1 Introduction
Let
$\mathcal {A}$
be a unital C
$^*$
-algebra and let G be a compact group that acts strongly continuously on
$\mathcal {A}$
by
$^*$
-automorphisms
$\alpha _g:\mathcal {A} \to \mathcal {A}$
,
$g \in G$
. In this article, we call this data a C
$^*$
-dynamical system, denote it briefly by
$(\mathcal {A},G,\alpha )$
, and customarily write
$\mathcal {B}:=\mathcal {A}^G:=\{x \in \mathcal {A} :(\forall g\in G) \, \alpha _g(x) = x\}$
for its fixed point algebra. Research into C
$^*$
-dynamical systems is inherently interesting and has always been an area of active research both in operator algebras and noncommutative geometry. It is desirable to identify properties of C
$^*$
-dynamical systems that are both commonly occurring and significant enough to obtain interesting results. To expedite matters, let us revisit the notion of freeness, which exemplifies one such property: A C
$^*$
-dynamical system
$(\mathcal {A},G,\alpha )$
is called free if the so-called Ellwood map
$$ \begin{align*} \Phi: \mathcal{A} \otimes_{\text{alg}} \mathcal{A} \rightarrow C(G,\mathcal{A}), && \Phi(x\otimes y)(g):=x \alpha_g(y) \end{align*} $$
has dense range with respect to the canonical C
$^*$
-norm on
$C(G,\mathcal {A})$
. This condition, first introduced by Ellwood [Reference Ellwood10] for actions of quantum groups on C
$^*$
-algebras, is known to be equivalent to Rieffel’s saturatedness [Reference Rieffel19] and the Peter–Weyl–Galois condition [Reference Baum, Commer and Hajac2]. By [Reference Phillips16, Proposition 7.1.12 and Theorem 7.2.6], a continuous action
$r:P\times G\rightarrow P$
of a compact group G on a compact space P is free in the classical sense, i. e., all stabilizer groups are trivial, if and only if the induced C
$^*$
-dynamical system
$(C(P),G,\alpha )$
, where
$\alpha _g(f)(p) := f(r(p,g))$
for all
$g\in G$
,
$f\in C(P)$
, and
$p\in P$
, is free in the sense of Ellwood. Free C
$^*$
-dynamical systems thus provide a natural framework for noncommutative principal bundles. Because of this and their wide range of applications, these objects have garnered widespread interest and have been extensively studied by many researchers in recent years (see, e. g., [Reference Antonini, Guido, Isola and Rubin1, Reference Baum, Commer and Hajac2, Reference Brzeziński and Szymański4–Reference Dabrowski, Sitarz and Zucca8, Reference Farsi and Latrémolière11, Reference Gootman, Lazar and Peligrad12, Reference Phillips17, Reference Schwieger and Wagner22, Reference Schwieger and Wagner24] and ref therein).
Free and ergodic C
$^*$
-dynamical systems, also known as full multiplicity ergodic actions, have been the focal point of [Reference Wassermann26]. In particular, Wassermann proved the interesting result that each free and ergodic C
$^*$
-dynamical system
$(\mathcal {A},G,\alpha )$
can be realized as the invariants of an equivariant coaction of G on
${\mathcal {K}}(L^2(G))$
. Noteworthily, this constitutes an important step in the classification of such C
$^*$
-dynamical systems by means of a generalized cocycle theory.
Now, let us consider a free C
$^*$
-dynamical system
$(\mathcal {A},G,\alpha )$
with a general fixed point algebra
$\mathcal {B}$
. The overall purpose of this paper is to extend Wassermann’s result by showing that
$(\mathcal {A},G,\alpha )$
can be realized as the G-continuous part of the invariants of an equivariant coaction of G on a corner of
$\mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}})$
for a certain Hilbert space
${\mathfrak {H}}$
that arises from the freeness property (Theorem 3.11). From this we derive the following representation theoretic result: Any faithful
$^*$
-representation of
$\mathcal {B}$
on a Hilbert space
${\mathfrak {H}}_{\mathcal {B}}$
gives rise to a faithful covariant representation of
$(\mathcal {A},G,\alpha )$
on some truncation of
${\mathfrak {H}}_{\mathcal {B}} \otimes {\mathfrak {H}}$
(Corollary 3.13). It is our hope that Theorem 3.11 can be further utilized to present conditions under which certain properties of the fixed point algebra
$\mathcal {B}$
pass over to
$(\mathcal {A},G,\alpha )$
. Note that we do not address the problem of classifying free C
$^*$
-dynamical systems in this paper. In fact, this problem has been thoroughly treated in [Reference Schwieger and Wagner22] using different methods.
Following this introduction, the fundamental definitions and notations are presented in Section 2. The proofs are provided in Section 3, which is in essence divided into three parts: first, constructing an equivariant coaction from a given free C
$^*$
-dynamical system; second, identifying the corresponding invariants as a free C
$^*$
-dynamical system; and third, proving that the original and the derived free C
$^*$
-dynamical systems are isomorphic.
2 Preliminaries and notations
2.1 About tensor products
In this article, tensor products of C
$^*$
-algebras are taken with respect to the minimal tensor product, which is denoted by
$\otimes $
. Let
$\mathcal {A}$
,
$\mathcal {B}$
, and
${\mathcal {C}}$
be unital C
$^*$
-algebras. If there is no ambiguity, then we consider each one of them as a C
$^*$
-subalgebra of
$\mathcal {A} \otimes \mathcal {B} \otimes {\mathcal {C}}$
and extend maps on
$\mathcal {A}$
,
$\mathcal {B}$
, or
${\mathcal {C}}$
canonically by tensoring with the respective identity map. For the sake of clarity, we also make use of the leg numbering notation, for instance, given
$x \in \mathcal {A} \otimes {\mathcal {C}}$
, we write
$x_{13}$
to denote the corresponding element in
$\mathcal {A} \otimes \mathcal {B} \otimes {\mathcal {C}}$
.
2.2 About multiplier algebras
Let
$\mathcal {A}$
be a C
$^*$
-algebra. A linear operator
$m: \mathcal {A} \to \mathcal {A}$
is said to be a multiplier of
$\mathcal {A}$
if for each
$a \in \mathcal {A}$
there exists
$c \in \mathcal {A}$
such that
$a^* m(b) = c^* b$
for all
$b \in \mathcal {A}$
. The set of all multipliers of
$\mathcal {A}$
is a unital C
$^*$
-algebra which is denoted by
${\mathcal {M}}(\mathcal {A})$
and called the multiplier algebra of
$\mathcal {A}$
. For a faithful and nondegenerate
$^*$
-representation
${\pi : \mathcal {A} \to {\mathcal {L}}({\mathfrak {H}})}$
it may be identified with the following operator algebra:
$$ \begin{align*} {\mathcal{M}}(\mathcal{A}) = \{ m \in {\mathcal{L}}({\mathfrak{H}}) : (\forall a \in \mathcal{A}) \, ma, am \in \mathcal{A}\}. \end{align*} $$
Note that if
$\mathcal {A}$
is a unital C
$^*$
algebra, then
${\mathcal {M}}(\mathcal {A}) \cong \mathcal {A}$
. For a thorough treatment of multipliers we refer to [Reference Pedersen15, Section 3.12].
