, as is usual, L1 (G) denotes the space of equivalence classes of Haar-integrable complex-valued functions on G, and g ∗ f stands for the convolution of g and f defined by Z g ∗ f (t) =
g(s)f (s−1 t) dmG (s)
G
(locally a. e. t ∈ G).
We recall that all locally compact Abelian groups are amenable and so are all compact, in particular finite, groups. Any closed subgroup of an amenable group is amenable. 2
Journal Pre-proof
Moreover, if a group has an amenable closed normal subgroup such that the quotient by this subgroup is amenable, then the group is amenable. Solvable groups are amenable. On the other hand, semisimple Lie groups are not amenable, unless they are compact. The free group group on two generators is not amenable. It is vital to realise that amenability is not a purely group-theoretical concept—it depends on the topology of the group considered. For example, SO(3) under the usual topology is compact, hence amenable, but SO(3) under the discrete topology is not amenable, as it contains the free group on two generators which fails to be amenable. Amenable locally compact groups can be characterised by a number of alternative conditions. Of interest to us here will be the so-called (P1 ) condition introduced by elements f of L1 (G) such that RReiter [10]. Let P (G) denote the set of non-negative 1 f dmG = 1. With the norm of a function f in L (G) being G Z kf k1 = |f | dmG , G
the statement of Reiter’s condition goes as follows:
(P1 ) there exists a net {gα }α∈A in P (G) such that for each compact set K ⊂ G.
t∈K
roo
α
f
lim max kgα − λt gα k1 = 0
lP
rep
A fundamental result is that a locally compact group G is amenable if and only if G has property (P1 ) (cf. [3, Theorem 3.2.1]). To get a feeling for nets involved in property (P1 ) we remark that in the case G = R Reiter’s condition holds with any sequence of the form {gTn }n∈N , where {Tn }n∈N is a sequence of positive numbers diverging to infinity and gTn = (2Tn )−1 1[−Tn ,Tn ] for each n ∈ N. Here, as usual, we denote by 1E the characteristic function of the set E. The underlying sequences {[−Tn , Tn ]}n∈N of subsets of R are examples of so-called Følner sequences. Generally, Følner nets can be defined for any amenable group G and used to form nets in P (G) satisfying condition (P1 ). For more information about amenable groups, see [3], [4, §17], [6], [9, Chapter 8], and [11].
rna
2.2. Representations of C ∗ -algebras Given a C ∗ -algebra A, let S+,≤1 (A) denote the set of all positive functionals ω on A such that kωk ≤ 1. As a subset of the dual A∗ of A, S+,≤1 (A) inherits the weak* topology from A∗ . We recall that a neighbourhood base for ω ∈ A∗ is given by {ω 0 ∈ A∗ | |ω 0 (Ak ) − ω(Ak )| < , Ak ∈ A, k = 1, . . . , K, > 0}
Jo u
as A1 , . . . , AK runs through all K-tuples in A, > 0, but otherwise arbitrary, and K = 1, 2, . . . . Let H be a Hilbert space and let L (H) be the algebra of all linear bounded operators in H. Let A be a closed ∗-subalgebra of L (H). We denote by N (A) the set of elements ω in S+,≤1 (A) of the form N X ω(A) = hAΘn , Θn i , where Θ1 , . . . , ΘN ∈ H and
PN
n=1
n=1
2
kΘn k ≤ 1. We have the following fundamental result: 3
Journal Pre-proof
Proposition 2.1. Let H be a Hilbert space and let A be a closed ∗-subalgebra of L (H). Then S+,≤1 (A) is the weak* closure of N (A). The conclusion means, of course, that if ω ∈ S+,≤1 (A), then for each finite set PN {A1 , . . . , AK } in A and each > 0, there exist Θ1 , . . . , ΘN ∈ H with n=1 kΘn k2 ≤ 1 such that N X hAk Θn , Θn i − ω(Ak ) < n=1
for each k = 1, . . . , K. The proof is a rather straightforward application of the bipolar theorem to N (A) considered as a subset of the dual space of the real Banach space Ah of all hermitian elements of A. For details, we refer to [2, Theorem 1.1] or [8, Lemma 8.1]. In what follows, we shall make use of the following variation on the above proposition.
n=1
for each k = 1, . . . , K.
roo
f
Proposition 2.2. Let π and σ be two representations of a C ∗ -algebra A on Hilbert spaces Hπ and Hσ , respectively, and suppose, in addition, that σ is faithful. Let Ω ∈ Hπ be such that kΩk = 1. Then, given A1 , . . . , AK ∈ A and > 0, there exist Θ1 , . . . , ΘN ∈ Hσ with PN 2 n=1 kΘn k ≤ 1 such that N X hσ(Ak )Θn , Θn i − hπ(Ak )Ω, Ωi <
rep
Proof. The mapping ωΩ : A 7→ hπ(A)Ω, Ωi is clearly a member of S+,≤1 (A). Being faithful, σ is isometric (cf. [5, Theorem 3.1.5]), and this implies first that the image of A by σ, which we denote by B, is a closed ∗-subalgebra of L (Hσ ), and second that σ is invertible with kσ −1 (B)k = kBk for each B ∈ B. Consequently, the mapping ω = ωΩ ◦ σ −1 , which satisfies
lP
ω(σ(A)) = hπ(A)Ω, Ωi
(A ∈ A),
is a member of S+,≤1 (B). By Proposition 2.1, ω is then also an element of the weak∗ closure of N (B), and this immediately yields the result.
rna
2.3. Dynamical systems and crossed products
Jo u
A C ∗ -algebraic system is a triple (A, G, α), where A is a C ∗ -algebra, G is a locally compact group, and α : t 7→ αt is a homomorphism from G into the group of ∗-automorphisms of A such that, for each A ∈ A, the function G 3 t 7→ αt (A) ∈ A is continuous. Let (A, G, α) be a C ∗ -algebraic system. A covariant representation of (A, G, α) is a triple (H, π, U ), where H is a Hilbert space, π is a representation of A on H, and U : t 7→ Ut is a strongly continuous unitary representation of G on H such that, for each A ∈ A and each t ∈ G, π(αt (A)) = Ut π(A)Ut∗ . (2.1) Here, for any S ∈ L (H), S ∗ denotes the adjoint operator of S. 4
Journal Pre-proof
Let ∆ denote the Haar modular function of G. We recall that Z Z ∆(t)−1 f (t−1 ) dmG (t) = f (t) dmG (t) G
(2.2)
G
for any (possibly vector-valued) Haar integrable function f on G. Let Cc (G, A) be the space of all continuous functions from G into A wih compact support. The space Cc (G, A) can be converted into a norm ∗-algebra by introducing a multiplication, involution, and norm according to the rules: Z (xy)(t) = x(s)αs (y(s−1 t)) dmG (s), G
x∗ (t) = ∆(t)−1 αt (x(t−1 ))∗ , Z kxk1 = kx(s)k dmG (s)
(x, y ∈ Cc (G, A), t ∈ G).
G
f
With any covariant representation (H, π, U ) of (A, G, α) there is associated a representation ρπ,U of Cc (G, A) on H defined by Z ρπ,U (x) = π(x(t))Ut dmG (t) (x ∈ Cc (G, A)),
roo
G
the integral being taken in the strong operator topology. The representation ρπ,U is called the integrated form of (H, π, U ). Exploiting the integrated forms of covariant representations, one can define a new norm on Cc (G, A), namely (x ∈ Cc (G, A)),
rep
kxku = sup kρπ,U (x)k
where the supremum is taken over all covariant representations (H, π, U ) of (A, G, α). It is not possible to take supremum over a class, but the right hand side of the above equation is a supremum of a subclass of a set, namely R, so it is a supremum of a subset of R. One easily sees that k · ku is a C ∗ -seminorm, but in fact k · ku is a C ∗ -norm, the so-called universal norm. Moreover,
lP
kxku ≤ kxk1
(x ∈ Cc (G, A)).
