数理解析研究所講究録
1534 巻 2007 年 119-131

119

SUBMODULES OF HILBERT$C^{*}$-MODULES AND

THEIR ORTHOGONAL COMPLEMENTED SUBSPACES

byMASAHARU KUSUDA (楠田雅治)

Department of Mathematics, Kansai University (関西大学)

1. IntroductionLet
be a -algebra and let $X$ be a Hilbert -module with an A-valued
inner product . ,. For any closed subspace of $X$ , we denote bythe$X$
orthogonally complemented subspace of in , i.e.,
$A$

$A$

$C^{*}$$\langle$$\mathrm{Y}^{\perp}$$\mathrm{Y}$$\rangle$$\mathrm{Y}$$\mathrm{Y}^{\perp}=${,$x\in X|\langle x$$y\rangle=0$

for all$y\in \mathrm{Y}$}.

We say that a closed -submoduleof a Hilbert -module $X$ is orthogonally$X$$X$
complemented in if coincides with, and that a closed -submodule ofa Hilbert -module $X$ is onhogonally closed in $X$ if. If is orthogonally$X$$X$
complemented in , then it is orthogonally closed in . But the converse is notnecessarily true. As is
known, every closed subspace of a Hilbert space isorthogonally complemented. This fact is a reason why it isier to work on Hilbert
spaces than on Banach spaces. Thus we have reached a question of when every
closed submodule of a Hilbert C’-module $X$ is orthogonally closed or orthogonallycomplemented in $X$ . The purpose of this article is to introduce complete answers(containing the author’s unpublished results) to the above question, which have beenobtained by Magajna [11], Schweitzer [12] and the author [4], [6]. Here it would besignificant to remark that although the Hilbert C’-modules to be considered inthis article are supposed to be full, the assumption to be full is not essential in
the subject (or the question) mentioned above. In the latter half part of \S 2, we
mention the author’s density theorem which says that a submodule of a Hilbert
C’-module $X$ is dense inin some topology.
In \S 3, as an easy application of results in \S 2 and the author’s results oncrossed products ([2], [5]), we discuss the orthogonal complementedness of closed
submodules in crossed products of Hilbert C’-modules.
$A$

$A$

$\mathrm{Y}$$A$$\mathrm{Y}\oplus \mathrm{Y}^{\perp}$$A$$(\mathrm{Y}^{\perp})^{\perp}=\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{w}\mathrm{e}\mathrm{U}$$\mathrm{e}\mathrm{a}s$$\mathrm{Y}$$\mathrm{Y}$$(\mathrm{Y}^{\perp})^{\perp}$$C^{*}-$

2. Recent Developments and Density Theorem
Recall the definition of a Hilbert C’-module. Let be a C’-algebra. By a rightHilbert -module, we mean a right -module $X$ equipped with an -valued pairing, called an -valued inner product, satisfying the following conditions:
$A$

$A$

$\langle\cdot, \rangle$

$A$

$A$

$A$

120

is sesquilinear. (We make the convention thatin the first variable and is linear in the second variable.)(1)$\langle\cdot, \cdot\rangle$(2)$\langle x , y\rangle=\langle y, x\rangle$(3)$\langle x :(4)$\langle x , x\rangle\geqq 0$’$\langle\cdot, \cdot\rangle$

is conjugate-linear

for all $x,$ $y\in X$ .

ya\rangle=\langle x, y\rangle a$for all $a\in A$ andfor all $x\in X$ , and$x,$ $y\in$\langle x , x\rangle=0$

X$.

implies that $x=0$ .

(5) $X$ is a Banach space with respect to the norm$||x||=||\langle x, x\rangle||\#$.