Lemma 2.1 Let
$\pi _{\mathcal {A}} : \mathcal {A} \to {\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}})$
and
$\pi _{\mathcal {B}} : \mathcal {B} \to {\mathcal {L}}({\mathfrak {H}}_{\mathcal {B}})$
be faithful and nondegenerate
$^*$
-homomorphisms of C
$^*$
-algebras
$\mathcal {A}$
and
$\mathcal {B,}$
respectively. Then
$$ \begin{align*} {\mathcal{M}}(\mathcal{A}) \otimes \mathcal{B} = ({\mathcal{L}}({\mathfrak{H}}_{\mathcal{A}}) \otimes \mathcal{B}) \cap {\mathcal{M}}(\mathcal{A} \otimes \mathcal{B}). \end{align*} $$
Proof Clearly,
${\mathcal {M}}(\mathcal {A}) \otimes \mathcal {B} \subseteq ({\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}}) \otimes \mathcal {B}) \cap {\mathcal {M}}(\mathcal {A} \otimes \mathcal {B})$
. Hence it suffices to establish the opposite inclusion. For this purpose, we first assume that
$\mathcal {B}$
is unital. Let
$x := \sum _{i=1}^n x_i \otimes b_i \in {\mathcal {M}}(\mathcal {A} \otimes \mathcal {B})$
for operators
$x_1,\ldots ,x_n \in {\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}})$
and linear independent elements
$b_1,\ldots b_n \in \mathcal {B}$
. Moreover, let
$\psi $
be a state on
$\mathcal {B}$
such that
$\psi (b_i^* b_i)> 0$
for all
$1 \le i \le n$
. Applying the Gram–Schmidt process, we may w. l. o. g. assume that
$\psi (b_i^* b_j) = \delta _{i,j}$
for all
$1 \le i,j \le n$
, i. e.,
$b_1, \dots , b_n$
is an orthonormal system. We write
$\omega _i(b) := \psi (b_i^* b)$
,
$b \in \mathcal {B}$
,
$1 \le i \le n$
, for the corresponding dual system of functionals on
$\mathcal {B}$
. Since
$x \in {\mathcal {M}}(\mathcal {A} \otimes \mathcal {B})$
, for each
$a \in \mathcal {A}$
we have
$x(a \otimes 1_{\mathcal {B}}) \in \mathcal {A} \otimes \mathcal {B}$
. Consequently,
$$ \begin{align*} x_i a = (\operatorname{\mathrm{id}} \otimes \omega_i(x)) a = \operatorname{\mathrm{id}} \otimes \omega_i (x (a \otimes 1_{\mathcal{B}})) \in \mathcal{A} \end{align*} $$
for all
$1 \le i \le n$
and
$a \in \mathcal {A}$
. Likewise, we find
$ax_i \in \mathcal {A}$
. It follows that
$x_i \in {\mathcal {M}}(\mathcal {A})$
for all
$1 \le i \le n$
, and hence
$x \in {\mathcal {M}}(\mathcal {A}) \otimes \mathcal {B}$
. Taking limits, we thus get
$$ \begin{align*} \bigl( {\mathcal{L}}({\mathfrak{H}}_A) \otimes \mathcal{B} \bigr) \cap {\mathcal{M}}(\mathcal{A} \otimes \mathcal{B}) \subseteq {\mathcal{M}}(\mathcal{A}) \otimes \mathcal{B.} \end{align*} $$
For non-unital
$\mathcal {B}$
we may replace
$1_{\mathcal {B}}$
by an approximate unit of
$\mathcal {B}$
and use similar arguments, the detailed verification being left to the reader.
2.3 About coactions of compact groups
Let G be a locally compact Hausdorff group. We denote by
$\lambda : G \to {\mathcal {U}} (L^2(G))$
the left regular representation given by
$(\lambda _gf)(h):=f(g^{-1}h)$
and by
$r : G \to {\mathcal {U}} (L^2(G))$
the right regular representation given by
$(r_gf)(h):=f(hg)$
. For
$f \in C_0(G)$
we write
$r(f)$
for the integrated form with respect to the right regular representation and consider
$C^*_r(G)$
as the norm closure of the
$^*$
-algebra
$r(C(G)) \subseteq {\mathcal {L}}(L^2(G))$
or, equivalently, as the fixed point algebra of
${\mathcal {K}}(L^2(G))$
under the adjoint action
$\operatorname {\mathrm {Ad}}[\lambda _g]$
,
$g\in G$
.
Usually, we consider
$C^*_r(G)$
as a quantum group with respect to the faithful and nondegenerate
$^*$
-homomorphism
$\delta _G: C^*_r(G) \to {\mathcal {M}}(C^*_r(G) \otimes C^*_r(G))$
defined by the integrated form of the diagonal representation
$G \ni g \mapsto r_g \otimes r_g \in {\mathcal {U}}(L^2(G) \otimes L^2(G))$
. Due to [Reference Landstad14, p. 255] (see also [Reference Stinespring25, p. 48]), the unitary
$W_G$
on
$L^2(G \times G) \cong L^2(G) \otimes L^2(G)$
defined by
$(W_G f)(g,h) := f(g,hg)$
implements the map
$\delta _G$
in the sense that
$\delta _G(x) = W_G^* \bigl ( x \otimes 1_G \bigr ) W_G$
for all
$x \in C^*_r(G)$
. We shall also utilize the fact that
$W_G \in {\mathcal {M}} \bigl ( C_0(G) \otimes C^*_r(G) \bigr )$
as well as the identities
$$ \begin{align} (W_G)_{23}(W_G)_{12}(W_G)_{13} &= (W_G)_{12}(W_G)_{23}, \end{align} $$
$$ \begin{align} (r_g \otimes 1_G) W_G &= W_G (r_g \otimes r_g) \qquad\qquad\ \forall g \in G, \end{align} $$
$$ \begin{align} (1_G \otimes \lambda_g) W_G &= W_G (1_G \otimes \lambda_g) \qquad\qquad \forall g \in G, \end{align} $$
$$ \begin{align}(\lambda_g \otimes r_g) W_G &= W_G (\lambda_g \otimes 1_G) \qquad\qquad \forall g \in G. \end{align} $$
More generally, we are concerned with coactions of compact groups on C
$^*$
-algebras and refer to [Reference Echterhoff, Kaliszewski, Quigg and Raeburn9, Appendix A] for a detailed discussion on the subject. However, for expediency we now repeat the basic definition.
Definition 2.1 A coaction of a locally compact Hausdorff group G on a C
$^*$
-algebra
$\mathcal {A}$
is a faithful and nondegenerate
$^*$
-homomorphism
$\delta : \mathcal {A} \to {\mathcal {M}}(\mathcal {A} \otimes C^*_r(G))$
satisfying the coaction identity
$$ \begin{align} (\delta \otimes \operatorname{\mathrm{id}}) \circ \delta = (\operatorname{\mathrm{id}} \otimes \delta_G) \circ \delta. \end{align} $$
As coactions take values in multiplier algebras, the coaction identity involves the extensions of the maps
$\delta \otimes \operatorname {\mathrm {id}}$
and
$\operatorname {\mathrm {id}} \otimes \delta _G$
to the multiplier algebras
${\mathcal {M}}(\mathcal {A} \otimes C^*_r(G))$
and
${\mathcal {M}}(\mathcal {A} \otimes C^*_r(G) \otimes C^*_r(G))$
in domain and codomain, respectively.
Remark 2.2 Let
$\mathbb {G}$
be a quantum group and let
$\mathcal {A}$
be a C
$^*$
-algebra. We wish to mention that in the quantum group literature, a map
$\delta : \mathcal {A} \to {\mathcal {M}}(\mathcal {A} \otimes C^*_r(\mathbb {G}))$
satisfying (2.5) is called “a coaction of
$C^*_r(\mathbb {G})$
on
$\mathcal {A}$
” or “a coaction of the dual quantum group
$\widehat {\mathbb {G}}$
on
$\mathcal {A}$
”.
2.4 About free C
$^*$
-dynamical systems
One of the key tools utilized in this article is a characterization of freeness that we previously introduced in [Reference Schwieger and Wagner22, Lemma 3.2], namely that a C
$^*$
-dynamical system
$(\mathcal {A},G,\alpha )$
is free if and only if for each irreducible representation
$(\sigma ,V_\sigma )$
of G there exists a finite-dimensional Hilbert space
${\mathfrak {H}}_\sigma $
and an isometry
$s(\sigma ) \in \mathcal {A} \otimes {\mathcal {L}}(V_\sigma ,{\mathfrak {H}}_\sigma )$
satisfying
$\alpha _g(s(\sigma ))=s(\sigma ) (1_{\mathcal {A}} \otimes \sigma _g)$
for all
$g \in G$
. However, to simplify notation we patch this family of isometries together and use the following characterization instead.