Jo u
rna
The completion of Cc (G, A) in k · ku is a C ∗ -algebra called the crossed product of A by G. It will be denoted by A oα G. For every covariant representation (H, π, U ) of (A, G, α), the representation ρπ,U can be uniquely extended by continuity to a representation of A oα G. This extension will be denoted by π o U . Let H be a Hilbert space with inner product h·, ·i. Let L2 (G, H) be the completion of Cc (G, H), where Cc (G, H) is the space of all continuous functions from G into H with compact support, in the norm derived from the inner product Z hξ, ηi = hξ(t), η(t)i dmG (t) (ξ, η ∈ Cc (G, H)). G
Suppose that π is a representation of A in H. Define a representation π ˜ of A in L2 (G, H) 2 and a unitary representation λ : t 7→ λt of G in L (G, H) by setting (˜ π (A)ξ)(s) = π(αs−1 (A))ξ(s), (λt ξ)(s) = ξ(t−1 s)
(A ∈ A, ξ ∈ L2 (G, H)), t ∈ G, a. e. s ∈ G). 5
Journal Pre-proof
The triple (L2 (G, H), π ˜ , λ) is a covariant representation of (A, G, α). The corresponding representations ρπ˜ ,λ and π ˜ o α are termed the regular representation of Cc (G, A) induced by π and the regular representation of A oα G induced by π, respectively. Alongside the universal norm, one can define another C ∗ -norm on Cc (G, A), namely kxkr = sup kρπ˜ ,λ (x)k
(x ∈ Cc (G, A)),
roo
f
where the supremum is taken over all regular representations of Cc (G, A) (which in turn correspond to all Hilbert space representations of A). This is termed the reduced norm. The completion of Cc (G, A) in k·kr is a C ∗ -algebra called the reduced crossed product of A by G. It will be denoted by Aoα,r G. If π is a representation of A, then the representation ρπ˜ ,λ of Cc (G, A) can be uniquely extended by continuity to a representation of A oα,r G. This extension will be denoted by π ˜ or λ. There exists a unique ∗-homomorphism h from Aoα G into Aoα,r G such that h(x) = x for every x ∈ Cc (G, A). The homomorphism is the unique extension by continuity of the identity mapping of Cc (G, A) ⊂ A oα G onto Cc (G, A) ⊂ A oα,r G which is continuous, given that kxkr ≤ kxku for each x ∈ Cc (G, A). The map h is surjective. Indeed, the image of A oα G by h is closed in A oα,r G (the image of any C ∗ -algebra via a ∗-homomorphism into a C ∗ -algebra is closed; cf. [5, Theorem 3.1.6]) and contains Cc (G, A) which is dense in A oα,r G, therefore this image is all of A oα,r G. If π is a representation of A, then π ˜ o λ = (˜ π or λ) ◦ h.
rep
This follows from the fact that (˜ π or λ) ◦ h is a representation of A oα G coinciding with ρπ˜ ,λ on Cc (G, A) and the fact that π ˜ o λ is the only extension by continuity of ρπ˜ ,λ to a representation of A oα G. For a comprehensive treatment of the theory of crossed products, see [7] and [12]. 3. Regular representations of reduced crossed products
lP
The starting point of our considerations is the following result.
rna
Theorem 3.1. Let (A, G, α) be a C ∗ -algebraic system. Let π and σ be Hilbert space representations of A and suppose, in addition, that σ is faithful. Then k˜ π or λ(x)k ≤ k˜ σ or λ(x)k for each x ∈ A oα,r G and k˜ π o λ(x)k ≤ k˜ σ o λ(x)k for each x ∈ A oα G.
Jo u
Proof. Since kρπ˜ ,λ (x)k ≤ kxkr and kρσ˜ ,λ (x)k ≤ kxkr for each x ∈ Cc (G, A), and since, by definition, A oα,r G is the completion of Cc (G, A) in k · kr , it is clear that to prove that k˜ π or λ(x)k ≤ k˜ σ or λ(x)k for each x ∈ A oα,r G, it suffices to show that kρπ˜ ,λ (x)k ≤ kρσ˜ ,λ (x)k for each x ∈ Cc (G, A). Replacing in this argument k · kr by k · ku we see that the estimate kρπ˜ ,λ (x)k ≤ kρσ˜ ,λ (x)k, x ∈ Cc (G, A), is also all that is needed to conclude that k˜ π o λ(x)k ≤ k˜ σ o λ(x)k for each x ∈ A oα G. Before continuing, let us recall that a representation τ of a C ∗ -algebra B on a Hilbert space H is trivial if τ (B) = 0 for each B ∈ B, and is non-degenerate if there is no non-zero ξ ∈ H such that τ (B)ξ = 0 for all B ∈ B. Let π = πtr ⊕πnd be the decomposition of π into the direct sum of a trivial representation and a non-degenerate representation of A (cf. [1, Proposition 2.2.6]). Then, as is easily seen, ρπ˜ ,λ = ρπ˜tr ,λ ⊕ ρπ˜nd ,λ and, moreover, ρπ˜tr ,λ is 6
Journal Pre-proof
trivial (ρπ˜nd ,λ is in fact non-degenerate, but this is irrelevant at this point). Consequently, kρπ˜ ,λ (x)k = kρπ˜nd ,λ (x)k for each x ∈ Cc (G, A), and we see that the proof reduces L to the case that π is non-degenerate. Assuming that π is non-degenerate, let π = ι∈I πι be a decomposition of π into the Ldirect sum of cyclic representations (cf. [1, Proposition 2.2.7]). Then, clearly, ρπ˜ ,λ = ι∈I ρπ˜ι ,λ , so in order to show that kρπ˜ ,λ (x)k ≤ kρσ˜ ,λ (x)k for each x ∈ Cc (G, A), it suffices to show that kρπ˜ι ,λ (x)k ≤ kρσ˜ ,λ (x)k for each ι ∈ I and each x ∈ Cc (G, A). Thus we may assume from the outset that π is cyclic. With Hπ denoting the Hilbert space of π, let Ω ∈ Hπ be a cyclic vector for π such that kΩk = 1. First, we observe that the functions of the form s 7→ π(x(s))Ω
(s ∈ G)
are dense in L2 (G, Hπ ). Indeed, any element of L2 (G, Hπ ) can be approximated arbitrarily closely in norm by functions of form s 7→
I X
fi (s)ηi
i=1
(s ∈ G),
rep
roo
f
where f1 , . . . , fI ∈ Cc (G) and η1 , . . . , ηI ∈ Hπ . Since, by assumption, any element of Hπ can be approximated arbitrarily closely in norm by elements of the form π(A)Ω, where A ∈ A, it follows that any element of L2 (G, Hπ ) can be approximated arbitrarily closely in norm by functions of the form s 7→ π(x(s))Ω, where x is an element of Cc (G, A) of the PI form x(s) = i=1 fi (s)Ai , s ∈ G, with f1 . . . , fI ∈ Cc (G) and A1 , . . . , AI ∈ A. Next, fix x ∈ Cc (G, A) arbitrarily and let > 0. Since ρπ˜ ,λ (x∗ x) is a self-adjoint operator in L (L2 (G, Hπ )), there exists ξ0 ∈ L2 (G, Hπ ) with kξ0 k = 1 such that kρπ˜ ,λ (x∗ x)k − ≤ hρπ˜ ,λ (x∗ x)ξ0 , ξ0 i. By the observation made above, there exists z ∈ Cc (G, A) such that the function ξ : s 7→ π(z(s))Ω treated as a member of L2 (G, Hπ ) satisfies kξk ≤ 1 and
lP
kρπ˜ ,λ (x∗ x)k − ≤ hρπ˜ ,λ (x∗ x)ξ, ξi. Define y ∈ Cc (G, A) by Clearly,
y(s) = αs (z(s))
rna
z(s) = αs−1 (y(s))
(s ∈ G).
and ξ(s) = π αs−1 (y(s)) Ω
for each s ∈ G. Let e denote the neutral element of G. We claim that
kξk2 = π (y ∗ y)(e) Ω, Ω hρπ˜ ,λ (x∗ x)ξ, ξi = hπ((y ∗ x∗ xy)(e))Ω, Ωi .
Jo u
and
(3.1)
(3.2)
(3.3)
Indeed, by applying (2.2) and the interchangeability of the Bochner integral with bounded linear operators (cf. [13, p. 134, Corollary 2]), we see that Z
2 kξk = π αs−1 (y(s)) Ω, π αs−1 (y(s)) Ω dmG (s) G
7
Journal Pre-proof
= = =
Z
ZG
ZG
ZG
∆(s)−1 π αs (y(s−1 )) Ω, π αs (y(s−1 )) Ω dmG (s) ∆(s)−1
D
π αs (y(s−1 ))
∗
E π αs (y(s−1 )) Ω, Ω dmG (s)
D E ∗ ∆(s)−1 π αs (y(s−1 )) αs (y(s−1 )) Ω, Ω dmG (s)
π y ∗ (s)αs (y(s−1 )) Ω, Ω dmG (s) G Z ∗ −1 = π y (s)αs (y(s )) Ω dmG (s), Ω GZ ∗ −1 = π y (s)αs (y(s )) dmG (s) Ω, Ω G
= π (y ∗ y)(e) Ω, Ω .