Furthermore, $X$ is said to be full if $X$ satisfies an additional condition:(6) the closed linear span of $\{\langle x , y\rangle|x, y\in X\}$ coincides with$A$.

be a C’-algebra. Left Hilbert -modules are defined similarly, except
requirethat wethat should act on the left of $X$ , that the -valued inner productshould be linear in the first variable, and thatfor all$\in A$$y\in
X$
and $x,$.
Let and $B$ be C’-algebras. We denote by
the -valued inner producton the left Hilbert -module and by
the $B$ -valued inner product on the rightHilbert $B$ -module, respectively. By an $A-B$-imprimitivity bimodule $X$ , we meana full left Hilbert -module and full right Hilbert $B$ -module $X$ satisfying

Let

$A$

$A$

$A$

$A$

$\langle ax, y\rangle=a\langle x , y\rangle$$\langle\cdot, \cdot\rangle$

$a$

$A$

$A\langle\cdot, \cdot\rangle$

$A$

$A$

$\langle\cdot, \cdot\rangle_{B}$$A$$z\in X$ .
for all
Here we remark that it follows from the above condntion (7) that the following
condition holds:(7)$A\langle x, y\rangle\cdot z=x\cdot\langle y , z\rangle_{B}$(8)$A\langle xb , y\rangle=A\langle x , yb^{*}\rangle$$x,$ $y\in X$.

$x,$ $y,$and$\langle ax, y\rangle_{B}=\langle x , a^{*}y\rangle_{B}$for all$a$$\in A,$$b\in B$andNow we consider the question of when every closed submodule of a Hilbert-module $X$ is orthogonally closed or orthogonaUy complemented in $X$ . Note that
there are known cases where a single closed submodule of a Hilbert -module $X$
becomes orthogonally closed (or orthogonally complemented) in $X$ . For example,
if is an adjointable linear operator with closed range on $X$ , each of the kernel ofand the range of is orthogonally complemented. But all closed submodules ofa Hilbert C’-module are not necessarily orthogonally closed (hence complemented)
in general, as the following examples show.
$\mathrm{Y}$$C^{*}$$C^{*}$$\mathrm{Y}$

$T$

$T$

$T$

Example 2.1. Let $A=C([0,1])$ be the C’-algebra of all continuous functions on
the closed interval $[0,1]$ .

.

$\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}\mathrm{b}\mathrm{y}\langle f,g\rangle=f^{*}g.\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{c}1\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}1\mathrm{e}\mathrm{Y}=\{f\in X|f(0)=0\}^{\rangle}(\mathrm{l})\mathrm{P}\mathrm{u}\mathrm{t}X=A\mathrm{a}\mathrm{s}\mathrm{a}\mathrm{H}\mathrm{i}1\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{t}A- \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}1\mathrm{e}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{e}A- \mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\langle\cdot,\cdot$

Then$\mathrm{Y}^{\perp}=\{0\}$. Hence$\mathrm{Y}\oplus \mathrm{Y}^{\perp}\neq X$.

(2) Put $J=\{f\in A|f(\mathrm{O})=0\}$ and let $X=A\oplus j$ as a Hilbert-valued inner productdefined by$A$$\langle\langle\cdot, \cdot\rangle\rangle$.

$\langle\langle f_{1}\oplus f_{2}, g_{1}\oplus g_{2}\rangle\rangle=\langle f_{1}, g_{1}\rangle+\langle f_{2}, g_{2}\rangle$$A$-module with the121Consider $\mathrm{Y}=\{f\oplus f|f\in J\}$ . Then $\mathrm{Y}^{\perp}=\{g\oplus(-g)|g\in J\}$ . Hence we see that$\mathrm{Y}\oplus \mathrm{Y}^{\perp}=\{(f+g)\oplus(f-g)|f,g\in J\}=J\oplus J\neq X(=A\oplus J)$.