Lemma 2.3 (see [Reference Schwieger and Wagner24, Lemma 3.1])
For a C
$^*$
-dynamical system
$({\mathcal {A}}, G, \alpha )$
the following statements are equivalent:
-
(a)
$(\mathcal {A}, G, \alpha )$
is free. -
(b) There is a unitary representation
$\mu : G \to {\mathcal {U}}({\mathfrak {H}})$
with finite-dimensional multiplicity spaces and, given any faithful covariant representation
$(\pi ,u)$
of
$(\mathcal {A},G,\alpha )$
on some Hilbert space
${\mathfrak {H}}_{\mathcal {A}}$
, an isometry
$S \in {\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}} \otimes L^2(G), {\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}})$
satisfying (2.6)
$$ \begin{align} \!S {\mathcal{A}} \otimes {\mathcal{K}}(L^2(G)) &\subseteq {\mathcal{A}} \otimes {\mathcal{K}}(L^2(G),{\mathfrak{H}}),\qquad\qquad \end{align} $$
(2.7)
$$ \begin{align}\quad\qquad (u_g \otimes 1_{\mathfrak{H}}) S &= S (u_g \otimes r_g) && \forall g \in G, \end{align} $$
(2.8)
$$ \begin{align} \quad\qquad(1_{\mathcal{A}} \otimes \mu_g) S &= S (1_{\mathcal{A}} \otimes \lambda_g) && \forall g \in G. \end{align} $$
Here, we do not distinguish between
$\mathcal {A}$
and
$\pi (\mathcal {A}) \subseteq {\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}})$
for sake of brevity. Furthermore, the tensor product
$\mathcal {A} \otimes {\mathcal {K}}(L^2(G),{\mathfrak {H}})$
is closed with respect to the operator norm, where
${\mathcal {K}}(L^2(G),{\mathfrak {H}})$
is regarded as the respective corner of
${\mathcal {K}}(L^2(G) \oplus {\mathfrak {H}})$
.
The adjoint
$S^* \in {\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}}, {\mathfrak {H}}_{\mathcal {A}} \otimes L^2(G))$
of the isometry S in Lemma 2.3 satisfies
$S^* \mathcal {A} \otimes {\mathcal {K}}({\mathfrak {H}}) \subseteq \mathcal {A} \otimes {\mathcal {K}}({\mathfrak {H}},L^2(G))$
or, equivalently,
$$ \begin{align} {\mathcal{A}} \otimes {\mathcal{K}}({\mathfrak{H}}) S \subseteq {\mathcal{A}} \otimes {\mathcal{K}}(L^2(G),{\mathfrak{H}}). \end{align} $$
For this reason, we can assert that S is, in fact, a multiplier for
$\mathcal {A} \otimes {\mathcal {K}}(L^2(G) \oplus {\mathfrak {H}})$
, i. e.,
$S \in {\mathcal {M}}(\mathcal {A} \otimes {\mathcal {K}}( L^2(G) \oplus {\mathfrak {H}}))$
, with
$(1_{\mathcal {A}} \otimes p_{L^2(G)}) S = 0 = S (1_{\mathcal {A}} \otimes p_{\mathfrak {H}})$
, where
$p_{\mathfrak {H}}$
and
$p_{L^2(G)}$
denote the canonical projections onto
${\mathfrak {H}}$
and
$L^2(G),$
respectively.
A particular simple class of free actions is given by so-called cleft actions (see [Reference Schwieger and Wagner21]). Regarded as noncommutative principal bundles, these actions are essentially characterized by the fact that all associated noncommutative vector bundles are trivial. For convenience of the reader we now recall the definition. Indeed, we call a C
$^*$
-dynamical system
$({\mathcal {A}},G,\alpha )$
cleft if there is a unitary
$U \in {\mathcal {M}}(\mathcal {A} \otimes C^*_r(G))$
such that
$\bar {\alpha }_g(U) = U (1_{\mathcal {A}} \otimes r_g)$
for all
$g \in G$
. Here,
$\bar {\alpha }_g$
,
$g\in G$
, denotes the strictly continuous extension of
$\alpha _g \otimes 1_G$
,
$g\in G$
, to
${\mathcal {M}}(\mathcal {A} \otimes C^*_r(G))$
(see, e. g., [Reference Pedersen15, Proposition 3.12.10]), which is continuous for the strict topology. It is clear that each cleft C
$^*$
-dynamical system is free with a possible choice for
$\mu $
and
${\mathfrak {H}}$
given by
$\lambda $
and
$L^2(G),$
respectively.
3 The realization
To commence, we fix a free C
$^*$
-dynamical system
$(\mathcal {A},G,\alpha )$
with fixed point algebra
$\mathcal {B}$
along with a faithful covariant representation
$(\pi ,u)$
thereof on some Hilbert space
${\mathfrak {H}}_{\mathcal {A}}$
. By Lemma 2.3, there is a unitary representation
$\mu : G \to {\mathcal {U}}({\mathfrak {H}})$
with finite-dimensional multiplicity spaces and an isometry
$S \in {\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}} \otimes L^2(G), {\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}})$
satisfying (2.6), (2.7), (2.8), and (2.9).
We see at once that
$P := SS^* \in {\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}})$
is a projection satisfying
$(u_g \otimes \mu _h) P = P (u_g \otimes \mu _h)$
for all
$g,h \in G$
. Moreover, we have
$P \in {\mathcal {M}}(\mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}}))$
, which is easily checked. In what follows, the central object of interest is the corresponding corner
$$ \begin{align} \mathcal{D} := P ( \mathcal{B} \otimes {\mathcal{K}}({\mathfrak{H}}) ) P \subseteq \mathcal{B} \otimes {\mathcal{K}}({\mathfrak{H}}) \cap P{\mathcal{L}}({\mathfrak{H}}_{\mathcal{A}} \otimes {\mathfrak{H}})P. \end{align} $$
Clearly,
$\mathcal {D}$
is pointwise fixed by
$\operatorname {\mathrm {Ad}}[u_g \otimes 1_{\mathfrak {H}}]$
,
$g\in G$
. Furthermore,
$\operatorname {\mathrm {Ad}}[1_{\mathcal {A}} \otimes \mu _g]$
,
${g\in G}$
, is a strongly continuous action on
$\mathcal {D}$
, i. e.,
$(\mathcal {D},G,\operatorname {\mathrm {Ad}}[1_{\mathcal {A}} \otimes \mu ])$
is a C
$^*$
-dynamical system. Our first task is to construct a coaction of G on
$\mathcal {D}$
in terms of S and
$W_G$
(see Section 2.3). We begin with a series of lemmas.
Lemma 3.1 For the element
$W_S := S_{12} (W_G)_{23} S_{12}^*$
in
${\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}} \otimes L^2(G))$
the following assertions hold:
-
1.
$W_S$
is a partial isometry with initial and final projection
$P \otimes 1_G$
. -
2.
$(u_g \otimes \mu _h \otimes r_h) W_S = W_S (u_g \otimes \mu _h \otimes r_g)$
for all
$g,h \in G$
. -
3.
$W_S \in {\mathcal {M}}(\mathcal {A} \otimes {\mathcal {K}}({\mathfrak {H}}) \otimes C^*_r(G))$
.Footnote
1
Proof
-
1. Since S is an isometry, we see at once that the initial and final projections are given by
$W_{S}^* W_S = P \otimes 1_G$
and
$W_S W_S^* = P \otimes 1_G$
, respectively. -
2. Let
$g,h \in G$
. Then an easy verification yields
$$ \begin{align*} (u_g \otimes \mu_h \otimes r_h) W_S &= (u_g \otimes \mu_h \otimes r_h) S_{12} (W_G)_{23} S_{12}^* \\\overset{({2.7}),({2.8})}{=}& S_{12} (u_g \otimes r_g \lambda_h \otimes r_h) (W_G)_{23} S_{12}^* \\\overset{({2.2}),({2.4})}{=}& S_{12} (W_G)_{23} (u_g \otimes r_g \lambda_h \otimes r_g) S_{12}^* \\\overset{({2.7}),({2.8})}{=}& S_{12} (W_G)_{23} S_{12}^* (u_g \otimes \mu_h \otimes r_g) = W_S (u_g \otimes \mu_h \otimes r_g). \end{align*} $$
-
3. From (2.6), we deduce that
$S \mathcal {A} \otimes {\mathcal {K}}({\mathfrak {H}},L^2(G)) \subseteq \mathcal {A} \otimes {\mathcal {K}}({\mathfrak {H}})$
. Combining this once more with (2.6) and the fact that
$W_G \in {\mathcal {L}}(L^2(G)) \otimes {\mathcal {L}}(L^2(G))$
yields
$$ \begin{align*} W_S \in {\mathcal{M}}({\mathcal{A}} \otimes {\mathcal{K}}({\mathfrak{H}}) \otimes {\mathcal{K}}(L^2(G))). \end{align*} $$
But since
$C^*_r(G)$
is the fixed point algebra of
${\mathcal {K}}(L^2(G))$
under
$\operatorname {\mathrm {Ad}}[\lambda _g]$
,
$g\in G$
, and
$W_G$
satisfies (2.3), it may further be concluded that
$W_S \in {\mathcal {M}}(\mathcal {A} \otimes {\mathcal {K}}({\mathfrak {H}}) \otimes C^*_r(G))$
.