=
rep
roo
f
To prove the other formula, we first note that Z Z hρπ˜ ,λ (x∗ x)ξ, ξi = π αs−1 ((x∗ x)(t)) ξ(t−1 s) dmG (t), ξ(s) dmG (s) G G Z Z
= π αs−1 ((x∗ x)(t)) ξ(t−1 s), ξ(s) dmG (t) dmG (s) G G Z Z
= π αs−1 ((x∗ x)(t)) π αs−1 t (y(t−1 s)) Ω, G G π αs−1 (y(s)) Ω dmG (t) dmG (s). Since
lP
we have
π αs−1 ((x∗ x)(t)) π αs−1 t (y(t−1 s)) = π αs−1 ((x∗ x)(t))αs−1 t (y(t−1 s)) = π αs−1 (x∗ x)(t)αt (y(t−1 s)) , Z Z D hρπ˜ ,λ (x x)ξ, ξi = π αs−1 (x∗ x)(t)αt (y(t−1 s)) Ω, G G i E π αs−1 (y(s)) Ω dmG (t) dmG (s).
rna
∗
Jo u
Now the internal integral simplifies as follows: Z D E π αs−1 (x∗ x)(t)αt (y(t−1 s)) Ω, π αs−1 (y(s)) Ω dmG (t) G Z = π αs−1 (x∗ x)(t)αt (y(t−1 s)) Ω dmG (t), π αs−1 (y(s)) Ω GZ = π αs−1 (x∗ x)(t)αt (y(t−1 s)) dmG (t) Ω, π αs−1 (y(s)) Ω G
= π αs−1 ((x∗ xy)(s)) Ω, π αs−1 (y(s)) Ω . 8
Journal Pre-proof
Therefore hρπ˜ ,λ (x∗ x)ξ, ξi = =
Z
π αs−1 ((x∗ xy)(s)) Ω, π αs−1 (y(s)) Ω dmG (s)
ZG D
π αs−1 (y(s))
∗
E π αs−1 ((x∗ xy)(s)) Ω, Ω dmG (s)
ZG D E ∗ = π αs−1 (y(s)) αs−1 ((x∗ xy)(s)) Ω, Ω dmG (s) ZG
= ∆(s)−1 π y ∗ (s−1 )αs−1 ((x∗ xy)(s)) Ω, Ω dmG (s) ZG D E = π y ∗ (s)αs (x∗ xy)(s−1 ) Ω, Ω dmG (s) G Z ∗ ∗ −1 = π y (s)αs (x xy)(s ) Ω dmG (s), Ω GZ ∗ ∗ −1 y (s)αs (x xy)(s ) dmG (s) Ω, Ω = π G
roo
f
= hπ((y ∗ x∗ xy)(e))Ω, Ωi ,
rep
and the claim is established. Let Hσ denote the Hilbert space of σ.P Since π is faithful, it follows from Proposition 2.2 N that there exist Θ1 , . . . , ΘN ∈ Hσ with n=1 kΘn k2 ≤ 1 such that N X (3.4) hσ((y ∗ y)(e))Θn , Θn i − hπ((y ∗ y)(e))Ω, Ωi < n=1
and
N X ∗ ∗ ∗ ∗ hσ((y x xy)(e))Θn , Θn i − hπ((y x xy)(e))Ω, Ωi < .
For 1 ≤ n ≤ N , let
lP
n=1
ηn (s) = σ(αs−1 y(s))Θn
(3.5)
(s ∈ G).
Then, by repeating the argument used to establish (3.2) and (3.3), we deduce that (3.6)
hσ((y ∗ x∗ xy)(e))Θn , Θn i = hρσ˜ ,λ (x∗ x)ηn , ηn i
(3.7)
rna
and
hσ((y ∗ y)(e))Θn , Θn i = kηn k2
for 1 ≤ n ≤ N . Now, by (3.1), (3.3), (3.5), and (3.7),
Jo u
kρπ˜ ,λ (x∗ x)k − 2 ≤ (ρπ˜ ,λ (x∗ x)ξ, ξ) − = hπ((y ∗ x∗ xy)(e))Ω, Ωi − ≤
N X
n=1
∗ ∗
hσ((y x xy)(e))Θn , Θn i =
9
N X
n=1
hρσ˜ ,λ (x∗ x)ηn , ηn i .
Journal Pre-proof
In view of (3.6), N X
n=1
hρσ˜ ,λ (x∗ x)ηn , ηn i ≤ kρσ˜ ,λ (x∗ x)k = kρσ˜ ,λ (x∗ x)k
N X
n=1 N X
n=1
kηn k2 hσ((y ∗ y)(e))Θn , Θn i .
Moreover, by (3.4) and (3.2), N X
n=1
hσ((y ∗ y)(e))Θn , Θn i ≤ hπ((y ∗ y)(e))Ω, Ωi + = kξk2 + ≤ 1 + .
Therefore
kρπ˜ ,λ (x∗ x)k − 2 ≤ kρσ˜ ,λ (x∗ x)k(1 + )
roo
f
and further, by letting → 0, kρπ˜ ,λ (x∗ x)k ≤ kρσ˜ ,λ (x∗ x)k. Since kρπ˜ ,λ (x)k2 = kρπ˜ ,λ (x∗ x)k and kρσ˜ ,λ (x)k2 = kρσ˜ ,λ (x∗ x)k, we see that kρπ˜ ,λ (x)k ≤ kρσ˜ ,λ (x)k, as was to be proved.
rep
Remark. The conclusion of Theorem 3.1 can be just confined to the statement that, under the assumptions of the theorem, k˜ π or λ(x)k ≤ k˜ σ or λ(x)k for all x ∈ Aoα,r G—the other statement follows from it as a consequence. Indeed, if k˜ π or λ(x)k ≤ k˜ σ or λ(x)k for all x ∈ A oα,r G, then in particular k˜ π or λ(h(x))k ≤ k˜ σ or λ(h(x))k for all x ∈ A oα,r G, where h is the canonical ∗-homomorphism from A oα G onto A oα G, and, since π ˜ oλ = (˜ π or λ) ◦ h and σ ˜ o λ = (˜ σ or λ) ◦ h, this in turn is equivalent to k˜ π o λ(x)k ≤ k˜ σ o λ(x)k for all x ∈ A oα G. We now present the first of the two results that are the main subject of exposition in this note (cf. [7, Theorem 7.7.5] and [12, Section 7.2]).
lP
Theorem 3.2. Let (A, G, α) be a C ∗ -algebraic system. If σ is a faithful representation of A, then σ ˜ or λ is a faithful representation of A oα,r G.
rna
Proof. Let x ∈ Aoα,r G. If π is a representation of A, then, by Theorem 3.1, k˜ π oλ(x)k ≤ k˜ σ o λ(x)k, and this implies that kxkr ≤ k˜ σ o λ(x)k. On the other hand, the reverse inequality k˜ σ o λ(x)k ≤ kxkr holds by definition. Consequently, k˜ σ o λ(x)k = kxkr and the theorem is proved. 4. Representations of crossed products by amenable groups
Jo u
The subsequent discussion will rely on the following result. Theorem 4.1. Let (A, G, α) be a C ∗ -algebraic system where G is amenable, and let (H, π, U ) be a covariant representation of (A, G, α). Then kπ o U (x)k ≤ k˜ π o λ(x)k for each x ∈ A oα G.
10
Journal Pre-proof
Proof. By the, by now, familiar argument, the theorem will be established once we show that kρπ,U (x)k ≤ kρπ˜ ,λ (x)k for every x ∈ Cc (G, A). Let x ∈ Cc (G, A). Then, by (2.1), for any s, t ∈ G, π(αs−1 x(t))Us−1 = Us−1 π(x(t)) so π(αs−1 x(t))Us−1 t = π(αs−1 x(t))Us−1 Ut = Us−1 π(x(t))Ut . Now, if Φ, Ψ ∈ H, then hπ(αs−1 x(t))Us−1 t Φ, Us−1 Ψi = hUs−1 π(x(t))Ut Φ, Us−1 Ψi = hπ(x(t))Ut Φ, Ψi .
Consequently, hρπ,U (x)Φ, Ψi =
Z
G
hπ(αs−1 x(t))Us−1 t Φ, Us−1 Ψi dmG (t)
(4.1)
roo
f
for each s ∈ G. By amenability of G, there exists a net {gα }α∈A in P (G) such that, for each compact subset K ⊂ G, lim max kgα − λt gα k1 = 0. α
t∈K
(4.2)
G
rep
Select α ∈ A arbitrarily. Multiplying both sides of (4.1) by gα (s) and integrating over s, we get Z Z hρπ,U (x)Φ, Ψi = hπ(αs−1 x(t))Us−1 t Φ, Us−1 Ψi dmG (t) gα (s) dmG (s). (4.3) G
Putting
ψα (s) =
p
gα (s)Us−1 Ψ
(a. e. s ∈ G),
lP
we can rewrite (4.3) as Z Z Dp E hρπ,U (x)Φ, Ψi = gα (s)π(αs−1 x(t))Us−1 t Φ, ψα (s) dmG (t) dmG (s). G
G
rna
The last equality can in turn be re-expressed as
where
Z Z Dp
Jo u
I1 (α) =
G
and
I2 (α) =
Z Z Dp G
G
G
hρπ,U (x)Φ, Ψi = I1 (α) + I2 (α),
(4.4)
E gα (t−1 s)π(αs−1 x(t))Us−1 t Φ, ψα (s) dmG (t) dmG (s)
E p −1 gα (s) − gα (t s) π(αs−1 x(t))Us−1 t Φ, ψα (s) dmG (t) dmG (s). 11
Journal Pre-proof
The two integrals are meaningful, as the following argument shows. Since π, just as any representation of a C ∗ -algebra, is contractive (cf. [1, Proposition 1.3.7]), we have kπ(αs−1 x(t))k ≤ kαs−1 x(t)k = kx(t)k ≤ max kx(t)k t∈supp x
for any s, t ∈ G. Also kψα (s)k = in I1 (α) is majorised by
p
(4.5)
gα (s)kΨk for a. e. s ∈ G. We see that the integrand
p p const × 1supp x (t) gα (t−1 s) gα (s),
where, of course, supp x denotes the support of t 7→ x(t). Since Z p G
gα
(t−1 s)
p gα (s) dmG (s) ≤
=
Z Z
gα (t
−1
s) dmG (s)
G
1/2 Z
G
1/2 gα (s) dmG (s)
gα (s) dmG (s) = 1
G
f
for each t ∈ G, we have G
supp x
G
roo
p p −1 1supp x (t) gα (t s) gα (s) dmG (t) dmG (s) G Z Z p p −1 = gα (t s) gα (s) dmG (s) dmG (t) ≤ mG (supp x),
Z Z
rep
which implies that the integrand in I1 (α) is integrable. A similar argument yields the integrability of the integrand in I2 (α). We shall exploit the representation (4.4) as follows. First we shall show that |I1 (α)| ≤ kρπ˜ ,λ (x)kkΦkkΨk
(4.6)
lP
for each α ∈ A, and next we shall show that
lim I2 (α) = 0. α
(4.7)
rna
This will then imply that
|hρπ,U (x)Φ, Ψi| ≤ kρπ˜ ,λ (x)kkΦkkΨk
Jo u
and further, in view of the arbitrariness of Φ and Ψ, that kρπ,U (x)k ≤ kρπ˜ ,λ (x)k, as desired. With a view to establishing (4.6), put p φα (s) = gα (s)Us−1 Φ (a. e. s ∈ G).
