On Equivariant Embedding of Hilbert C^* modules

TL;DR

This paper proves that any Hilbert G-C*-module over an ergodic action of a compact Lie group can be embedded into a trivial module, extending the understanding of equivariant embeddings in operator algebras.

Contribution

It establishes the existence of equivariant embeddings for all Hilbert G-C*-modules under ergodic actions of compact Lie groups, regardless of countable generation.

Findings

Any Hilbert G-C*-module admits an equivariant embedding into a trivial module.

The result applies to modules over ergodic actions of compact Lie groups.

It generalizes previous results to non-countably generated modules.

Abstract

We prove that an arbitrary (not necessarily countably generated) Hilbert $G$-$\cla$ module on a G-C^* algebra $\cla$ admits an equivariant embedding into a trivial $G-\cla$ module, provided G is a compact Lie group and its action on $\cla$ is ergodic.

Equations

$$\Vert \xi {\Vert }_{\infty ,n}:=\sum _{i_1,i_2,...i_k;k\le n,i_t\in \{1,...,N\}}\Vert {\partial }_{i_1}{\partial }_{i_2}...{\partial }_{i_k}\xi \Vert ,$$

$$\Vert \xi {\Vert }_{2,n}^2:=\sum _{i_1,i_2,...i_k;k\le n,i_t\in \{1,...,N\}}\Vert d_{i_1}{d}_{i_2}...d_{i_k}\xi {\Vert }_2^2,$$

$$\Vert \xi {\Vert }_n^{(q)}:=\sum _{i_1,i_2,...i_k;k\le n,i_t\in \{1,...,N\}}\Vert {\partial }_{i_1}{\partial }_{i_2}...{\partial }_{i_k}\xi {\Vert }^{(q)}.$$

$$\Vert <{\beta }_1,{\beta }_2>{\Vert }_2^2=\tau ({\beta }_2^{\ast }{\beta }_1{\beta }_1^{\ast }{\beta }_2)\le \Vert {\beta }_1{\Vert }^2\tau ({\beta }_2^{\ast }{\beta }_2),$$

$$\Vert <\gamma (f)(\beta ),\gamma (f)(\beta )>{\Vert }_2\le C(f{)}^2\Vert \beta \Vert \Vert \beta {\Vert }_2,$$

$${\partial }_J\gamma (f)(\beta )=(-1{)}^k\gamma (f_J)(\beta ).$$

$$\Vert {\partial }_I{x}_{mn}{\Vert }_2$$

$$<(\xi ,w),({\xi }^{\prime },w^{\prime })>=<w,<\xi ,{\xi }^{\prime }>w^{\prime }>,$$

$$({\Gamma }_0(\xi ))w:=[\xi ,w],$$

$$V_g[\xi ,w]:=[{\beta }_g(\xi ),u_{g}w],$$

Full text

1 Introduction

Let $G$ be a locally compact group, ${\cal A}$ be a $C^{*}$-algebra, and assume that there is a strongly continuous representation $\alpha:G\rightarrow Aut({\cal A})$. Following the terminology of [6], we introduce the concept of a Hilbert $C^{*}$ $G-{\cal A}$-module as follows :

Definition 1.1

A Hilbert $C^{*}$ $G-{\cal A}$ module (or $G-{\cal A}$* module* for short) is a pair $(E,\beta)$ where $E$ is a Hilbert $C^{*}$ ${\cal A}$-module and $\beta$ is a map from $G$ into the set of $\mathbb{C}$-linear (caution : not ${\cal A}$-linear !) maps from $E$ to $E$, such that $\beta_{g}\equiv\beta(g),g\in G$ satisfies the following :

(i) $\beta_{gh}=\beta_{g}\circ\beta_{h}$ for $g,h\in G,$ $\beta_{e}={\rm Id},$ where $e$ is the identity element of $G$;

(ii) $\beta_{g}(\xi a)=\beta_{g}(\xi)\alpha_{g}(a)$ for $\xi\in E,a\in{\cal A};$

(iii) $g\mapsto\beta_{g}(\xi)$ is continuous for each fixed $\xi\in E;$

(iv) $\langle\beta_{g}(\xi),\beta_{g}(\eta)\rangle=\alpha_{g}(\langle\xi,\eta\rangle)$ for all $\xi,\eta\in E,$ where $\langle\cdot,\cdot\rangle$ denotes the ${\cal A}$-valued inner product of $E.$ **