Here and subsequently, we do not distinguish between
${\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}} \otimes L^2(G))$
and
${\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}}) \otimes {\mathcal {L}}(L^2(G))$
for simplicity of notation.
Lemma 3.2 For the map
$$ \begin{align*} \delta: {\mathcal{L}}({\mathfrak{H}}_{\mathcal{A}} \otimes {\mathfrak{H}}) \to {\mathcal{L}}({\mathfrak{H}}_{\mathcal{A}} \otimes {\mathfrak{H}} \otimes L^2(G)), && \delta(x) : = \operatorname{\mathrm{Ad}}[W_S](x \otimes 1_G) := W_S (x \otimes 1_G) W_S^* \end{align*} $$
the following assertions hold:
-
1.
$\delta $
is equivariant w.r.t.
$\operatorname {\mathrm {Ad}}[u_g \otimes \mu _h]$
,
$g,h \in G$
on
${\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}})$
and
$\operatorname {\mathrm {Ad}}[u_g \otimes \mu _h \otimes r_h]$
,
$g,h \in G$
, on
${\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}} \otimes L^2(G))$
. -
2.
$\delta (x) \in {\mathcal {M}}(\mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}}) \otimes C^*_r(G))$
for all
$x \in \mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}})$
. -
3.
$\delta $
satisfies the coaction identity
$(\delta \otimes \operatorname {\mathrm {id}}) \circ \delta = (\operatorname {\mathrm {id}} \otimes \delta _G) \circ \delta $
.
Proof
-
1. Let
$g,h \in G$
and let
$x \in {\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}})$
. Then where we have used Lemma 3.1.2 for the third-to-last equality.
$$ \begin{align*} \delta(\operatorname{\mathrm{Ad}}[u_g \otimes \mu_h](x)) &= \operatorname{\mathrm{Ad}}[W_S](\operatorname{\mathrm{Ad}}[u_g \otimes \mu_h \otimes 1_G](x \otimes 1_G)) \\ &= \operatorname{\mathrm{Ad}}[W_S (u_g \otimes \mu_h \otimes 1_G)](x \otimes 1_G) \\ &= \operatorname{\mathrm{Ad}}[W_S (u_g \otimes \mu_h \otimes r_g)](x \otimes 1_G) \\ &= \operatorname{\mathrm{Ad}}[(u_g \otimes \mu_h \otimes r_h ) W_S](x \otimes 1_G) \\ &= \operatorname{\mathrm{Ad}}[u_g \otimes \mu_h \otimes r_h ](\operatorname{\mathrm{Ad}}[W_S](x \otimes 1_G)) \\ &= \operatorname{\mathrm{Ad}}[u_g \otimes \mu_h \otimes r_h] (\delta(x)), \end{align*} $$
-
2. Let
$x \in \mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}})$
and let
$y \in \mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}}) \otimes C^*_r(G)$
. Applying Lemma 3.1.3 yields
$\delta (x)y \in \mathcal {A} \otimes {\mathcal {K}}({\mathfrak {H}}) \otimes C^*_r(G)$
. Moreover, the equivariance of
$\delta $
implies that for all
$$ \begin{align*} \operatorname{\mathrm{Ad}}[u_g \otimes 1_{\mathfrak{H}} \otimes 1_G](\delta(x)y) &= \operatorname{\mathrm{Ad}}[u_g \otimes 1_{\mathfrak{H}} \otimes 1_G](\delta(x)) \operatorname{\mathrm{Ad}}[u_g \otimes 1_{\mathfrak{H}} \otimes 1_G](y) \\ &= \delta(\operatorname{\mathrm{Ad}}[u_g \otimes 1_{\mathfrak{H}}](x)) y = \delta(x)y \end{align*} $$
$g \in G$
, i. e.,
$\delta (x)y \in \mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}}) \otimes C^*_r(G)$
. Likewise, we obtain
$y\delta (x) \in \mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}}) \otimes C^*_r(G)$
. It follows that
$\delta (x) \in {\mathcal {M}}(\mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}}) \otimes C^*_r(G))$
as required.
-
3. To demonstrate that
$(\delta \otimes \operatorname {\mathrm {id}}) \circ \delta = (\operatorname {\mathrm {id}} \otimes \delta _G) \circ \delta $
, we first evaluate both sides of the identity. Indeed, for each
$x \in {\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}})$
we have
$$ \begin{align*} \delta \otimes \operatorname{\mathrm{id}} (\delta(x)) &= \delta \otimes \operatorname{\mathrm{id}} (\operatorname{\mathrm{Ad}}[W_S] (x \otimes 1_G)) \\ &= \operatorname{\mathrm{Ad}}[(W_S)_{123}(W_S)_{124}](x \otimes 1_G) \\ &= \operatorname{\mathrm{Ad}}[S_{12}(W_G)_{23}(W_G)_{24} S_{12}^*](x \otimes 1_G). \end{align*} $$
Moreover, the right hand side of the identity becomes
$$ \begin{align*} \operatorname{\mathrm{id}} \otimes \delta_G (\delta(x)) &= \operatorname{\mathrm{id}} \otimes \delta_G (\operatorname{\mathrm{Ad}}[W_S](x \otimes 1_G)) \\ &= \operatorname{\mathrm{Ad}}[(W_G)_{34}^*(W_S)_{123}](x \otimes 1_G) \\ &= \operatorname{\mathrm{Ad}}[(W_G)_{34}^*(W_S)_{123}(W_G)_{34}](x \otimes 1_G) \\ &= \operatorname{\mathrm{Ad}}[(W_G)_{34}^*S_{12}(W_G)_{23}S_{12}^*(W_G)_{34}](x \otimes 1_G) \\ &= \operatorname{\mathrm{Ad}}[S_{12}(W_G)_{34}^*(W_G)_{23}(W_G)_{34}S_{12}^*](x \otimes 1_G). \end{align*} $$
Comparing these two expressions, we see that the claim will be proved once we show that
$(W_G)_{12}(W_G)_{13} = (W_G)_{23}^*(W_G)_{12}(W_G)_{23}$
. But this is clear from (2.1).
Remark 3.3 We stress that the map
$\delta $
is, in general, not a
$^*$
-homomorphism.
To proceed, we put
${\mathfrak {H}}_P := P ({\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}})$
, identify
${\mathcal {L}}({\mathfrak {H}}_P)$
with
$P {\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}}) P$
by means of the map
$P {\mathcal {L}}({\mathfrak {H}}_{\mathcal {A}} \otimes {\mathfrak {H}}) P \to {\mathcal {L}}({\mathfrak {H}}_P)$
,
$x \mapsto x |_{{\mathfrak {H}}_P}^{{\mathfrak {H}}_P}$
, and note that
$\mathcal {D} \subseteq {\mathcal {L}}({\mathfrak {H}}_P)$
is nondegenerate.
Lemma 3.4 Restricting
$\delta $
to
${\mathcal {D}}$
(see (3.1)) yields a map
$$ \begin{align} \delta_{\mathcal{D}} : {\mathcal{D}} \rightarrow {\mathcal{M}}({\mathcal{D}} \otimes C^*_r(G)) \subseteq {\mathcal{L}}({\mathfrak{H}}_P \otimes L^2(G)), \end{align} $$
for which the following assertions hold:
-
1.
$\delta _{\mathcal {D}}$
is a
$^*$
-homomorphism. -
2.
$\delta _{\mathcal {D}}$
is faithful. -
3.
$\delta _{\mathcal {D}}$
is nondegenerate. -
4.
$\delta _{\mathcal {D}}$
satisfies the coaction identity
$(\delta _{\mathcal {D}} \otimes \operatorname {\mathrm {id}}) \circ \delta _{\mathcal {D}} = (\operatorname {\mathrm {id}} \otimes \delta _G) \circ \delta _{\mathcal {D}}$
. -
5.
$\delta _{\mathcal {D}}$
is equivariant w.r.t.
$\operatorname {\mathrm {Ad}}[1_{\mathcal {A}} \otimes \mu _g]$
,
$g\in G$
, on
${\mathcal {D}}$
and
$\operatorname {\mathrm {Ad}}[1_{\mathcal {A}} \otimes \mu _g \otimes r_g]$
,
$g\in G$
, on
${\mathcal {M}}({\mathcal {D}} \otimes C^*_r(G))$
.