We denote by the spectrum of , that is, the set of (unitary) equivalenceclasses of nonzero irreducible representations of equipped with the Jacobson topology. We note that is a locally compact space, not necessarily a -space.
The first answer to our question above was given by Magajna [11]. Here recall
that a -algebra is called dual if it is a type I -algebra with discrete spectrum,of allor equivalently if is isomorphic to a -subalgebra of the -algebra
compact linear operators on some Hilbert space .
$\hat{A}$

$A$

$A$

$\hat{A}$$T_{0}$$A$$C^{*}$$C^{*}$$A$$C^{*}$$C^{*}$$C(\mathcal{H})$$\mathcal{H}$-algebras and let $X$ be an
beTheorem 2.2 ([11, Theorem 1]). Let
andA–B-imprimitivity bimodule. Then every closed right -submodule of $X$ is or-algebra.
thogonally complemented if and only if is isomorp $hic$ to a
$B$

$A$

$C^{*}$$B$$d\mathrm{u}alC^{*}$$B$

Soon after the above result was shown, Schweitzer [12] elaborated on Magajna’s
theorem, that is, he showed the following theorem:-algebras and let $X$ be an $A-$
Theorem 2.3 ([12, Theorem 1]). Let and $B$ be$B$ -imprimitivity bimodule.
If every closed right $B$ -submodule of $X$ is orthogonallyclosed, then there are familiesof Hilbert spaces such that $A\cong$$A$$C^{*}$$\{\mathcal{H}_{i}\}_{i\in If}\{\mathcal{K}_{i}\}_{i\in I}$$\sum_{i\in I}^{\oplus}C(\mathcal{H}_{1}),$$B \cong\sum_{i\in I}^{\oplus}C(\mathcal{K}_{i})$means isomo rphic.

and$X \cong\sum_{i\in I}^{\oplus}C(\mathcal{K}_{i}, \mathcal{H}_{i})$, where the symbol$”\underline{\simeq}$“

Remark that it is trivial that the converse holds in Theorem 2.3. As a corollary,furthermore we immediately have the following:–B-algebras and let $X$ be an
Corollary 2.4 ([12]). Let
and $B$ be
imp rimitivity bimodule. Then every closed right $B$ -submodule of $X$ is orthogonallyclosed if and only if every closed right $B$ -submodule of $X$ is orthogonally complemented in $X$ .
$A$$\mathrm{Y}$$A$$C^{*}$

Let $X$ be an $A-B$-imprimitivity bimodule. For a closed $A-B$-subbimoduleof $X$ , it is not difficult to prove that for $x\in X$ ,$A\langle x , y\rangle=0$

for all$y\in \mathrm{Y}\Leftrightarrow\langle x , y\rangle_{B}=0$

for all$y\in$Y.

Thus we see that$\mathrm{Y}^{\perp}=${$x\in X|_{A}\langle x$,$y\rangle=\langle x$,$y\rangle_{B}=0$

for all$y\in \mathrm{Y}$

}.

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Theorem 2.5 ([4, Theorem 2.3]). Let
andbe C’-algebras and let $X$ be an$A-B$ -imprimitivity bimodule. Consider the following conditions:
$A$

$B$

of is discrete in the Jacobson topology.
(2) The spectrumof is discrete in the Jacobson topology.
(3) Every closed $A-B$ -subbimodule of is comlemented in .
. If either or is a -sPace, then conditions
Then we have (1)(1) The spectrum$\hat{A}$$A$$\hat{B}$

$B$

$X$

$X$

$\hat{B}$$\hat{A}$$\Leftrightarrow(2)\Rightarrow(3)$$T_{1}$(1) $-(3)$ are equivalent.

In the above theorem, the implication (3)is not true in general. Hencethe assumption that either or be a -space is
to show the implication.
(3)
Recall that the primitive spectrum Prim $(A)$ of a C’-algebra is the topological space, consisting of all primitive ideals of , endowed with the Jacobson
topology.
$\Rightarrow(2)$$\hat{A}$$\hat{B}$$T_{1}$$\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{e}8\mathrm{s}\mathrm{a}\mathrm{r}\mathrm{y}$$\Rightarrow(2)$

$A$

$A$

Theorem 2.6 ([4, Theorem 2.6]). Let and be-algebras and let $X$ be an$A-B$ -imprimitivity bimodule. Consider the following conditions:$A$(1) The primitive spectrum Prim $(A)$ of

$A$

$B$

$C^{*}$

is discrete in the Jacobson topology.

(2) The primitive spectrum Prim $(E)$ of $B$ is discrete in the Jacobson topology.