When $\beta$ is understood from the context, we may refer to $E$ as a $G-{\cal A}$ module, without explicitly mentioning the pair $(E,\beta).$ Given two $G-{\cal A}$ modules $(E_{1},\beta)$ and $(E_{2},\gamma)$, there is a natural $G$-action induced on ${\cal L}(E_{1},E_{2})$, given by $\pi_{g}(T)(\xi):=\gamma_{g}(T(\beta_{g^{-1}}(\xi)))$ for $g\in G,\xi\in E_{1},T\in{\cal L}(E_{1},E_{2}).$ $T\in{\cal L}(E_{1},E_{2})$ is said to be $G$-equivariant if $\pi_{g}(T)=T{\rm~{}for~{}all~{}}g\in G.$ It is clear that for each fixed $T\in{\cal L}(E_{1},E_{2})$ and $\xi\in E_{1}$, $g\mapsto\pi_{g}(T)\xi$ is continuous. We say that $T$ is $G$-continuous if $g\mapsto\pi_{g}(T)$ is continuous with respect to the norm topology on ${\cal L}(E_{1},E_{2}).$ We say that $E_{1}$ and $E_{2}$ are isomorphic as $G-{\cal A}$-modules, or that they are equivariantly isomorphic if there is a $G$-equivariant unitary map $T\in{\cal L}(E_{1},E_{2}).$ We call a $(G-{\cal A})$ module of the form $({\cal A}\otimes{\cal H},\alpha_{g}\otimes\gamma_{g})$ (where ${\cal H}$ is some Hilbert space) a trivial $G-{\cal A}$ module. We say that $(E,\beta)$ is *embeddable * if there is an equivariant isometry from $E$ to ${\cal A}\otimes{\cal H}$ for some Hilbert space ${\cal H}$ with a $G$-action $\gamma$, or in other words, $(E,\beta)$ is equivariantly isomorphic with a sub-$G-{\cal A}$ module of $({\cal A}\otimes{\cal H},\beta\otimes\gamma)$. Note that ${\cal A}\otimes{\cal H}$ is the closure of ${\cal A}\otimes_{\rm alg}{\cal H}$ under the norm inherited from ${\cal B}({\cal H}_{0},{\cal H}_{0}\otimes{\cal H})$ where ${\cal H}_{0}$ is any Hilbert space such that ${\cal A}$ is isometrically embedded into ${\cal B}({\cal H}_{0})$. The following result on the embeddabiity is due to Mingo and Phillips ([6]).

Theorem 1.2

Let $(E,\beta)$ be a Hilbert $C^{*}$ $G-{\cal A}$ module and assume that $E$ is countably generated as a Hilbert ${\cal A}$-module, that is, there is a countable set $S=\{e_{1},e_{2},...\}$ of elements of $E$ such that the right ${\cal A}$-linear span of $S$ is dense in $E$. Assume furthermore that $G$ is compact. Then $(E,\beta)$ is embeddable.

When $G$ is the trivial singleton group, the above result was proved by Kasparov.

If the $C^{*}$ algebra ${\cal A}$ is replaced by a von Neumann algebra ${\cal B}\subseteq{\cal B}(h)$ for some Hilbert space $h$ and $G$ is a locally compact group with a strongly continuous unitary representation $g\mapsto u_{g}\in{\cal B}(h)$, one can define Hilbert von Neumann $G$-${\cal B}$ module $(E,\beta)$. The only difference is that $E$ is now a Hilbert von Neumann ${\cal A}$-module equipped with the natural locally convex strong operator topology, and that we replace the norm-continuity in (iii) of the above definition by a weaker continuity : namely, the continuity of $g\mapsto\beta_{g}(\xi)$ (for fixed $\xi\in E$) with respect to the locally convex topology of $E$. In this case, we have a stronger version of the Theorem 1.2 (see [3] and [2], Theorem 4.3.5, page 99), namely without the condition of $E$ being countably generated and without the compactness of $G$. It should be remarked here that the trivial Hilbert von Neumann ${\cal B}$ module ${\cal B}\otimes{\cal H}$ is defined to be the closure of ${\cal B}\otimes_{\rm alg}{\cal H}$ with respect to the strong-operator-topology inherited from ${\cal B}(h,h\otimes{\cal H})$.