Proof
-
0. We first show that
$\delta _{\mathcal {D}}$
is well defined. For this purpose, let
${x=PyP \in \mathcal {D}}$
for some
$y \in \mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}})$
and let
$z \in {\mathcal {D}} \otimes C^*_r(G) \subseteq \mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}}) \otimes C^*_r(G)$
. By Lemma 3.2.2,
$\delta (y) z \in \mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}}) \otimes C^*_r(G)$
. Moreover,
$$ \begin{align*} \delta(x)z = (P \otimes 1_G) \delta(y) (P \otimes 1_G) z = (P \otimes 1_G) \delta(y) z (P \otimes 1_G). \end{align*} $$
Consequently,
$$ \begin{align*} \delta(x)z \in (P \otimes 1_G) \mathcal{B} \otimes {\mathcal{K}}({\mathfrak{H}}) \otimes C^*_r(G) (P \otimes 1_G) = {\mathcal{D}} \otimes C^*_r(G). \end{align*} $$
Because the same conclusion can be drawn for the element
$z\delta (x)$
, it follows that
$\delta (x) \in {\mathcal {M}}({\mathcal {D}} \otimes C^*_r(G))$
. -
1. It is immediate that
$\delta _{\mathcal {D}}$
is a
$^*$
-map. To see that
$\delta _{\mathcal {D}}$
is an algebra map, let
$x,y \in {\mathcal {D}}$
. Then
$$ \begin{align*} \delta_{{\mathcal{D}}}(xy) &= W_S (xy \otimes 1_G ) W_S^* = W_S (xPy \otimes 1_G ) W_S^* \\ &= W_S (x \otimes 1_G) (P \otimes 1_G) (y \otimes 1_G ) W_S^* \\ &=\underbrace{W_S (x \otimes 1_G) W_S^*}_{=\delta_{{\mathcal{D}}}(x)} \underbrace{W_S (y \otimes 1_G ) W_S^*}_{=\delta_{{\mathcal{D}}}(y)}. \end{align*} $$
-
2. Let
$x \in {\mathcal {D}}$
such that
$\delta (x) =0$
. Multiplying the latter equation by
$W_S^*$
from the left and by
$W_S$
from the right gives
$PxP \otimes 1_G = x \otimes 1_G = 0$
. Hence
$x=0$
as required. -
3. Let
$\{ T_i \}$
be a bounded approximate identity for
${\mathcal {K}}({\mathfrak {H}})$
. Then
$\{\delta (P(1_{\mathcal {B}} \otimes T_i)P)\}$
is a bounded approximate identity for
${\mathcal {D}} \otimes C^*_r(G)$
, i. e.,
$\delta (P(1_{\mathcal {B}} \otimes T_i)P)$
converges to
$1$
in the strict topology of
${\mathcal {M}} ({\mathcal {D}} \otimes C^*_r(G))$
. In particular,
$\delta ({\mathcal {D}}) ({\mathcal {D}} \otimes C^*_r(G))$
is dense in
${\mathcal {D}} \otimes C^*_r(G)$
which amounts to saying that
$\delta _{\mathcal {D}}$
is nondegenerate. As a matter of fact, we even have equality by the Cohen factorization theorem (see, e. g., [Reference Raeburn and Williams18, Proposition 2.33]). -
4. Since
$\delta _{\mathcal {D}}$
is nondegenerate,
$\delta _D \otimes \operatorname {\mathrm {id}} : \mathcal {D} \otimes C^*_r(G) \to {\mathcal {M}}(\mathcal {D} \otimes C^*_r(G) \otimes C^*_r(G))$
uniquely extends to a
$^*$
-homomorphisms with domain
${\mathcal {M}}(\mathcal {D} \otimes C^*_r(G))$
which, by uniqueness, must agree with the restriction–corestriction
$$ \begin{align*} \delta \otimes \operatorname{\mathrm{id}} \restriction_{{\mathcal{M}}(\mathcal{D} \otimes C^*_r(G))}^{{\mathcal{M}}(\mathcal{D} \otimes C^*_r(G) \otimes C^*_r(G))}. \end{align*} $$
This establishes the statement when combined with Lemma 3.2.3
-
5. By Lemma 3.2.1, it suffices to note that
$\mathcal {D}$
is invariant under
$\operatorname {\mathrm {Ad}}[1_{\mathcal {A}} \otimes \mu _g]$
,
$g\in G$
, and that
${\mathcal {M}}(\mathcal {D} \otimes C^*_r(G))$
is invariant under
$\operatorname {\mathrm {Ad}}[1_{\mathcal {A}} \otimes \mu _g \otimes r_g]$
,
$g\in G$
.
Remark 3.5 Much the same proof as above also works for the restriction of
$\delta $
to
${\mathcal {L}}({\mathfrak {H}}_P)$
in domain and codomain, i. e., the map
${\mathcal {L}}({\mathfrak {H}}_P) \to {\mathcal {L}}({\mathfrak {H}}_P \otimes L^2(G))$
,
$x \mapsto \delta (x)$
.
Corollary 3.6
$\delta _{\mathcal {D}}$
is a coaction of G on
$\mathcal {D}$
that is equivariant w.r.t.
$\operatorname {\mathrm {Ad}}[1_{\mathcal {A}} \otimes \mu _g]$
,
$g\in G$
, on
${\mathcal {D}}$
and
$\operatorname {\mathrm {Ad}}[1_{\mathcal {A}} \otimes \mu _g \otimes r_g]$
,
$g\in G$
, on
${\mathcal {M}}({\mathcal {D}} \otimes C^*_r(G))$
.
The task is now to recover
$(\mathcal {A},G,\alpha )$
from
$\delta _{\mathcal {D}}$
. For this purpose, we recall that
${\mathcal {M}}(\mathcal {D})$
may be identified with
$P {\mathcal {M}}(\mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}})) P$
(see [Reference Blackadar3, Section II.7.3.14]) and consider
$\delta _{\mathcal {D}}$
’s fixed point algebra:
$$ \begin{align*} \text{Fix}(\delta_{\mathcal{D}}) := \{ x \in {\mathcal{M}}(\mathcal{D}) : \delta_{\mathcal{D}}(x) = x \otimes 1_G \} \subseteq {\mathcal{L}}({\mathfrak{H}}_P). \end{align*} $$
Since
$P \in {\mathcal {M}}(\mathcal {D})$
and
$\delta _{\mathcal {D}}(P) = P \otimes 1_G$
, we see that
$\text {Fix}(\delta _{\mathcal {D}})$
is a unital C
$^*$
-subalgebra of
${\mathcal {M}}(\mathcal {D})$
. Furthermore,
$\text {Fix}(\delta _{\mathcal {D}})$
is invariant under
$\operatorname {\mathrm {Ad}} [1_{\mathcal {A}} \otimes \mu _g]$
,
$g\in G$
, as is easy to check. Unfortunately, the resulting action
$G \to \operatorname {\mathrm {Aut}}(\text {Fix}(\delta _{\mathcal {D}}))$
is unlikely to be continuous for the norm topology on
${\mathcal {M}}(\mathcal {D})$
. The best we can do is say that it is continuous for the strict topology. This inconvenience can be avoided by only taking into account those elements
$x \in \text {Fix}(\delta _{\mathcal {D}})$
for which the map
$G \to \text {Fix}(\delta _{\mathcal {D}})$
,
$g \mapsto \operatorname {\mathrm {Ad}} [1_{\mathcal {A}} \otimes \mu _g](x)$
is norm continuous, i. e.,
$$ \begin{align*} \tilde{\mathcal{A}} := \{ x \in \text{Fix}(\delta_{\mathcal{D}}) : G \ni g \mapsto \operatorname{\mathrm{Ad}}[1_{\mathcal{A}} \otimes \mu_g](x) \in \text{Fix}(\delta_{\mathcal{D}}) \,\,\, \text{norm continuous} \}. \end{align*} $$
Now, a straightforward verification shows that
$\tilde {\mathcal {A}}$
is a unital C
$^*$
-subalgebra of
$\text {Fix}(\delta _{\mathcal {D}})$
with unit
$1_{\tilde {\mathcal {A}}} := P$
on which
$\operatorname {\mathrm {Ad}} [1_{\mathcal {A}} \otimes \mu _g]$
,
$g \in G$
, acts strongly continuous. In particular, writing
$\tilde {\alpha }_g$
,
$g \in G$
, for the restriction of
$\operatorname {\mathrm {Ad}} [1_{\mathcal {A}} \otimes \mu _g]$
,
$g \in G$
, to
$\tilde {\mathcal {A}}$
in both domain and codomain, we conclude that
$(\tilde {\mathcal {A}},G,\tilde {\alpha })$
is a concrete C
$^*$
-dynamical system on
${\mathfrak {H}}_P$
.