Then, clearly, φα is an element of L2 (G, H) with kφα k = kΦk. Moreover Z p (ρπ˜ ,λ (x)φα )(s) = gα (t−1 s)π(αs−1 x(t))Us−1 t Φ dmG (t) (a. e. s ∈ G). G
12
Journal Pre-proof
Changing the order of integration in the expression for I1 (α) and using the equality above, we obtain Z Z p −1 gα (t s)π(αs−1 x(t))Us−1 t Φ dmG (t), ψα (s) dmG (s) I1 (α) = G ZG = h(ρπ˜ ,λ (x)φα )(s), ψα (s)i dmG (s) = hρπ˜ ,λ (x)φα , ψα i . G
Hence |I1 (α)| ≤ kρπ˜ ,λ (x)kkφα kkψα k and now (4.6) follows given that kφα k = kΦk and kψα k = kΨk. To establish (4.7), note that a change of the order of integration in the expression for I2 (α) and an application of (4.5) lead to Z Z Dp E p −1 gα (s) − gα (t s) π(αs−1 x(t))Us−1 t Φ, ψα (s) dmG (s) dmG (t) |I2 (α)| = G G Z Z p p p −1 ≤ gα (s) − gα (t s) gα (s) dmG (s) dmG (t) t∈supp x
Now, for each t ∈ G,
f
G
× kΦkkΨk max kx(t)k.
roo
supp x
G
rep
Z p p p gα (s) − gα (t−1 s) gα (s) dmG (s)
Z 1/2 Z 1/2 2 p p −1 ≤ gα (s) − gα (t s) dmG (s) gα (s) dmG (s) G
1/2 Z 2 p p −1 . gα (s) − gα (t s) dmG (s) =
Moreover, since
we have
gα (s) −
p
gα (t−1 s)
rna
p
lP
G
G
2
p p p p ≤ gα (s) − gα (t−1 s) gα (s) + gα (t−1 s) = gα (s) − gα (t−1 s) ,
Z 1/2 2 p p 1/2 −1 gα (s) − gα (t s) dmG (s) ≤ kgα − λt gα k1 . G
Jo u
Therefore
1/2
|I2 (α)| ≤ mG (supp x) max kgα − λt gα k1 kΦkkΨk max kx(t)k, t∈supp x
t∈supp x
which, in view of (4.2), immediately yields (4.7) and finishes the proof. The following result is an immediate consequence of Theorems 3.1 and 4.1. 13
Journal Pre-proof
Theorem 4.2. Let (A, G, α) be a C ∗ -algebraic system where G is amenable. If σ is a faithful representation of A, then σ ˜ o λ is a faithful representation of A oα G. Proof. Let x ∈ A oα G. If (H, π, U ) is a covariant representation of (A, G, α), then, by Theorem 4.1, kπ o U (x)k ≤ k˜ π o λ(x)k and, by Theorem 3.1, k˜ π o λ(x)k ≤ k˜ σ o λ(x)k, so that kπ o U (x)k ≤ k˜ σ o λ(x)k. This implies that kxku ≤ k˜ σ o λ(x)k. On the other hand, the reverse inequality k˜ σ o λ(x)k ≤ kxku holds by definition. Thus k˜ σ o λ(x)k = kxku and the theorem is established. We finally present the second of the two results that we have aimed to expose in this note (cf. [7, Theorem 7.7.7] and [12, Theorem 7.13]). Theorem 4.3. Let (A, G, α) be a C ∗ -algebraic system where G is amenable. Then Aoα G is canonically and isometrically ∗-isomorphic with A oα,r G.
roo
f
Proof. Let h be the canonical ∗-homomorphim from A oα G onto A oα,r G. By the Gelfand–Naimark theorem (cf. [5, Theorem 3.4.1]), there is a faithful Hilbert space representation of A, say σ. Since σ ˜ o λ = (˜ σ or λ) ◦ h and since, by Theorem 4.2, σ ˜oλ is a faithful representation of A oα G, it follows that h is injective. Thus h is a ∗isomorphism of C ∗ -algebras, which is then necessarily isometric (cf. [5, Theorem 3.1.5]), and the theorem follows. References
Jo u
rna
lP
rep
[1] J. Dixmier, C ∗ -algebras, North-Holland, Amsterdam, 1977. [2] J.M.G. Fell, The dual spaces of C ∗-algebras, Trans. Amer. Math. Soc. 94 (1960) 365–403. [3] F.P. Greenleaf, Invariant Means on Topological Groups and their Applications, Van Nostrand, New York, 1969. [4] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, volume 1, 2nd ed., Springer, Berlin, 1979. [5] G.J. Murphy, C ∗ -algebras and Operator Theory, Academic Press, Boston, 1990. [6] A.L.T. Paterson, Amenability, Amer. Math. Society, Providence, RI, 1988. [7] G.K. Pedersen, C ∗-algebras and their Automorphism Groups, Academic Press, London, 1979. [8] J.P. Pier, Amenable Locally Compact Groups, Wiley, New York, 1984. [9] H. Reiter, J.D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, 2nd ed., Clarendon Press, Oxford, 2000. [10] H.J. Reiter, Investigations in harmonic analysis, Trans. Amer. Math. Soc. 73 (1952) 401–427. [11] V. Runde, Lectures on Amenability, volume 1774 of Lecture Notes in Math., Springer, Berlin, 2002. [12] D.P. Williams, Crossed Products of C ∗ -algebras, Amer. Math. Society, Providence, RI, 2007. [13] K. Yosida, Functional Analysis, 6th ed., Springer, 1980.
14
PII: DOI: Reference:
S0723-0869(19)30045-3 https://doi.org/10.1016/j.exmath.2019.08.001 EXMATH 25374
To appear in:
Expositiones Mathematicae
Received date : 29 March 2019 Revised date : 30 August 2019 Accepted date : 30 August 2019 Please cite this article as: W. Chojnacki, A note on crossed products, Expositiones Mathematicae (2019), doi: https://doi.org/10.1016/j.exmath.2019.08.001. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2019 Elsevier GmbH. All rights reserved. ⃝
Journal Pre-proof
A note on crossed products Wojciech Chojnackia,b a School b Faculty
of Computer Science, The University of Adelaide, SA 5005, Australia of Mathematics and Natural Sciences, College of Sciences, Cardinal Stefan Wyszy´ nski University, ul. Dewajtis 5, 01-815 Warszawa, Poland
Abstract
f
We give an exposition of two fundamental results of the theory of crossed products. One of these states that every regular representation of a reduced crossed product is faithful whenever the underlying Hilbert space representation of the C ∗ -algebra that together with an automorphism group gives rise to the crossed product is faithful. The other result states that a full and a reduced crossed products coincide whenever their common underlying automorphism group is amenable.