(3) Every closed $A-B$ -subbimodule of $X$ is comlemented in $X$ .

. If either Prim $(A)$ or Prim $(B)$ is a
Then we have (1)$-(3)$
are equivalent.
conditions (1)$\Leftrightarrow(2)\Rightarrow(3)$$T_{1}$-space, then-algebra is dual if and only if
Note that a separable
is discrete ([3]).
But even though a nonsepamble C’-algebra has discrete spectrum
is notnecessariry dual.
$\hat{A}$$C^{*}$$A$$\hat{A},$$A$-algebras and let $X$ be an
Theorem 2.7 ([6, Theorem 2.3]). Let
andbe$A-B$ -imprimitivity bimodule. Then every closed $A-B$ -submodule of $X$ is orthogonally closed in $X$ if and only if eve closed $A-B$ -submodule of $X$ is orthogonallycomplemented in $X$ .
$A$

$B$

$C^{*}$$\mathrm{r}y$Corollary 2.8 ([6, Corollay 2.4]). Let
and-algebras and let $X$ be an
be$A-B$ -imprimitivity bimodule. Consider the following conditions (1) $-(4)$ :
$A$

$B$

$C^{*}$

of is discrete in the Jacobson topology.
(2) The spectrumof is discrete in the Jacobson topology.
(3) Every closed $A-B$ -submodule ofis complemented in .
(4) Every closed $A-B$ -submodule of is orthogonally closed in(1) The spectrum$\hat{A}$$A$$\hat{B}$

$B$

$X$

$X$

$X$

$X$

.

123

Then we have (1)conditions (1) $-(4)$ are equivalent.

$\Leftrightarrow(2)\supset(3)\Leftrightarrow(4)$. If either$\hat{A}$or$\hat{B}$is a$T_{1}$-space, thenRemark 2.9. Let and be -algebras and let $X$ be an $A-B$-imprimitivity
bimodule. Consider the following conditions (1) $-(4)$ .
$A$

$B$

$C^{*}$(1) Every closed $B$ -submodule of $X$ is orthogonally closed in $X$ .
(2) Every closed $B$ -submodule of $X$ is orthogonally comlemented in $X$ .

(3) Every closed $A-B$ -submodule of $X$ is orthogonally closed in $X$ .

(4) Every closed $A-B$ -submodule of $X$ is orthogonally comlemented in $X$ .

does. The implication (2)Then we have (1)not hold in general. For example, consider a UHF -algebra which is not of typeI. Then is not -space, hence not discrete. Since is simple, $X$ is automaticallyfull. Then $X$ has no nontrivial $A-B$ -subbimodules, hence condition (4) above
holds. Since is not a dual -algebra, condition (2) does not hold.
$\Leftarrow(3)$$\Leftrightarrow(2)\Rightarrow(3)\Leftrightarrow(4)$$B$$C^{*}$$B$$\hat{B}$$T_{1}$$B$$C^{*}$

Recall here that if $H$ is a Hilbert space, every subspace $K$ of $H$ is dense inin the norm topology. Thus we have a problem of how such a result is generalizedfor Hilbert C’-modules. In the rest of this section, we mention the density theorem
with respect tothat any submoduleof a Hilbert-module is dense insome topology. Its proof will appear in [10]. For this, we must intoroduce such a.
topology on Hilbert$K^{\perp\perp}$$\mathrm{Y}^{\perp\perp}$$C^{*}$$\mathrm{Y}$$C^{*}- \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}8$Deflnition 2.10. Let be a C’-algebra and let $X$ be a Hilbert -module. The$\sigma(X, S(A))$ -topology on $X$ is the topology generated by a basis consisting of thesets$O=$ { $x\in X||\varphi_{i}(\langle x-x_{0},$for $i=1,2,$
}
$A$

$A$

$\cdots,$ $n$$y\rangle)|<\epsilon$

for all choices of $x_{0}\in X,$
we denote by the closure of$\varphi_{1},$$\overline{\mathrm{Y}}$$\varphi_{2},$$\mathrm{Y}$. For a subset
andin the $\sigma(X, S(A))$ -topology.