Lemma 3.7 For the element
$\tilde {S} := S_{12} S_{13} (W_G)_{23}^* S_{12}^* \in {\mathcal {L}}({\mathfrak {H}}_P \otimes L^2(G), {\mathfrak {H}}_P \otimes {\mathfrak {H}})$
the following assertions hold:
-
1.
$\tilde {S}$
is an isometry satisfying
$\delta \otimes \operatorname {\mathrm {id}} (\tilde {S}) = \tilde {S}_{124}$
. -
2.
$(u_g \otimes \mu _h \otimes \mu _k) \tilde {S} = \tilde {S} (u_g \otimes \mu _h \otimes r_h \lambda _k)$
for all
$g,h,k \in G$
. -
3.
$\tilde {S} \mathcal {D} \otimes {\mathcal {K}}(L^2(G)) \subseteq \mathcal {D} \otimes {\mathcal {K}}(L^2(G),{\mathfrak {H}})$
and
$\mathcal {D} \otimes {\mathcal {K}}({\mathfrak {H}}) \tilde {S} \subseteq \mathcal {D} \otimes {\mathcal {K}}(L^2(G),{\mathfrak {H}})$
, i.e.,
$\tilde {S} \in {\mathcal {M}}(\mathcal {D} \otimes {\mathcal {K}}(L^2(G) \oplus {\mathfrak {H}}))$
such that
$(1_{{\mathfrak {H}}_P} \otimes p_{L^2(G)}) \tilde {S} = 0 = \tilde {S} (1_{{\mathfrak {H}}_P} \otimes p_{{\mathfrak {H}}})$
, where
$p_{{\mathfrak {H}}}$
and
$p_{L^2(G)}$
denote the canonical projections onto
${\mathfrak {H}}$
and
$L^2(G)$
, respectively (see the discussion below Lemma 2.3). -
4.
$\tilde {S} {\mathcal {M}}(\mathcal {D}) \otimes {\mathcal {K}}(L^2(G)) \subseteq {\mathcal {M}}(\mathcal {D}) \otimes {\mathcal {K}}(L^2(G),{\mathfrak {H}})$
.
Proof
-
1. We have
$\tilde {S}^*\tilde {S} = S_{12} (W_G)_{23} S_{13}^* S_{12}^* S_{12} S_{13} (W_G)_{23}^* S_{12}^* = S_{12} S_{12}^* = 1_{\tilde {\mathcal {A}}} \otimes 1_G$
, i. e.,
$\tilde {S}$
is an isometry. Furthermore, it is easily deduced that where the second-to-last equality is due to the fact that
$$ \begin{align*} \delta \otimes \operatorname{\mathrm{id}} (\tilde{S}) &= \operatorname{\mathrm{Ad}}[ S_{12} (W_G)_{23} S_{12}^*] (\tilde{S}_{124}) \\ &=S_{12} (W_G)_{23} S_{12}^* S_{12} S_{14} (W_G)_{24}^* S_{12}^* S_{12} (W_G)_{23}^* S_{12}^* \\ &= S_{12} S_{14} (W_G)_{23} (W_G)_{24}^* (W_G)_{23}^* S_{12}^* \\ &= S_{12} S_{14} (W_G)_{24}^* S_{12}^* = \tilde{S}_{124}, \end{align*} $$
$W_G \in L^\infty (G) \otimes {\mathcal {L}}(L^2(G))$
.
-
2. Let
$g,h,k \in G$
. Then
$$ \begin{align*} (u_g \otimes \mu_h \otimes \mu_k) \tilde{S} &= (u_g \otimes \mu_h \otimes \mu_k) S_{12} S_{13} (W_G)_{23}^* S_{12}^* \\\overset{({2.7}),({2.8})}{=}& S_{12} S_{13} (u_g \otimes r_g \lambda_h \otimes r_g \lambda_k) (W_G)_{23}^* S_{12}^* \\\overset{({2.2}),({2.3}),({2.4})}{=}& S_{12} S_{13} (W_G)_{23}^* (u_g \otimes r_g \lambda_h \otimes r_h \lambda_k) S_{12}^* \\\overset{({2.7}),({2.8})}{=}& S_{12} S_{13} (W_G)_{23}^* S_{12}^* (u_g \otimes \mu_h \otimes r_h \lambda_k) = \tilde{S} (u_g \otimes \mu_h \otimes r_h \lambda_k). \end{align*} $$
-
3. Using successively the identity
$\tilde {S} = S_{12} S_{13} S_{12}^* W_S^*$
, the inclusion
$\mathcal {D} \subseteq \mathcal {A} \otimes {\mathcal {K}}({\mathfrak {H}})$
, Lemma 3.1.3, and three times (2.6) yields
$$ \begin{align*} \tilde{S} \mathcal{D} \otimes {\mathcal{K}}(L^2(G)) &= S_{12} S_{13} S_{12}^* W_S\mathcal{D} \otimes {\mathcal{K}}(L^2(G)) \\ &\subseteq S_{12} S_{13} S_{12}^* \mathcal{A} \otimes {\mathcal{K}}({\mathfrak{H}}) \otimes {\mathcal{K}}(L^2(G)) \\ &\subseteq \mathcal{A} \otimes {\mathcal{K}}({\mathfrak{H}}) \otimes {\mathcal{K}}(L^2(G),{\mathfrak{H}}). \end{align*} $$
As
$(u_g \otimes 1_{\mathfrak {H}} \otimes 1_{\mathfrak {H}}) \tilde {S} x= \tilde {S} x (u_g \otimes 1_{\mathfrak {H}} \otimes 1_G)$
for all
$g \in G$
and
$x \in \mathcal {D} \otimes {\mathcal {K}}(L^2(G))$
, which is due to the second part of the lemma and the fact that
$\mathcal {D}$
is fixed under
$\operatorname {\mathrm {Ad}}[u_g \otimes 1_{\mathfrak {H}}]$
,
$g \in G$
, we further conclude that and hence that
$$ \begin{align*} \tilde{S} \mathcal{D} \otimes {\mathcal{K}}(L^2(G)) \subseteq \mathcal{B} \otimes {\mathcal{K}}({\mathfrak{H}}) \otimes {\mathcal{K}}(L^2(G),{\mathfrak{H}}), \end{align*} $$
$$ \begin{gather*} \tilde{S} \mathcal{D} \otimes {\mathcal{K}}(L^2(G)) = (P)_{12} \tilde{S} \mathcal{D} \otimes {\mathcal{K}}(L^2(G)) (P)_{12} \\ \subseteq (P)_{12} \mathcal{B} \otimes {\mathcal{K}}({\mathfrak{H}}) \otimes {\mathcal{K}}(L^2(G),{\mathfrak{H}}) (P)_{12} = \mathcal{D} \otimes {\mathcal{K}}(L^2(G),{\mathfrak{H}}). \end{gather*} $$
In the same manner, we can see that
$\mathcal {D} \otimes {\mathcal {K}}({\mathfrak {H}}) \tilde {S} \subseteq \mathcal {D} \otimes {\mathcal {K}}(L^2(G),{\mathfrak {H}})$
. From this it follows that
$\tilde {S} \in {\mathcal {M}}(\mathcal {D} \otimes {\mathcal {K}}(L^2(G) \oplus {\mathfrak {H}}))$
such that
$(1_{{\mathfrak {H}}_P} \otimes p_{L^2(G)}) S = 0 =S (1_{{\mathfrak {H}}_P} \otimes p_{\mathfrak {H}})$
as claimed. -
4. This is clear from Lemma 2.1, because
by the third part of the lemma.
$$ \begin{align*} \tilde{S} {\mathcal{M}}(D) \otimes {\mathcal{K}}(L^2(G)) &\subseteq {\mathcal{L}}({\mathfrak{H}}_P) \otimes {\mathcal{K}}(L^2(G) \oplus {\mathfrak{H}})\\ &\mathrm{and}\\ \tilde{S} {\mathcal{M}}(D) \otimes {\mathcal{K}}(L^2(G)) &\subseteq {\mathcal{M}}(\mathcal{D} \otimes {\mathcal{K}}(L^2(G) \oplus {\mathfrak{H}})) {\mathcal{M}}(\mathcal{D} \otimes {\mathcal{K}}(L^2(G) \oplus {\mathfrak{H}})) \\ &\subseteq {\mathcal{M}}(\mathcal{D} \otimes {\mathcal{K}}(L^2(G) \oplus {\mathfrak{H}})) \end{align*} $$
Corollary 3.8 The C
$^*$
-dynamical system
$(\tilde {\mathcal {A}},G,\tilde {\alpha })$
is free.