roo
Keywords: C ∗ -dynamical system, crossed product, representation, amenable group 2010 MSC: Primary 47L65, Secondary 43A07, 46L55
rep
1. Introduction
Jo u
rna
lP
The theory of crossed products of C ∗ -algebras by automorphism groups is a deep and interesting area of the modern theory of operator algebras, with links to such disciplines as non-commutative geometry and mathematical physics. The mathematical significance of the theory is attested, among others, by the fact that a substantial portion of the classic treatise by Pedersen [7] and the entire recent monograph by Williams [12] are devoted to crossed products. For physicists the crossed product construction is important as it allows exhibition of various simple and primitive (in algebraic sense) C ∗ -algebras that can serve as algebras of quantum observables. With applications in mind, including specifically those in mathematical physics, crossed products are treated in numerous introductory lecture notes and tutorials. However, as a rule, rudimentary texts on the subject stop short of presenting the proofs of two fundamental results on crossed products, namely that (1) every regular representation of a reduced crossed product is faithful whenever the underlying Hilbert space representation of the C ∗ -algebra that together with an automorphism group gives rise to the crossed product is faithful; and that (2) a full and a reduced crossed products coincide whenever their common underlying automorphism group is amenable. This state of affairs is reflective of the fact that the relevant proofs, as presented, say, in [7] and [12], require combining together a number of disparate elements that are somewhat hard to trace and follow. The purpose of this note is to provide a Email address: [email protected], [email protected] (Wojciech Chojnacki)
1
Journal Pre-proof
self-contained exposition of the above-mentioned fundamental results. It is hoped that our presentation will serve, among others, to alleviate the difficulties that writers of introductory texts on crossed products typically encounter when they attempt to strike a balance between conciseness and completeness. 2. Preliminaries We first recall some concepts and facts that will be of relevance throughout the paper. A basic familiarity with locally compact groups and C ∗ -algebras will be assumed. 2.1. Amenable groups Let G be a locally compact group, and let mG denote a left-invariant Haar measure on the group. Let f be a function on G. The left translate of f by t ∈ G is the function λt f defined by λt f (s) = f (t−1 s) (s ∈ G). The right translate of f by t ∈ G is the function ρt f defined by
f
(s ∈ G).
roo
ρt f (s) = f (st)
Let L∞ (G) denote the Banach space of equivalence classes of essentially bounded Haarmeasurable complex-valued functions on G which agree locally almost everywhere. A linear functional m : L∞ (G) → C is called a mean on L∞ (G) if
rep
(i) m(f ) = m(f ), (ii) m(1) = 1,
(iii) f ≥ 0 implies m(f ) ≥ 0.
A mean is called left-invariant if
lP
(iv) m(λt (f )) = m(f ) for all t ∈ G and f ∈ L∞ (G).
rna
A locally group G is called amenable if there exists a left-invariant mean on L∞ (G). As it turns out, L∞ (G) has a left-invariant mean if and only if it has a right-invariant mean and this is equivalent to L∞ (G) having a two-sided invariant mean. It is also equivalent to L∞ (G) having a topologically invariant mean, that is, a mean m such that Z m(g ∗ f ) = g dmG m(f ) (g ∈ L1 (G), f ∈ L∞ (G)). G
Jo u
Here, as is usual, L1 (G) denotes the space of equivalence classes of Haar-integrable complex-valued functions on G, and g ∗ f stands for the convolution of g and f defined by Z g ∗ f (t) =
g(s)f (s−1 t) dmG (s)
G
(locally a. e. t ∈ G).
We recall that all locally compact Abelian groups are amenable and so are all compact, in particular finite, groups. Any closed subgroup of an amenable group is amenable. 2
Journal Pre-proof
Moreover, if a group has an amenable closed normal subgroup such that the quotient by this subgroup is amenable, then the group is amenable. Solvable groups are amenable. On the other hand, semisimple Lie groups are not amenable, unless they are compact. The free group group on two generators is not amenable. It is vital to realise that amenability is not a purely group-theoretical concept—it depends on the topology of the group considered. For example, SO(3) under the usual topology is compact, hence amenable, but SO(3) under the discrete topology is not amenable, as it contains the free group on two generators which fails to be amenable. Amenable locally compact groups can be characterised by a number of alternative conditions. Of interest to us here will be the so-called (P1 ) condition introduced by elements f of L1 (G) such that RReiter [10]. Let P (G) denote the set of non-negative 1 f dmG = 1. With the norm of a function f in L (G) being G Z kf k1 = |f | dmG , G
the statement of Reiter’s condition goes as follows:
(P1 ) there exists a net {gα }α∈A in P (G) such that for each compact set K ⊂ G.
t∈K
roo
α
f
lim max kgα − λt gα k1 = 0
lP
rep
A fundamental result is that a locally compact group G is amenable if and only if G has property (P1 ) (cf. [3, Theorem 3.2.1]). To get a feeling for nets involved in property (P1 ) we remark that in the case G = R Reiter’s condition holds with any sequence of the form {gTn }n∈N , where {Tn }n∈N is a sequence of positive numbers diverging to infinity and gTn = (2Tn )−1 1[−Tn ,Tn ] for each n ∈ N. Here, as usual, we denote by 1E the characteristic function of the set E. The underlying sequences {[−Tn , Tn ]}n∈N of subsets of R are examples of so-called Følner sequences. Generally, Følner nets can be defined for any amenable group G and used to form nets in P (G) satisfying condition (P1 ). For more information about amenable groups, see [3], [4, §17], [6], [9, Chapter 8], and [11].
rna
2.2. Representations of C ∗ -algebras Given a C ∗ -algebra A, let S+,≤1 (A) denote the set of all positive functionals ω on A such that kωk ≤ 1. As a subset of the dual A∗ of A, S+,≤1 (A) inherits the weak* topology from A∗ . We recall that a neighbourhood base for ω ∈ A∗ is given by {ω 0 ∈ A∗ | |ω 0 (Ak ) − ω(Ak )| < , Ak ∈ A, k = 1, . . . , K, > 0}
Jo u
as A1 , . . . , AK runs through all K-tuples in A, > 0, but otherwise arbitrary, and K = 1, 2, . . . . Let H be a Hilbert space and let L (H) be the algebra of all linear bounded operators in H. Let A be a closed ∗-subalgebra of L (H). We denote by N (A) the set of elements ω in S+,≤1 (A) of the form N X ω(A) = hAΘn , Θn i , where Θ1 , . . . , ΘN ∈ H and
PN
n=1
n=1
2
kΘn k ≤ 1. We have the following fundamental result: 3
Journal Pre-proof
Proposition 2.1. Let H be a Hilbert space and let A be a closed ∗-subalgebra of L (H). Then S+,≤1 (A) is the weak* closure of N (A). The conclusion means, of course, that if ω ∈ S+,≤1 (A), then for each finite set PN {A1 , . . . , AK } in A and each > 0, there exist Θ1 , . . . , ΘN ∈ H with n=1 kΘn k2 ≤ 1 such that N X hAk Θn , Θn i − ω(Ak ) < n=1
for each k = 1, . . . , K. The proof is a rather straightforward application of the bipolar theorem to N (A) considered as a subset of the dual space of the real Banach space Ah of all hermitian elements of A. For details, we refer to [2, Theorem 1.1] or [8, Lemma 8.1]. In what follows, we shall make use of the following variation on the above proposition.
n=1
for each k = 1, . . . , K.
roo
f
Proposition 2.2. Let π and σ be two representations of a C ∗ -algebra A on Hilbert spaces Hπ and Hσ , respectively, and suppose, in addition, that σ is faithful. Let Ω ∈ Hπ be such that kΩk = 1. Then, given A1 , . . . , AK ∈ A and > 0, there exist Θ1 , . . . , ΘN ∈ Hσ with PN 2 n=1 kΘn k ≤ 1 such that N X hσ(Ak )Θn , Θn i − hπ(Ak )Ω, Ωi <
rep
Proof. The mapping ωΩ : A 7→ hπ(A)Ω, Ωi is clearly a member of S+,≤1 (A). Being faithful, σ is isometric (cf. [5, Theorem 3.1.5]), and this implies first that the image of A by σ, which we denote by B, is a closed ∗-subalgebra of L (Hσ ), and second that σ is invertible with kσ −1 (B)k = kBk for each B ∈ B. Consequently, the mapping ω = ωΩ ◦ σ −1 , which satisfies
lP
ω(σ(A)) = hπ(A)Ω, Ωi
(A ∈ A),
is a member of S+,≤1 (B). By Proposition 2.1, ω is then also an element of the weak∗ closure of N (B), and this immediately yields the result.
rna
2.3. Dynamical systems and crossed products
Jo u
A C ∗ -algebraic system is a triple (A, G, α), where A is a C ∗ -algebra, G is a locally compact group, and α : t 7→ αt is a homomorphism from G into the group of ∗-automorphisms of A such that, for each A ∈ A, the function G 3 t 7→ αt (A) ∈ A is continuous. Let (A, G, α) be a C ∗ -algebraic system. A covariant representation of (A, G, α) is a triple (H, π, U ), where H is a Hilbert space, π is a representation of A on H, and U : t 7→ Ut is a strongly continuous unitary representation of G on H such that, for each A ∈ A and each t ∈ G, π(αt (A)) = Ut π(A)Ut∗ . (2.1) Here, for any S ∈ L (H), S ∗ denotes the adjoint operator of S. 4
Journal Pre-proof
Let ∆ denote the Haar modular function of G. We recall that Z Z ∆(t)−1 f (t−1 ) dmG (t) = f (t) dmG (t) G
(2.2)
G
for any (possibly vector-valued) Haar integrable function f on G. Let Cc (G, A) be the space of all continuous functions from G into A wih compact support. The space Cc (G, A) can be converted into a norm ∗-algebra by introducing a multiplication, involution, and norm according to the rules: Z (xy)(t) = x(s)αs (y(s−1 t)) dmG (s), G
x∗ (t) = ∆(t)−1 αt (x(t−1 ))∗ , Z kxk1 = kx(s)k dmG (s)
(x, y ∈ Cc (G, A), t ∈ G).