$\cdots,$$\varphi_{n}\in S(A)$Theorem 2.11. (Density) Let be aThen every -submodule is dense in
$A$

$A$

$\mathrm{Y}$$C^{*}$$\epsilon>0$$\mathrm{Y}$of $X$ ,-algebra and let $X$ be a Hilbert A-module.
the $\sigma(X, S(A))$ -topology, that is,$\mathrm{Y}^{\perp\perp}in$$\overline{\mathrm{Y}}=\mathrm{Y}^{\perp\perp}$.

If we apply our density theorem above to Hilbert spaces $H$ , we easily obtainin the norm
the well known result that every subspace $K$ of $H$ is dense in
topology.
Corollary 2.12. Let be a-algebra and let $X$ be a Hilbert -module. For any
dense in $X$ in the $\sigma(X, S(A))$ -topology.
-submodule , we have$K^{\perp\perp}$

$A$

$A$

$\mathrm{Y}$$C^{*}$$A$$\mathrm{Y}\oplus \mathrm{Y}^{\perp}is$

The following result characterizes the closedness of submodules in terms of the-topology.

$\sigma(X, S(A))$

124
Corollary 2.13. Let be a-algebm and let $X$ be a Hilbert$\sigma(X,
S(A))$ -topology, that is,
submodule
is closed in thethere exists a submodule.
of $X$ such that$A$$C^{*}$$A$-module. An A, if and only if$\mathrm{Y}=\overline{\mathrm{Y}}$$\mathrm{Y}$$\mathrm{Y}=\mathrm{Y}_{0}^{\perp}$$\mathrm{Y}_{0}$The following result characterizes dual
topology.

$C^{*}$-algberas in terms of the $\sigma(X, S(A))-$

Corollary 2.14. Let-algebra and let $X$ be a Hilbert -module. Thenbe aevery closed -submodule of $X$ is closed in $X$ in the $\sigma(X, S(A))$ -topology, thatis,
if and only if is a dual C’-algebra.
$A$

$A$

$C^{*}$$A$$\mathrm{Y}$$A$$\mathrm{Y}=\overline{\mathrm{Y}}$3. Applications to Crossed Products of Hilbert C’-modules
Let $(A, G, \alpha)$ be a C’-dynamical system. By a C’-dynamical system, we meana triple $(A, G, \alpha)$ consisting of a -algebra , a locally compact group withleft invariant Haar measure $ds$ and a group homomorphism a from
into theautomorphism group of such that $G\ni tarrow\alpha_{t}(x)$ is continuous for each in in
the norm topology. Denote by $K(A, G)$ the linear space of all continuous functions
from into with compact support and by $L^{1}(A, G)$ the completion of $K(A, G)$
by the -norm. Note that $L^{1}(A, G))$ admits the Banach*-algebra structure. Thenthe -crossed productof by is the enveloping -algebra of $L^{1}(A, G)$ .
Recall that for any covariant representation, the representationofis defined by$A$$C^{*}$

$G$

$G$

$A$

$x$

$A$

$A$

$G$

$L^{1}$$C^{*}$$G$$A$$A\cross_{\alpha}G$$C^{*}$$(\pi, u, \mathcal{H})$$u,\mathcal{H})$$(\pi\cross$$A\cross_{\alpha}G$$( \pi\cross u)(x)=\int_{G}\pi(x(t))u_{t}dt$

For a given representationof on the Hilbert space$A$$(\pi_{A}, \mathcal{H})$,$x\in L^{1}(A, G)$of , we always denote bydefined by$A$.