Proof Our strategy of proof is to apply Lemma 2.3. Due to the first two statements of Lemma 3.7, we can already assert that
$\tilde {S} \in {\mathcal {L}}({\mathfrak {H}}_P \otimes L^2(G), {\mathfrak {H}}_P \otimes {\mathfrak {H}})$
is an isometry satisfying (2.7) and (2.8). What is left is to establish that
$$ \begin{align*} \tilde{S} \tilde{\mathcal{A}} \otimes {\mathcal{K}}(L^2(G)) \subseteq \tilde{\mathcal{A}} \otimes {\mathcal{K}}(L^2(G),{\mathfrak{H}}). \end{align*} $$
For this purpose, let
$x \in \tilde {\mathcal {A}} \otimes {\mathcal {K}}(L^2(G))$
. By Lemma 3.7, we have
$\tilde {S} x \in {\mathcal {M}}(\mathcal {D}) \otimes {\mathcal {K}}(L^2(G),{\mathfrak {H}})$
, and hence we may apply the map
$\delta _{\mathcal {D}} \otimes \operatorname {\mathrm {id}}$
which gives
$$ \begin{align*} \delta_{\mathcal{D}} \otimes \operatorname{\mathrm{id}} (\tilde{S} x) = \delta_{\mathcal{D}} \otimes \operatorname{\mathrm{id}} (\tilde{S}) \delta_{\mathcal{D}} \otimes \operatorname{\mathrm{id}} (x) = \tilde{S}_{124} x_{124} = (\tilde{S} x)_{124} \end{align*} $$
when combined with Lemma 3.7.1 That is,
$\tilde {S} x \in \text {Fix}(\delta _{\mathcal {D}}) \otimes {\mathcal {K}}(L^2(G),{\mathfrak {H}})$
. To see that, in fact,
$\tilde {S} x \in \tilde {\mathcal {A}} \otimes {\mathcal {K}}(L^2(G),{\mathfrak {H}})$
, we prove that the map
$G \ni g \mapsto \operatorname {\mathrm {Ad}} [1_{\mathcal {A}} \otimes \mu _g] \otimes \operatorname {\mathrm {id}} (\tilde {S} x) \in \text {Fix}(\delta _{\mathcal {D}}) \otimes {\mathcal {K}}(L^2(G),{\mathfrak {H}})$
is norm continuous. For the latter assertion, we use the identity
$$ \begin{align*} \operatorname{\mathrm{Ad}} [1_{\mathcal{A}} \otimes \mu_g] \otimes \operatorname{\mathrm{id}} (\tilde{S} x) = \tilde{S} (1_{\mathcal{A}} \otimes 1_{\mathfrak{H}} \otimes r_g) \operatorname{\mathrm{Ad}} [1_{\mathcal{A}} \otimes \mu_g \otimes 1_G](x), && g \in G, \end{align*} $$
which is a consequence of Lemma 3.7.2, together with the immediate norm continuity of the maps
$G \ni g \mapsto \operatorname {\mathrm {Ad}} [1_{\mathcal {A}} \otimes \mu _g \otimes 1_G](x)$
and
$G \ni g \mapsto (1_{\mathcal {A}} \otimes 1_{\mathfrak {H}} \otimes r_g) y$
,
$y \in {\mathcal {L}}({\mathfrak {H}}_P) \otimes {\mathcal {K}}(L^2(G))$
.
The fixed point algebra of
$(\tilde {\mathcal {A}},G,\tilde {\alpha })$
is characterized by our next result.
Lemma 3.9 The fixed point algebra of the C
$^*$
-dynamical system
$(\tilde {\mathcal {A}},G,\tilde {\alpha })$
is the image of the faithful unital
$^*$
-homomorphism
$\gamma : \mathcal {B} \rightarrow \tilde {\mathcal {A}}$
,
$\gamma (b) := S(b \otimes 1_G) S^*$
.
Proof We first establish that
$\gamma $
is well defined. To do this, let
$b \in \mathcal {B}$
and let
$x \in \mathcal {D}$
. By (2.6) and (2.9), we have
$\gamma (b)x \in \mathcal {A} \otimes {\mathcal {K}}({\mathfrak {H}})$
. Combining (2.7) with the fact that
$\mathcal {D}$
is fixed under
$\operatorname {\mathrm {Ad}}[u_g \otimes 1_{\mathfrak {H}}]$
,
$g \in G$
, further yields
$\gamma (b)x \in \mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}})$
. In consequence,
$\gamma (b)x \in \mathcal {D}$
, because
$\gamma (b)x = S^*S \gamma (b)x S^*S$
. In the same way we see that
$x\gamma (b) \in \mathcal {D}$
. Hence
$\gamma (b) \in {\mathcal {M}}(\mathcal {D})$
. Since
$$ \begin{align*} \delta_{\mathcal{D}}(\gamma(b)) &= \operatorname{\mathrm{Ad}}[ S_{12} (W_G)_{23} S_{12}^* ] ( S_{12} ( b \otimes 1_G \otimes 1_G ) S_{12}^* ) \\ &= \operatorname{\mathrm{Ad}}[ S_{12} (W_G)_{23} S_{12}^* S_{12} ] ( b \otimes 1_G \otimes 1_G) \\ &= \operatorname{\mathrm{Ad}}[ S_{12}] (b \otimes 1_G \otimes 1_G) = \gamma(b) \otimes 1_G, \end{align*} $$
i.e.,
$\gamma (b) \in \text {Fix}(\delta _{\mathcal {D}})$
, and
$\operatorname {\mathrm {Ad}}[1_{\mathcal {A}} \otimes \mu _g](\gamma (b)) = \gamma (b)$
for all
$g \in G$
, which is due to (2.8), it may finally be concluded that
$\gamma (b) \in \tilde {\mathcal {A}}$
. Note that we have actually proved more, namely that
$\gamma (b)$
is fixed under
$\tilde {\alpha }_g$
,
$g \in G$
, which amounts to saying that
${\gamma (\mathcal {B}) \subseteq \tilde {\mathcal {A}}^G}$
. We thus proceed to show that
$\tilde {\mathcal {A}}^G \subseteq \gamma (\mathcal {B})$
. For this, let
$x \in \tilde {\mathcal {A}}^G$
. We put
$y := S^* x S$
and observe that
$(W_G)_{23} (y \otimes 1_G) = (y \otimes 1_G) (W_G)_{23}$
or, equivalently,
$\operatorname {\mathrm {id}} \otimes \delta _G (y \otimes 1_G) = y \otimes 1_G$
. As the coaction
$\delta _G$
is ergodic, we can assert that
${y = b \otimes 1_G}$
for some
$b \in \mathcal {B}$
. It follows that
$x = P x P= S y S^* = S (b \otimes 1_G) S^* = \gamma (b)$
, i. e.,
${\tilde {\mathcal {A}}^G \subseteq \gamma (\mathcal {B})}$
as required. That
$\gamma $
is a faithful unital
$^*$
-homomorphism is clear, and so the proof is complete.
We proceed with a technical no-go result which is also of independent interest. For its proof we make use of the fact that each C
$^*$
-dynamical system
$(\mathcal {A},G,\alpha )$
can be decomposed into its isotypic components, say
$A(\sigma )$
,
$\sigma \in \operatorname {\mathrm {Irr}}(G)$
, which amounts to saying that their algebraic direct sum forms a dense
$^*$
-subalgebra of
$\mathcal {A}$
(see, e. g., [Reference Hofmann and Morris13, Theorem 4.22]).
Lemma 3.10 Let
$(\mathcal {A},G,\alpha )$
be a free C
$^*$
-dynamical system with fixed point algebra
$\mathcal {B}$
. Furthermore, let
$\mathcal {A}_0$
be a G-invariant unital C
$^*$
-subalgebra. If the induced C
$^*$
-dynamical system
$(\mathcal {A}_0,G,\alpha )$
is free and
$\mathcal {A}_0^G=\mathcal {B}$
, then
$\mathcal {A}_0 = \mathcal {A}$
.