In Theorem 1.2, the assumption that $E$ is countably generated restricts the applicability of the result, since it is not always easy to check the property of being countably generated. However, under some special assumption on the $G$-$C^{*}$ algebra ${\cal A}$, i.e. conditions on the group $G$, the $C^{*}$ algebra ${\cal A}$ and also on the nature of the action, it may be possible to prove the embedability for an arbitrary Hilbert $G$-${\cal A}$ module. The aim of the present article is to give some such sufficient conditions.

2 Ergodic action and its implication

We say that the action $\alpha$ of $G$ on a unital $C^{*}$-algebra ${\cal A}$ is ergodic if $\alpha_{g}(a)=a~{}{\rm~{}for~{}all~{}}g\in G$ if and only if $a$ is a scalar multiplie of $1$. There is a considerable amount of literature on ergodic action of compact groups, and we shall quote one interesting structure theorem which will be useful for us.

Proposition 2.1

*Let $G$ be a compact group acting ergodically on a unital $C^{*}$-algebra ${\cal A}.$ Then there is a set of elements $t^{\pi}_{ij},\pi\in\hat{G},i=1,...,d_{\pi},j=1,...,m_{\pi}$ of ${\cal A}$, where $\hat{G}$ is the set of equivalence classes of irreducible representations of $G$, $d_{\pi}$ is the dimension of the irreducible representation space denoted by $\pi$, $m_{\pi}\leq d_{\pi}$ is a natural number, such that the followings hold :

(i) There is a unique faithful $G$-invariant state $\tau$ on ${\cal A}$, which is in fact a trace,

(ii) The linear span of $\{t^{\pi}_{ij}\}$ is norm-dense in ${\cal A}$,

(iii) $\{t^{\pi}_{ij}\}$ is an orthonormal basis of $h=L^{2}({\cal A},\tau)$,

(iv) The action of $u_{g}$ coincides with the $\pi$th irreducible representation of $G$ on the vector space spanned by $t^{\pi}_{ij},i=1,...,d_{\pi}$ for each fixed $j$ and $\pi$,

(v) $\sum_{i=1,...d_{\pi}}(t^{\pi}_{ij})^{*}t^{\pi}_{ik}=\delta_{jk}d_{\pi}1$, where $\delta_{jk}$ denotes the Kronecker delta symbol. Thus, in particular, $\|t^{\pi}_{ij}\|\leq\sqrt{d_{\pi}}~{}{\rm~{}for~{}all~{}}\pi,i,j$.*

The proof can be obtained by combining the results of [7],[4] and [1].

Let now $h=L^{2}({\cal A},\tau)$, where $\tau$ is the unique invariant faithful trace described in Proposition 2.1. Let $u_{g}$ be the unitary in $h$ induced by the action of $G$, that is, on the dense set ${\cal A}\subseteq h,$ $u_{g}(a):=\alpha_{g}(a),$ where $\alpha_{g}$ denotes the $G$-action on ${\cal A}$. Denote also by $\alpha$ the action $g\mapsto u_{g}\cdot u_{g}^{*}$ on $\tilde{{\cal A}}$, which is the weak closure of ${\cal A}$ in ${\cal B}(h)$.

Let us now specialize to the case of a compact Lie group. If $G$ is such a group, with a basis of the Lie algebra given by $\{\chi_{1},...,\chi_{N}\}$, which has a strongly continuous action $\theta$ on a Banach space $F$, we can consider the space of ‘smooth’ or $C^{\infty}$-elements of $F$, denoted by $F^{\infty}$, consisting of all $\xi\in f$ such that $G\ni g\mapsto\theta_{g}(\eta)$ is $C^{\infty}$. It is easy to prove that $F^{\infty}$ is dense in $F$, and it is a $\ast$-subalgebra if $F$ is a locally convex $\ast$-algebra. Moreover, we equip $E^{\infty}$ with a family of seminorms $\|\cdot\|_{\infty,n}$, $n=0,1,...$ given by