G
f
With any covariant representation (H, π, U ) of (A, G, α) there is associated a representation ρπ,U of Cc (G, A) on H defined by Z ρπ,U (x) = π(x(t))Ut dmG (t) (x ∈ Cc (G, A)),
roo
G
the integral being taken in the strong operator topology. The representation ρπ,U is called the integrated form of (H, π, U ). Exploiting the integrated forms of covariant representations, one can define a new norm on Cc (G, A), namely (x ∈ Cc (G, A)),
rep
kxku = sup kρπ,U (x)k
where the supremum is taken over all covariant representations (H, π, U ) of (A, G, α). It is not possible to take supremum over a class, but the right hand side of the above equation is a supremum of a subclass of a set, namely R, so it is a supremum of a subset of R. One easily sees that k · ku is a C ∗ -seminorm, but in fact k · ku is a C ∗ -norm, the so-called universal norm. Moreover,
lP
kxku ≤ kxk1
(x ∈ Cc (G, A)).
Jo u
rna
The completion of Cc (G, A) in k · ku is a C ∗ -algebra called the crossed product of A by G. It will be denoted by A oα G. For every covariant representation (H, π, U ) of (A, G, α), the representation ρπ,U can be uniquely extended by continuity to a representation of A oα G. This extension will be denoted by π o U . Let H be a Hilbert space with inner product h·, ·i. Let L2 (G, H) be the completion of Cc (G, H), where Cc (G, H) is the space of all continuous functions from G into H with compact support, in the norm derived from the inner product Z hξ, ηi = hξ(t), η(t)i dmG (t) (ξ, η ∈ Cc (G, H)). G
Suppose that π is a representation of A in H. Define a representation π ˜ of A in L2 (G, H) 2 and a unitary representation λ : t 7→ λt of G in L (G, H) by setting (˜ π (A)ξ)(s) = π(αs−1 (A))ξ(s), (λt ξ)(s) = ξ(t−1 s)
(A ∈ A, ξ ∈ L2 (G, H)), t ∈ G, a. e. s ∈ G). 5
Journal Pre-proof
The triple (L2 (G, H), π ˜ , λ) is a covariant representation of (A, G, α). The corresponding representations ρπ˜ ,λ and π ˜ o α are termed the regular representation of Cc (G, A) induced by π and the regular representation of A oα G induced by π, respectively. Alongside the universal norm, one can define another C ∗ -norm on Cc (G, A), namely kxkr = sup kρπ˜ ,λ (x)k
(x ∈ Cc (G, A)),
roo
f
where the supremum is taken over all regular representations of Cc (G, A) (which in turn correspond to all Hilbert space representations of A). This is termed the reduced norm. The completion of Cc (G, A) in k·kr is a C ∗ -algebra called the reduced crossed product of A by G. It will be denoted by Aoα,r G. If π is a representation of A, then the representation ρπ˜ ,λ of Cc (G, A) can be uniquely extended by continuity to a representation of A oα,r G. This extension will be denoted by π ˜ or λ. There exists a unique ∗-homomorphism h from Aoα G into Aoα,r G such that h(x) = x for every x ∈ Cc (G, A). The homomorphism is the unique extension by continuity of the identity mapping of Cc (G, A) ⊂ A oα G onto Cc (G, A) ⊂ A oα,r G which is continuous, given that kxkr ≤ kxku for each x ∈ Cc (G, A). The map h is surjective. Indeed, the image of A oα G by h is closed in A oα,r G (the image of any C ∗ -algebra via a ∗-homomorphism into a C ∗ -algebra is closed; cf. [5, Theorem 3.1.6]) and contains Cc (G, A) which is dense in A oα,r G, therefore this image is all of A oα,r G. If π is a representation of A, then π ˜ o λ = (˜ π or λ) ◦ h.
rep
This follows from the fact that (˜ π or λ) ◦ h is a representation of A oα G coinciding with ρπ˜ ,λ on Cc (G, A) and the fact that π ˜ o λ is the only extension by continuity of ρπ˜ ,λ to a representation of A oα G. For a comprehensive treatment of the theory of crossed products, see [7] and [12]. 3. Regular representations of reduced crossed products
lP
The starting point of our considerations is the following result.
rna
Theorem 3.1. Let (A, G, α) be a C ∗ -algebraic system. Let π and σ be Hilbert space representations of A and suppose, in addition, that σ is faithful. Then k˜ π or λ(x)k ≤ k˜ σ or λ(x)k for each x ∈ A oα,r G and k˜ π o λ(x)k ≤ k˜ σ o λ(x)k for each x ∈ A oα G.
Jo u
Proof. Since kρπ˜ ,λ (x)k ≤ kxkr and kρσ˜ ,λ (x)k ≤ kxkr for each x ∈ Cc (G, A), and since, by definition, A oα,r G is the completion of Cc (G, A) in k · kr , it is clear that to prove that k˜ π or λ(x)k ≤ k˜ σ or λ(x)k for each x ∈ A oα,r G, it suffices to show that kρπ˜ ,λ (x)k ≤ kρσ˜ ,λ (x)k for each x ∈ Cc (G, A). Replacing in this argument k · kr by k · ku we see that the estimate kρπ˜ ,λ (x)k ≤ kρσ˜ ,λ (x)k, x ∈ Cc (G, A), is also all that is needed to conclude that k˜ π o λ(x)k ≤ k˜ σ o λ(x)k for each x ∈ A oα G. Before continuing, let us recall that a representation τ of a C ∗ -algebra B on a Hilbert space H is trivial if τ (B) = 0 for each B ∈ B, and is non-degenerate if there is no non-zero ξ ∈ H such that τ (B)ξ = 0 for all B ∈ B. Let π = πtr ⊕πnd be the decomposition of π into the direct sum of a trivial representation and a non-degenerate representation of A (cf. [1, Proposition 2.2.6]). Then, as is easily seen, ρπ˜ ,λ = ρπ˜tr ,λ ⊕ ρπ˜nd ,λ and, moreover, ρπ˜tr ,λ is 6
Journal Pre-proof
trivial (ρπ˜nd ,λ is in fact non-degenerate, but this is irrelevant at this point). Consequently, kρπ˜ ,λ (x)k = kρπ˜nd ,λ (x)k for each x ∈ Cc (G, A), and we see that the proof reduces L to the case that π is non-degenerate. Assuming that π is non-degenerate, let π = ι∈I πι be a decomposition of π into the Ldirect sum of cyclic representations (cf. [1, Proposition 2.2.7]). Then, clearly, ρπ˜ ,λ = ι∈I ρπ˜ι ,λ , so in order to show that kρπ˜ ,λ (x)k ≤ kρσ˜ ,λ (x)k for each x ∈ Cc (G, A), it suffices to show that kρπ˜ι ,λ (x)k ≤ kρσ˜ ,λ (x)k for each ι ∈ I and each x ∈ Cc (G, A). Thus we may assume from the outset that π is cyclic. With Hπ denoting the Hilbert space of π, let Ω ∈ Hπ be a cyclic vector for π such that kΩk = 1. First, we observe that the functions of the form s 7→ π(x(s))Ω
(s ∈ G)
are dense in L2 (G, Hπ ). Indeed, any element of L2 (G, Hπ ) can be approximated arbitrarily closely in norm by functions of form s 7→
I X
fi (s)ηi
i=1
(s ∈ G),
rep
roo
f
where f1 , . . . , fI ∈ Cc (G) and η1 , . . . , ηI ∈ Hπ . Since, by assumption, any element of Hπ can be approximated arbitrarily closely in norm by elements of the form π(A)Ω, where A ∈ A, it follows that any element of L2 (G, Hπ ) can be approximated arbitrarily closely in norm by functions of the form s 7→ π(x(s))Ω, where x is an element of Cc (G, A) of the PI form x(s) = i=1 fi (s)Ai , s ∈ G, with f1 . . . , fI ∈ Cc (G) and A1 , . . . , AI ∈ A. Next, fix x ∈ Cc (G, A) arbitrarily and let > 0. Since ρπ˜ ,λ (x∗ x) is a self-adjoint operator in L (L2 (G, Hπ )), there exists ξ0 ∈ L2 (G, Hπ ) with kξ0 k = 1 such that kρπ˜ ,λ (x∗ x)k − ≤ hρπ˜ ,λ (x∗ x)ξ0 , ξ0 i. By the observation made above, there exists z ∈ Cc (G, A) such that the function ξ : s 7→ π(z(s))Ω treated as a member of L2 (G, Hπ ) satisfies kξk ≤ 1 and
lP
kρπ˜ ,λ (x∗ x)k − ≤ hρπ˜ ,λ (x∗ x)ξ, ξi. Define y ∈ Cc (G, A) by Clearly,
y(s) = αs (z(s))
rna
z(s) = αs−1 (y(s))
(s ∈ G).
and ξ(s) = π αs−1 (y(s)) Ω
for each s ∈ G. Let e denote the neutral element of G. We claim that
kξk2 = π (y ∗ y)(e) Ω, Ω hρπ˜ ,λ (x∗ x)ξ, ξi = hπ((y ∗ x∗ xy)(e))Ω, Ωi .