Proof We shall have established the lemma if we prove that the respective isotypic components are equal, i. e.,
$A_0(\sigma ) = A(\sigma )$
for all
$\sigma \in \operatorname {\mathrm {Irr}}(G)$
. To prove the latter assertion, we fix
$\sigma \in \operatorname {\mathrm {Irr}}(G)$
and recall that the freeness of
$(\mathcal {A},G,\alpha )$
implies that
$A(\sigma )$
is a Morita equivalence bimodule between the unital C
$^*$
-algebras
$\mathcal {B} \otimes {\mathcal {L}}(V_\sigma )$
and
$$ \begin{align*} \mathcal{C} := \{ x \in \mathcal{A} \otimes {\mathcal{L}}(V_\sigma) : (\forall g \in G) \, (1_{\mathcal{A}} \otimes \sigma_g) \alpha_g(x) = x (1_{\mathcal{A}} \otimes \sigma_g) \} \end{align*} $$
(see [Reference Schwieger and Wagner20, Section 3]). Likewise, the freeness of
$(\mathcal {A}_0,G,\alpha )$
implies that
$A_0(\sigma )$
is a Morita equivalence bimodule between the unital C
$^*$
-algebras
$\mathcal {B} \otimes {\mathcal {L}}(V_\sigma )$
and
$$ \begin{align*} \mathcal{C}_0 := \{ x \in \mathcal{A}_0 \otimes {\mathcal{L}}(V_\sigma) : (\forall g \in G) \, (1_{\mathcal{A}} \otimes \sigma_g) x = x (1_{\mathcal{A}} \otimes \sigma_g)\}. \end{align*} $$
This makes it possible to apply [Reference Schwieger and Wagner22, Lemma A.1] which gives elements
$s_1,\ldots ,s_n \in A_0(\sigma )$
such that 
. Since the inner products of
$A_0(\sigma )$
are exactly the restrictions in domain and codomain of the respective inner products of
$A(\sigma )$
, we conclude that 
for all
$1 \leq k \leq n$
. Hence for each
$x \in A(\sigma )$
we have
i. e.,
$A(\sigma ) \subseteq A_0(\sigma )$
. This completes the proof, as the other inclusion is obvious.
Now, we are almost in a position to state and prove the main result of this section. The only preparatory point remaining concerns the following map:
$$ \begin{gather*} j_\alpha : \mathcal{A} \rightarrow C(G,\mathcal{A}) = \mathcal{A} \otimes C(G) \subseteq {\mathcal{L}}({\mathfrak{H}}_{\mathcal{A}} \otimes L^2(G)), \\ j_\alpha(x)(g) = \alpha_{g^{-1}}(x) = u_g^* x u_g. \end{gather*} $$
It is clear that
$j_\alpha $
is injective. Moreover, straightforward computations reveal that
$$ \begin{align} &\hspace{-3pt}\operatorname{\mathrm{Ad}}[u_g \otimes r_g] \circ j_\alpha = j_\alpha \ \kern5pt\qquad\qquad\qquad\qquad \forall g \in G , \end{align} $$
$$ \begin{align} \operatorname{\mathrm{Ad}}[1_{\mathcal{A}} \otimes \lambda_g] \circ j_\alpha &= j_\alpha \circ \alpha_g && \forall g \in G. \end{align} $$
Theorem 3.11 The map
$$ \begin{align} \pi_S : \mathcal{A} \rightarrow \tilde{\mathcal{A}} \subseteq{\mathcal{M}}(\mathcal{D}) \subseteq {\mathcal{L}}({\mathfrak{H}}_P), && \pi_S(x) := S j_\alpha(x) S^* \end{align} $$
is a G-equivariant
$^*$
-isomorphism.
Proof We start again by proving that the map under consideration is well defined. For this purpose, let
$x \in \mathcal {A}$
. The verification of
$\pi _S(x) \in {\mathcal {M}}(\mathcal {D})$
can be handled in much the same way as in the proof of Lemma 3.9, the only difference being that (2.7) needs to be combined with (3.3) to establish that
$\pi _S(x) d, d \pi _S(x) \in \mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}})$
for all
${d \in \mathcal {D}}$
. Moreover, the identity
$(W_G)_{23} j_\alpha (x)_{12} (W_G)_{23}^* = j_\alpha (x)_{12}$
, which is easy to check, implies that
$$ \begin{align*} \delta_{\mathcal{D}}(\pi_S(x)) &= \operatorname{\mathrm{Ad}}[S_{12} (W_G)_{23} S_{12}^*] (S_{12} j_\alpha(x)_{12} S_{12}^*) \\ &= \operatorname{\mathrm{Ad}}[S_{12}] ((W_G)_{23} j_\alpha(x)_{12} (W_G)_{23}^*) \\ &= \operatorname{\mathrm{Ad}}[S_{12}] (j_\alpha(x)_{12}) = \pi_S(x)_{12}, \end{align*} $$
i. e.,
$\pi _S(x) \in \text {Fix}(\delta _{\mathcal {D}})$
. That, in fact,
$\pi _S(x) \in \tilde {\mathcal {A}}$
as claimed now follows from
$$ \begin{align} \tilde{\alpha}_g (\pi_S(x)) \overset{({2.8})}= S (\operatorname{\mathrm{Ad}}[1_{\mathcal{A}} \otimes \lambda_g]( j_\alpha(x)) S^* \overset{({3.4})}= \pi_S(\alpha_g(x)), && g \in G. \end{align} $$
Of course, (3.6) also establishes that
$\pi _S$
is G-equivariant, and so it remains to show that
$\pi _S$
is a
$^*$
-isomorphism. Clearly,
$\pi _S$
is a
$^*$
-homomorphism. Moreover, it is injective, because
$j_\alpha $
is injective and S is an isometry. With this, we can assert that the induced C
$^*$
-dynamical system
$(\pi _S(\mathcal {A}),G,\tilde {\alpha })$
is free with fixed point algebra
${\pi _S(\mathcal {B}) = \gamma (\mathcal {B}) = \tilde {\mathcal {A}}^G}$
(see Lemma 3.9 for the latter equalities). Hence Lemma 3.10 implies that
$\pi _S$
is surjective, i. e.,
$\pi _S(\mathcal {A}) = \tilde {\mathcal {A}}$
, and therefore the proof is complete.
Remark 3.12 An alternative strategy for proofing Theorem 3.11 would be to establish that the isometries S and
$\tilde {S}$
are conjugated in the sense of [Reference Schwieger and Wagner22, Theorem 4.4]. With some technical effort this can be done.
Corollary 3.13 (cf. [Reference Schwieger and Wagner23, Corollary 4.2])
The pair
$(\pi _S, \operatorname {\mathrm {Ad}}[1_{\mathcal {A}} \otimes \mu ])$
is a faithful generalized covariant representation of
$(\mathcal {A},G,\alpha )$
on
${\mathcal {M}}(\mathcal {D}) = P{\mathcal {M}}(\mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}})) P$
. Moreover, any faithful
$^*$
-representation
$\pi _{\mathcal {B}} : \mathcal {B} \to {\mathcal {L}}({\mathfrak {H}}_{\mathcal {B}})$
gives rise to an honest faithful covariant representation on
$P({\mathfrak {H}}_{\mathcal {B}} \otimes {\mathfrak {H}})$
.
Corollary 3.14 (cf. [Reference Wassermann27, Theorem 10])
Let
$(\mathcal {A},G,\alpha )$
be a cleft C
$^*$
-dynamical system (see Section 2.4). Then
$(\mathcal {A},G,\alpha )$
can be realized as the invariants of an equivariant coaction of G on
$\mathcal {B} \otimes {\mathcal {K}}(L^2(G))$
. Moreover, any faithful
$^*$
-representation
${\pi _{\mathcal {B}} : \mathcal {B} \to {\mathcal {L}}({\mathfrak {H}}_{\mathcal {B}})}$
gives rise to an honest faithful covariant representation on
${\mathfrak {H}}_{\mathcal {B}} \otimes L^2(G)$
.
Outlook
As pointed out by the referee, a reformulation of our main result in terms of groups and their quantum duals could provide a pathway for generalizing our findings within the framework of quantum groups, potentially offering valuable insights in relation to Wassermann’s work. We leave this extension as a direction for future research.
Acknowledgements
The authors thank the anonymous referee for their clear and constructive feedback, which greatly contributed to the manuscript’s improvement. The first author also expresses gratitude to iteratec GmbH.
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