[TABLE]

with the convention $\|\cdot\|_{\infty,0}=\|\cdot\|$ and where $\partial_{j}(\xi):=\frac{d}{dt}|_{t=0}\theta_{{\rm exp}(t\chi_{j})}(\xi)$. The space $F^{\infty}$ is complete under this family of seminorms, and thus is a Fr$\acute{\rm e}$chet space. When $F$ is Hilbert space or a Hilbert module, we shall also consider a map $d_{j}$ given by essentially the same expression as that of $\partial_{j}$, with $\chi_{j}$ replaced by $i\chi_{j}$, and the Hilbertian seminorms $\{\|\cdot\|_{2,n}\}$ are given by

[TABLE]

with $\|\cdot\|_{2}$ denoting the norm of the Hilbert space (or Hilbert module) $F$.

More generally, if $F$ is a complete locally convex space given by a family of seminorms $\{\|\cdot\|^{(q)}\}$, then we can consider the smooth subspace $F^{\infty}$ and the maps $\partial_{j}$ as above, and make it a complete locally convex space with respect to a larger family of seminorms $\{\|\cdot\|^{(q)}_{n}\}$ where

[TABLE]

In case $F$ is a von Neumann algebra equipped with the locally convex strong operator topology, the locally convex space $F^{\infty}$ is a topological $\ast$-algebra, strongly dense in $F$.

Lemma 2.2

[2]** Let $G$ be a compact Lie group acting ergodically on a unital $C^{*}$-algebra ${\cal A}.$ Then $h^{\infty}={\cal A}^{\infty}$ as Fr$\acute{\rm e}$chet spaces.

*Proof :

*The fact that ${\cal A}^{\infty}=h^{\infty}$ as sets is contained in Lemma 8.1.20 of [2] (page 200-201). We only prove that the identity map is a topological homeomorphism.

Since the trace $\tau$ is finite, the Fr$\acute{\rm e}$chet topology of ${\cal A}^{\infty}$ is stronger than that of $h_{\infty}$. This implies that the identity map $I$, viewed as a linear map from the Fr$\acute{\rm e}$chet space $h^{\infty}$ to the Fr$\acute{\rm e}$chet space ${\cal A}^{\infty}$ is closable, hence continuous. This completes the proof that the two Fr$\acute{\rm e}$chet topologies on ${\cal A}^{\infty}=h^{\infty}$ are equivalent, i.e. ${\cal A}^{\infty}=h^{\infty}$ as topological spaces. $\Box$

Lemma 2.3

Let ${\cal H}$ be a (not necessarily separable) Hilbert space with a unitary representation $w\equiv w_{g}$ of $G$, and let us consider the Fr$\acute{\rm e}$chet modules $(\tilde{{\cal A}}\otimes{\cal H})^{\infty}$ and $({\cal A}\otimes{\cal H})^{\infty}$ corresponding to the action $\gamma_{g}:=\alpha_{g}\otimes w_{g}$. Let $\xi$ be an element of $(\tilde{{\cal A}}\otimes{\cal H})^{\infty}$ such that $g\mapsto\gamma_{g}(\xi)$ is continuous in the operator-norm topology. Then $\xi$ actually belongs to ${\cal A}\otimes{\cal H}$.