Jo u
and
(3.1)
(3.2)
(3.3)
Indeed, by applying (2.2) and the interchangeability of the Bochner integral with bounded linear operators (cf. [13, p. 134, Corollary 2]), we see that Z
2 kξk = π αs−1 (y(s)) Ω, π αs−1 (y(s)) Ω dmG (s) G
7
Journal Pre-proof
= = =
Z
ZG
ZG
ZG
∆(s)−1 π αs (y(s−1 )) Ω, π αs (y(s−1 )) Ω dmG (s) ∆(s)−1
D
π αs (y(s−1 ))
∗
E π αs (y(s−1 )) Ω, Ω dmG (s)
D E ∗ ∆(s)−1 π αs (y(s−1 )) αs (y(s−1 )) Ω, Ω dmG (s)
π y ∗ (s)αs (y(s−1 )) Ω, Ω dmG (s) G Z ∗ −1 = π y (s)αs (y(s )) Ω dmG (s), Ω GZ ∗ −1 = π y (s)αs (y(s )) dmG (s) Ω, Ω G
= π (y ∗ y)(e) Ω, Ω .
=
rep
roo
f
To prove the other formula, we first note that Z Z hρπ˜ ,λ (x∗ x)ξ, ξi = π αs−1 ((x∗ x)(t)) ξ(t−1 s) dmG (t), ξ(s) dmG (s) G G Z Z
= π αs−1 ((x∗ x)(t)) ξ(t−1 s), ξ(s) dmG (t) dmG (s) G G Z Z
= π αs−1 ((x∗ x)(t)) π αs−1 t (y(t−1 s)) Ω, G G π αs−1 (y(s)) Ω dmG (t) dmG (s). Since
lP
we have
π αs−1 ((x∗ x)(t)) π αs−1 t (y(t−1 s)) = π αs−1 ((x∗ x)(t))αs−1 t (y(t−1 s)) = π αs−1 (x∗ x)(t)αt (y(t−1 s)) , Z Z D hρπ˜ ,λ (x x)ξ, ξi = π αs−1 (x∗ x)(t)αt (y(t−1 s)) Ω, G G i E π αs−1 (y(s)) Ω dmG (t) dmG (s).
rna
∗
Jo u
Now the internal integral simplifies as follows: Z D E π αs−1 (x∗ x)(t)αt (y(t−1 s)) Ω, π αs−1 (y(s)) Ω dmG (t) G Z = π αs−1 (x∗ x)(t)αt (y(t−1 s)) Ω dmG (t), π αs−1 (y(s)) Ω GZ = π αs−1 (x∗ x)(t)αt (y(t−1 s)) dmG (t) Ω, π αs−1 (y(s)) Ω G
= π αs−1 ((x∗ xy)(s)) Ω, π αs−1 (y(s)) Ω . 8
Journal Pre-proof
Therefore hρπ˜ ,λ (x∗ x)ξ, ξi = =
Z
π αs−1 ((x∗ xy)(s)) Ω, π αs−1 (y(s)) Ω dmG (s)
ZG D
π αs−1 (y(s))
∗
E π αs−1 ((x∗ xy)(s)) Ω, Ω dmG (s)
ZG D E ∗ = π αs−1 (y(s)) αs−1 ((x∗ xy)(s)) Ω, Ω dmG (s) ZG
= ∆(s)−1 π y ∗ (s−1 )αs−1 ((x∗ xy)(s)) Ω, Ω dmG (s) ZG D E = π y ∗ (s)αs (x∗ xy)(s−1 ) Ω, Ω dmG (s) G Z ∗ ∗ −1 = π y (s)αs (x xy)(s ) Ω dmG (s), Ω GZ ∗ ∗ −1 y (s)αs (x xy)(s ) dmG (s) Ω, Ω = π G
roo
f
= hπ((y ∗ x∗ xy)(e))Ω, Ωi ,
rep
and the claim is established. Let Hσ denote the Hilbert space of σ.P Since π is faithful, it follows from Proposition 2.2 N that there exist Θ1 , . . . , ΘN ∈ Hσ with n=1 kΘn k2 ≤ 1 such that N X (3.4) hσ((y ∗ y)(e))Θn , Θn i − hπ((y ∗ y)(e))Ω, Ωi < n=1
and
N X ∗ ∗ ∗ ∗ hσ((y x xy)(e))Θn , Θn i − hπ((y x xy)(e))Ω, Ωi < .
For 1 ≤ n ≤ N , let
lP
n=1
ηn (s) = σ(αs−1 y(s))Θn
(3.5)
(s ∈ G).
Then, by repeating the argument used to establish (3.2) and (3.3), we deduce that (3.6)
hσ((y ∗ x∗ xy)(e))Θn , Θn i = hρσ˜ ,λ (x∗ x)ηn , ηn i
(3.7)
rna
and
hσ((y ∗ y)(e))Θn , Θn i = kηn k2
for 1 ≤ n ≤ N . Now, by (3.1), (3.3), (3.5), and (3.7),
Jo u
kρπ˜ ,λ (x∗ x)k − 2 ≤ (ρπ˜ ,λ (x∗ x)ξ, ξ) − = hπ((y ∗ x∗ xy)(e))Ω, Ωi − ≤
N X
n=1
∗ ∗
hσ((y x xy)(e))Θn , Θn i =
9
N X
n=1
hρσ˜ ,λ (x∗ x)ηn , ηn i .
Journal Pre-proof
In view of (3.6), N X
n=1
hρσ˜ ,λ (x∗ x)ηn , ηn i ≤ kρσ˜ ,λ (x∗ x)k = kρσ˜ ,λ (x∗ x)k
N X
n=1 N X
n=1
kηn k2 hσ((y ∗ y)(e))Θn , Θn i .
Moreover, by (3.4) and (3.2), N X
n=1
hσ((y ∗ y)(e))Θn , Θn i ≤ hπ((y ∗ y)(e))Ω, Ωi + = kξk2 + ≤ 1 + .
Therefore
kρπ˜ ,λ (x∗ x)k − 2 ≤ kρσ˜ ,λ (x∗ x)k(1 + )
roo
f
and further, by letting → 0, kρπ˜ ,λ (x∗ x)k ≤ kρσ˜ ,λ (x∗ x)k. Since kρπ˜ ,λ (x)k2 = kρπ˜ ,λ (x∗ x)k and kρσ˜ ,λ (x)k2 = kρσ˜ ,λ (x∗ x)k, we see that kρπ˜ ,λ (x)k ≤ kρσ˜ ,λ (x)k, as was to be proved.
rep
Remark. The conclusion of Theorem 3.1 can be just confined to the statement that, under the assumptions of the theorem, k˜ π or λ(x)k ≤ k˜ σ or λ(x)k for all x ∈ Aoα,r G—the other statement follows from it as a consequence. Indeed, if k˜ π or λ(x)k ≤ k˜ σ or λ(x)k for all x ∈ A oα,r G, then in particular k˜ π or λ(h(x))k ≤ k˜ σ or λ(h(x))k for all x ∈ A oα,r G, where h is the canonical ∗-homomorphism from A oα G onto A oα G, and, since π ˜ oλ = (˜ π or λ) ◦ h and σ ˜ o λ = (˜ σ or λ) ◦ h, this in turn is equivalent to k˜ π o λ(x)k ≤ k˜ σ o λ(x)k for all x ∈ A oα G. We now present the first of the two results that are the main subject of exposition in this note (cf. [7, Theorem 7.7.5] and [12, Section 7.2]).
lP
Theorem 3.2. Let (A, G, α) be a C ∗ -algebraic system. If σ is a faithful representation of A, then σ ˜ or λ is a faithful representation of A oα,r G.
rna
Proof. Let x ∈ Aoα,r G. If π is a representation of A, then, by Theorem 3.1, k˜ π oλ(x)k ≤ k˜ σ o λ(x)k, and this implies that kxkr ≤ k˜ σ o λ(x)k. On the other hand, the reverse inequality k˜ σ o λ(x)k ≤ kxkr holds by definition. Consequently, k˜ σ o λ(x)k = kxkr and the theorem is proved. 4. Representations of crossed products by amenable groups
Jo u
The subsequent discussion will rely on the following result. Theorem 4.1. Let (A, G, α) be a C ∗ -algebraic system where G is amenable, and let (H, π, U ) be a covariant representation of (A, G, α). Then kπ o U (x)k ≤ k˜ π o λ(x)k for each x ∈ A oα G.
10
Journal Pre-proof
Proof. By the, by now, familiar argument, the theorem will be established once we show that kρπ,U (x)k ≤ kρπ˜ ,λ (x)k for every x ∈ Cc (G, A). Let x ∈ Cc (G, A). Then, by (2.1), for any s, t ∈ G, π(αs−1 x(t))Us−1 = Us−1 π(x(t)) so π(αs−1 x(t))Us−1 t = π(αs−1 x(t))Us−1 Ut = Us−1 π(x(t))Ut . Now, if Φ, Ψ ∈ H, then hπ(αs−1 x(t))Us−1 t Φ, Us−1 Ψi = hUs−1 π(x(t))Ut Φ, Us−1 Ψi = hπ(x(t))Ut Φ, Ψi .