*Proof :-

*We shall denote by $\|\cdot\|_{p}$ ($p\geq 1$) the $L^{p}$-norm coming from the trace $\tau$ on ${\cal A}$. The identity $1$ of ${\cal A}$ will also be viewed as a unit vector in $L^{2}(\tau)$. Fix an orthonormal basis $\{e_{\alpha},\alpha\in T\}$ of ${\cal H}$ (which need not be separable), with each $e_{\alpha}\in{\cal H}^{\infty}$. Fix $\xi\in(\tilde{{\cal A}}\otimes{\cal H})^{\infty}$ satisfying the hytothesis of the lemma. Since $L^{2}(\tau)$ is separable, say with an orthonormal basis given by $\{x_{1},x_{2},...\}$, we can find, for each $i$, a counteble subset $T_{i}$ of $T$ such that $<\xi 1,x_{i}\otimes e_{\alpha}>=0$ for all $\alpha\notin T_{i}$. Denoting by $T_{\infty}$ the countable set $\bigcup_{i}T_{i}$, we have $<\xi 1,v\otimes e_{\alpha}>=0~{}\forall v\in L^{2}(\tau)$, for all $\alpha\notin T_{\infty}$. Write $T_{\infty}=\{e_{\alpha_{1}},e_{\alpha_{2}},...\}$. Denote by $\xi_{n}$ the element in ${\cal A}^{\infty}\otimes_{\rm alg}{\cal H}^{\infty}$ given by $\xi_{n}=(I\otimes P_{n})\xi$, where $P_{n}$ denotes the orthogonal projection onto the linear span of $\{e_{\alpha_{1}},...,e_{\alpha_{n}}\}$. It is clear that $\xi_{n}1\rightarrow\xi 1$ as $n\rightarrow\infty$. Now, for a $C^{\infty}$ complex-valued function $f$ on $G$ and an element $\eta\in\tilde{{\cal A}}\otimes{\cal H}$, denote by $\gamma(f)(\eta)$ the element $\int_{G}f(g)\gamma_{g}(\eta)dg~{}\in\tilde{{\cal A}}\otimes{\cal H}\subseteq L^{2}(\tau)\otimes{\cal H}$, where $dg$ stands for the normalised Haar measure on $G$ and the integral is convergent in the strong-operator topology. We claim that it is enough to prove that $\gamma(f)(\xi)\in{\cal A}\otimes{\cal H}$ for all $f\in C^{\infty}(G)$. Let us first prove this claim. Since $g\mapsto\gamma_{g}(\xi)$ is norm-continuous, given $\epsilon>0$, we can find a nonempty open subset $U$ of $G$ such that $\|\gamma_{g}(\xi)-\xi\|\leq\epsilon$ for all $g\in G$, and then choose $f\in C^{\infty}(G)$ with ${\rm supp}(f)\subseteq U$, $f\geq 0$ and $\int_{G}fdg=1$. It is easy to see that $\|\gamma(f)(\xi)-\xi\|\leq\epsilon$. Thus, $\xi$ is the operator-norm limit of a sequence of elements of the form $\gamma(f)(\xi)$, which proves the claim.

Let us now complete the proof of the lemma by showing that $\eta:=\gamma(f)(\xi)$ indeed belongs to ${\cal A}\otimes{\cal H}$ for every $f\in C^{\infty}(G)$. To this end, first observe that $\eta_{n}:=\gamma(f)(\xi_{n})$ belongs to ${\cal A}^{\infty}\otimes_{\rm alg}{\cal H}^{\infty}\subseteq{\cal A}\otimes{\cal H}$ for all $n$. Moreover, since $u_{g}1=1$ for all $g$ and $\gamma_{g}(\cdot)={\rm ad}_{u_{g}}\otimes w_{g}$, it is clear that $\eta_{n}1\rightarrow\gamma(f)(\xi)1=\eta 1$ as $n\rightarrow\infty$. Since each $\eta_{m,n}$ belongs to ${\cal A}\otimes{\cal H}$, for proving $\eta\in{\cal A}\otimes{\cal H}$ it is enough to prove that $\eta_{m,n}\rightarrow 0$ in the topology of ${\cal A}\otimes{\cal H}$, i.e. $x_{mn}:=<\eta_{m,n},\eta_{m,n}>\rightarrow 0$ in the norm-topology of ${\cal A}$. We shall prove that $x_{mn}\rightarrow 0$ in the Fr$\acute{\rm e}$chet topology of $h^{\infty}$, which will prove that it converges to [math] also in the topology of ${\cal A}^{\infty}.$

To this end, first note that for $\beta_{1},\beta_{2}\in(\tilde{{\cal A}}\otimes{\cal H})$, we have

[TABLE]

hence $\|<\beta_{1},\beta_{2}>\|_{2}\leq\|\beta_{1}\|\|\beta_{2}\|_{2}$. Moreover, $\|<\beta_{2},\beta_{1}>\|_{2}=\|<\beta_{1},\beta_{2}>^{*}\|_{2}=\|<\beta_{1},\beta_{2}>\|_{2}$ (since $\tau(x^{*}x)=\tau(xx^{*}),$ we have $\|x\|=\|x^{*}\|$). From this, we have