Consequently, hρπ,U (x)Φ, Ψi =
Z
G
hπ(αs−1 x(t))Us−1 t Φ, Us−1 Ψi dmG (t)
(4.1)
roo
f
for each s ∈ G. By amenability of G, there exists a net {gα }α∈A in P (G) such that, for each compact subset K ⊂ G, lim max kgα − λt gα k1 = 0. α
t∈K
(4.2)
G
rep
Select α ∈ A arbitrarily. Multiplying both sides of (4.1) by gα (s) and integrating over s, we get Z Z hρπ,U (x)Φ, Ψi = hπ(αs−1 x(t))Us−1 t Φ, Us−1 Ψi dmG (t) gα (s) dmG (s). (4.3) G
Putting
ψα (s) =
p
gα (s)Us−1 Ψ
(a. e. s ∈ G),
lP
we can rewrite (4.3) as Z Z Dp E hρπ,U (x)Φ, Ψi = gα (s)π(αs−1 x(t))Us−1 t Φ, ψα (s) dmG (t) dmG (s). G
G
rna
The last equality can in turn be re-expressed as
where
Z Z Dp
Jo u
I1 (α) =
G
and
I2 (α) =
Z Z Dp G
G
G
hρπ,U (x)Φ, Ψi = I1 (α) + I2 (α),
(4.4)
E gα (t−1 s)π(αs−1 x(t))Us−1 t Φ, ψα (s) dmG (t) dmG (s)
E p −1 gα (s) − gα (t s) π(αs−1 x(t))Us−1 t Φ, ψα (s) dmG (t) dmG (s). 11
Journal Pre-proof
The two integrals are meaningful, as the following argument shows. Since π, just as any representation of a C ∗ -algebra, is contractive (cf. [1, Proposition 1.3.7]), we have kπ(αs−1 x(t))k ≤ kαs−1 x(t)k = kx(t)k ≤ max kx(t)k t∈supp x
for any s, t ∈ G. Also kψα (s)k = in I1 (α) is majorised by
p
(4.5)
gα (s)kΨk for a. e. s ∈ G. We see that the integrand
p p const × 1supp x (t) gα (t−1 s) gα (s),
where, of course, supp x denotes the support of t 7→ x(t). Since Z p G
gα
(t−1 s)
p gα (s) dmG (s) ≤
=
Z Z
gα (t
−1
s) dmG (s)
G
1/2 Z
G
1/2 gα (s) dmG (s)
gα (s) dmG (s) = 1
G
f
for each t ∈ G, we have G
supp x
G
roo
p p −1 1supp x (t) gα (t s) gα (s) dmG (t) dmG (s) G Z Z p p −1 = gα (t s) gα (s) dmG (s) dmG (t) ≤ mG (supp x),
Z Z
rep
which implies that the integrand in I1 (α) is integrable. A similar argument yields the integrability of the integrand in I2 (α). We shall exploit the representation (4.4) as follows. First we shall show that |I1 (α)| ≤ kρπ˜ ,λ (x)kkΦkkΨk
(4.6)
lP
for each α ∈ A, and next we shall show that
lim I2 (α) = 0. α
(4.7)
rna
This will then imply that
|hρπ,U (x)Φ, Ψi| ≤ kρπ˜ ,λ (x)kkΦkkΨk
Jo u
and further, in view of the arbitrariness of Φ and Ψ, that kρπ,U (x)k ≤ kρπ˜ ,λ (x)k, as desired. With a view to establishing (4.6), put p φα (s) = gα (s)Us−1 Φ (a. e. s ∈ G).
Then, clearly, φα is an element of L2 (G, H) with kφα k = kΦk. Moreover Z p (ρπ˜ ,λ (x)φα )(s) = gα (t−1 s)π(αs−1 x(t))Us−1 t Φ dmG (t) (a. e. s ∈ G). G
12
Journal Pre-proof
Changing the order of integration in the expression for I1 (α) and using the equality above, we obtain Z Z p −1 gα (t s)π(αs−1 x(t))Us−1 t Φ dmG (t), ψα (s) dmG (s) I1 (α) = G ZG = h(ρπ˜ ,λ (x)φα )(s), ψα (s)i dmG (s) = hρπ˜ ,λ (x)φα , ψα i . G
Hence |I1 (α)| ≤ kρπ˜ ,λ (x)kkφα kkψα k and now (4.6) follows given that kφα k = kΦk and kψα k = kΨk. To establish (4.7), note that a change of the order of integration in the expression for I2 (α) and an application of (4.5) lead to Z Z Dp E p −1 gα (s) − gα (t s) π(αs−1 x(t))Us−1 t Φ, ψα (s) dmG (s) dmG (t) |I2 (α)| = G G Z Z p p p −1 ≤ gα (s) − gα (t s) gα (s) dmG (s) dmG (t) t∈supp x
Now, for each t ∈ G,
f
G
× kΦkkΨk max kx(t)k.
roo
supp x
G
rep
Z p p p gα (s) − gα (t−1 s) gα (s) dmG (s)
Z 1/2 Z 1/2 2 p p −1 ≤ gα (s) − gα (t s) dmG (s) gα (s) dmG (s) G
1/2 Z 2 p p −1 . gα (s) − gα (t s) dmG (s) =
Moreover, since
we have
gα (s) −
p
gα (t−1 s)
rna
p
lP
G
G
2
p p p p ≤ gα (s) − gα (t−1 s) gα (s) + gα (t−1 s) = gα (s) − gα (t−1 s) ,
Z 1/2 2 p p 1/2 −1 gα (s) − gα (t s) dmG (s) ≤ kgα − λt gα k1 . G
Jo u
Therefore
1/2
|I2 (α)| ≤ mG (supp x) max kgα − λt gα k1 kΦkkΨk max kx(t)k, t∈supp x
t∈supp x
which, in view of (4.2), immediately yields (4.7) and finishes the proof. The following result is an immediate consequence of Theorems 3.1 and 4.1. 13
Journal Pre-proof
Theorem 4.2. Let (A, G, α) be a C ∗ -algebraic system where G is amenable. If σ is a faithful representation of A, then σ ˜ o λ is a faithful representation of A oα G. Proof. Let x ∈ A oα G. If (H, π, U ) is a covariant representation of (A, G, α), then, by Theorem 4.1, kπ o U (x)k ≤ k˜ π o λ(x)k and, by Theorem 3.1, k˜ π o λ(x)k ≤ k˜ σ o λ(x)k, so that kπ o U (x)k ≤ k˜ σ o λ(x)k. This implies that kxku ≤ k˜ σ o λ(x)k. On the other hand, the reverse inequality k˜ σ o λ(x)k ≤ kxku holds by definition. Thus k˜ σ o λ(x)k = kxku and the theorem is established. We finally present the second of the two results that we have aimed to expose in this note (cf. [7, Theorem 7.7.7] and [12, Theorem 7.13]). Theorem 4.3. Let (A, G, α) be a C ∗ -algebraic system where G is amenable. Then Aoα G is canonically and isometrically ∗-isomorphic with A oα,r G.
roo
f
Proof. Let h be the canonical ∗-homomorphim from A oα G onto A oα,r G. By the Gelfand–Naimark theorem (cf. [5, Theorem 3.4.1]), there is a faithful Hilbert space representation of A, say σ. Since σ ˜ o λ = (˜ σ or λ) ◦ h and since, by Theorem 4.2, σ ˜oλ is a faithful representation of A oα G, it follows that h is injective. Thus h is a ∗isomorphism of C ∗ -algebras, which is then necessarily isometric (cf. [5, Theorem 3.1.5]), and the theorem follows. References
Jo u
rna
lP
rep
[1] J. Dixmier, C ∗ -algebras, North-Holland, Amsterdam, 1977. [2] J.M.G. Fell, The dual spaces of C ∗-algebras, Trans. Amer. Math. Soc. 94 (1960) 365–403. [3] F.P. Greenleaf, Invariant Means on Topological Groups and their Applications, Van Nostrand, New York, 1969. [4] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, volume 1, 2nd ed., Springer, Berlin, 1979. [5] G.J. Murphy, C ∗ -algebras and Operator Theory, Academic Press, Boston, 1990. [6] A.L.T. Paterson, Amenability, Amer. Math. Society, Providence, RI, 1988. [7] G.K. Pedersen, C ∗-algebras and their Automorphism Groups, Academic Press, London, 1979. [8] J.P. Pier, Amenable Locally Compact Groups, Wiley, New York, 1984. [9] H. Reiter, J.D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, 2nd ed., Clarendon Press, Oxford, 2000. [10] H.J. Reiter, Investigations in harmonic analysis, Trans. Amer. Math. Soc. 73 (1952) 401–427. [11] V. Runde, Lectures on Amenability, volume 1774 of Lecture Notes in Math., Springer, Berlin, 2002. [12] D.P. Williams, Crossed Products of C ∗ -algebras, Amer. Math. Society, Providence, RI, 2007. [13] K. Yosida, Functional Analysis, 6th ed., Springer, 1980.
14