[TABLE]

where $C(f):=\int|f|dg$. Let us now fix an ordered $k$-tuple $I=(i_{1},...,i_{k})$ ($k$ nonnegative integer), and let $C$ denote the maximum of $C(\chi_{j_{1}}...\chi_{j_{p}}f)$ where $J=(j_{1},...,j_{p})$ varies over all (including the empty set) ordered subsets of $I$. Let us abbreviate $\partial_{j_{1}}...\partial_{j_{p}}\beta$ and $\chi_{j_{1}}...\chi_{j_{p}}f$ by $\partial_{J}\beta$ and $f_{J}$ respectively, for $\beta\in({\cal A}\otimes{\cal H})^{\infty}$. Note that

[TABLE]

Using this as well as the Leibniz formula $\partial_{I}<\beta,\beta>=\sum_{J}<\partial_{J}\beta,\partial_{I-J}\beta>$ (with $J$ varying over all ordered substes of $I$), we have the following :

[TABLE]

since the number of ordered subsets of $I$ is $2^{k}$ and it is clear from the definition of $\xi_{n}$ that $\|\xi_{n}\|\leq\|\xi\|$ for all $n$. We also have $\|\xi_{m}-\xi_{n}\|^{2}_{2}=\tau(<\xi_{m}-\xi_{n},\xi_{m}-\xi_{n}>)=<(\xi_{m}-\xi_{n})(1),(\xi_{m}-\xi_{n})(1)>\rightarrow 0$ as $m,n\rightarrow\infty$. This proves $x_{m,n}\rightarrow 0$ in the topology of $h_{\infty}$, thereby completing the proof of the lemma. $\Box$

3 Main results on equivariant embedding of Hilbert modules

Let $(E,\beta)$ be a $G-{\cal A}$ module, where ${\cal A}$ and $G$ are as in the previous section, i.e. $G$ is a compact Lie group acting ergodically on the $C^{*}$ algebra ${\cal A}$. In this final section, we shall prove that any such $(E,\beta)$ is embeddable.

Lemma 3.1

We can find a Hilbert space ${\cal K}$, a strongly continuous unitary representation $g\mapsto V_{g}\in{\cal B}({\cal K})$ and a ${\cal A}$-linear isometry $\Gamma_{0}:E\rightarrow{\cal B}(h,{\cal K})$, such that $\Gamma_{0}\beta_{g}(\xi)=V_{g}(\Gamma_{0}\xi)u_{g}^{-1}$, and moreover, the complex linear span of elements of the form $\Gamma\xi w$ where $\xi\in E$ and $w\in h$ is dense in ${\cal K}$.

*Proof :

*The proof of this result is adapted from [3] and [2], Theorem 4.3.5 (page 99-101). We shall give only a brief sketch of the arguments involved, omitting the details. We consider first the formal vector space (say ${\cal V}$) spanned by symbols $(\xi,w)$, with $\xi\in E$ and $w\in h$, and define a semi-inner product on this formal vector space by setting

[TABLE]

where $<\xi,\xi^{\prime}>$ denotes the ${\cal A}$-valued inner product on $E$. By extending this semi-inner product by linearity and then taking quotient by the subspace (say ${\cal V}_{0}$) consisting of elements of zero norm we get a pre-Hilbert space, and its completion under the pre-inner product is denoted by ${\cal K}$. We also define $\Gamma_{0}:E\rightarrow{\cal B}(h,{\cal K})$ by setting

[TABLE]

where $[\xi,w]$ represents the equivalence class of $(\xi,w)$ in ${\cal S}\equiv{\cal V}/{\cal V}_{0}\subseteq{\cal K}$. That it is an isometry is verified by straightforward calculations. Next, we define $V_{g}$ on ${\cal S}$ by

[TABLE]

and verify that it is indeed an isometry, and since its range clearly contains a total subset, $V_{g}$ extends to a unitary on ${\cal K}$. Furthermore, $V_{g}V_{h}=V_{gh}$ and $V_{e}={\rm Id}$ (where $e$ is the identity of $G$) on ${\cal S}$,and hence on the whole of ${\cal K}.$ The strong continuity of $g\mapsto V_{g}$ is also easy to see. Indeed, it is enough to prove that $g\mapsto V_{g}X$ is continuous for any $X$ of the form $[\xi,v]$, $\xi\in E,v\in h.$ But $\|V_{g}([\xi,v])-[\xi,v]\|^{2}=2\langle[\xi,v],[\xi,v]\rangle-\langle V_{g}([\xi,v]),[\xi,v]\rangle-\langle[\xi,v],V_{g}([\xi,v])\rangle,$ and we have, $\langle V_{g}([\xi,v]),[\xi,v]\rangle-\langle[\xi,v],[\xi,v]\rangle=\langle(u_{g}v-v)\langle\beta_{g}(\xi),\xi\rangle v\rangle+\langle v,\langle(\beta_{g}(\xi)-\xi),\xi\rangle v\rangle.$ By assumption $\lim_{g\rightarrow e}(\beta_{g}(\xi)-\xi)=0$ in the norm topology of $E$, so $\langle(\beta_{g}(\xi)-\xi)v,\xi v\rangle\rightarrow 0$ as $g\rightarrow e$. Furthermore, $g\mapsto u_{g}v$ is continuous. This completes the proof of strong continuity of $V_{g}$. $\Box$

In view of the above result, we assume without loss of genetrality that $E\subset{\cal B}(h,{\cal K})$ (with the natural Hilbert module structure inherited from that of ${\cal B}(h,{\cal K})$), and $\beta_{g}(\cdot)=V_{g}\cdot u_{g}^{-1}$. Consider the strong operator closure of $\tilde{E}$ of $E$ in ${\cal B}(h,{\cal K})$. It is a Hilbert von Neumann $\tilde{{\cal A}}$ module (where $\tilde{{\cal A}}$ is the weak closure of ${\cal A}$ in $h$) . Moreover, the $G$-action $\beta_{g}=V_{g}\cdot u_{g}^{-1}$ can be extended to the whole of ${\cal B}(h,{\cal K})$, and denoted again by $\beta_{g}$. Clearly, this action leaves $\tilde{E}$ invariant, hence $(\tilde{E},\beta)$ is a Hilbert von Neumann $G$-$\tilde{{\cal A}}$ module. Let us recall that by ${\tilde{E}}^{\infty}$ we denote the locally convex space of elements $\xi$ in $\tilde{E}$ such that $g\mapsto\beta_{g}(\xi)$ is $C^{\infty}$ in the strong operator topology of $\tilde{E}$.

Theorem 3.2

*There exist a Hilbert space $k_{0}$, a unitary representation $w_{g}$ of $G$ in $k_{0}$ and an isometry $\Sigma$ from ${\cal K}$ to $h\otimes k_{0}$ such that

(i) $\Sigma$ is equivariant in the sense that $\Sigma V_{g}=(u_{g}\otimes w_{g})\Sigma$ for all $g$;

(ii) $\Sigma\xi\in{\cal A}\otimes k_{0}$ for all $\xi\in E$.

*Proof :

*The statement (i) is contained in the Theorem 4.3.5 of [2] (page 99). For proving (ii), we note that $E^{\infty}$ (w.r.t. the action $\beta$) is mapped by $\Sigma$ into $(\tilde{{\cal A}}\otimes k_{0})^{\infty}$ (w.r.t. the action $\gamma_{g}:={\rm ad}_{u_{g}}\otimes w_{g}$), and moreover, for $\xi\in E$, $g\mapsto\gamma_{g}(\Sigma(\xi))=\Sigma\beta_{g}(\xi)$ is norm-continuous since $g\mapsto\beta_{g}(\xi)$ is so and $\Sigma$ is isometry. Thus, (ii) follows from Lemma 2.3. $\Box$

It follows from the above theorem that $E$ can be equivariantly embedded in the trivial $G-{\cal A}$ module $({\cal A}\otimes k_{0},\alpha\otimes w)$. In particular, we have that

Theorem 3.3

If a compact Lie group $G$ has an ergodic action on a $C^{*}$-algebra ${\cal A}$, then every $G-{\cal A}$ module $(E,\beta)$ is embeddable.

Acknowledgement : The author would like to thank the anonymous referee who pointed out a crucial mistake in the earlier version, which has led to substantial revision of the paper.