$\overline{\pi_{A}}$the representation$L^{2}(\mathcal{H}_{A}, G)$$(\overline{\pi_{A}}(a)\xi)(t)=\pi_{A}(\alpha_{t^{-1}}(a))\xi(t)$for $a\in A,$
grable functions fromby, whereis the Hilbert space of all square inte. Define a unitary representation on
into$\xi\in L^{2}(\mathcal{H}_{A}, G)$$G$$L^{2}(\mathcal{H}_{A}, G)$$(\lambda^{A}s\xi)(t)=\xi(s^{-1}t)$Then$\lambda^{A}$$\mathcal{H}_{A}$.

is a covariant representation of . If
is faithful, thenis called the reduced C’-crossed product of by
and we$A$$(\overline{\pi_{A}}, \lambda^{A}, L^{2}(\mathcal{H}_{A}, G))$$(\overline{\pi_{A}}\cross\lambda^{A})(A\cross_{\alpha}G)$$L^{2}(\mathcal{H}_{A}, G)$$\pi_{A}$

$A$

$G$

denote it by.
Let $(A, G, \alpha)$ and $(B, G, \beta)$ be C’-dynamical systems and let $X$ be a left AHilbert module (resp. a right -Hilbert module). Suppose that there exists an-compatible action (resp. a -compatible action) of
on $X$ , that is, a grouphomomorphism from
into the group of invertible linear transformations on $X$
such that$A\cross_{\alpha,\mathrm{r}}G$$B$$\beta$$\alpha$$\eta$

$G$

$G$

125

(E1) $\eta_{t}(a\cdot x)=a_{t}(a)\eta_{t}(x)$ (resp.
(E2)$\eta_{t}(x\cdot b)=\eta_{t}(x)\beta_{t}(b)$(resp.

$A\langle\eta_{t}(x), \eta_{t}(y)\rangle=a_{t}(_{A}\langle x, y\rangle)$$\langle\eta_{t}(x),$);$\eta_{t}(y)\rangle_{B}=\beta_{t}(\langle x,$ $y\rangle_{B})$)for each $t\in G$ , $\in A,$ $b\in B,$ $y\in X$ ; and such thatis continuous from
into $X$ for each $x\in X$ in norm.
Let $(A, G, a)$ and $(B, G, \beta)$ be-dynamical systems. Let be ancompatible action of on a left A- and right -Hilbert module $X$ , that is, is an-compatible and -compatible action of . Then there exists a left
-and
right
-Hilbert module
containing a dense subspace $K(X, G)$ such
that
$a$

$tarrow\eta_{t}(x)$

$x,$

$G$

$C^{*}$$(a, \beta)-$$\eta$

$G$

$a$

$B$

$\eta$$G$$\beta$$(B\cross_{\beta}G)$$(A\cross_{\alpha}G)$$X\cross_{\eta}G$$(f \cdot x)(s)=\int_{G}f(t)\eta_{t}(x(t^{-1}s))dt$,$(x \cdot g)(s)=\int_{G}x(t)\beta_{t}(g(t^{-1}\mathit{8}))dt$,$A \mathrm{x}_{\alpha}G\langle x, y\rangle(s)=\int_{G}A\langle x(st^{-1}), \eta_{s}(y(t^{-1}))\rangle dt$,$\langle x, y\rangle_{B\mathrm{x}_{\beta}G}(s)=\int_{G}\beta_{t^{-1}}(\langle x(t), y(ts)\rangle_{B})dt$

for $f\in K(A, G),$ $x,$ $y\in K(X, G)$ , and $g\in K(B, G)$ . We callthe (flll)
crossed product of $X$ by . Here $K(X, G)$ (resp. $K(A,$ $G)$ and $K(B,$ $G)$ ) denotes
the set of continuous functions from
into $X$ (resp.
and $B$ ) with compact
support.
$X\cross_{\eta}G$

$G$

$G$

$A$

Let
and
be
-algebras and let $X$ be an A-B-Hilbert bimodule. We
say that a representation of $X$ as an A-B-Hilbert bimodule is a triple
consisting of nondegenerate representations
andof and on Hilbert spaces, respectively, together with a linear map
andsuch that$A$$B$$C^{*}$$(\pi_{A}, \pi_{X}, \pi_{B})$$\pi_{B}$$\pi_{A}$$\mathcal{H}_{A}$

$A$

$B$

$\pi_{X}:Xarrow B(\mathcal{H}_{B}, \mathcal{H}_{A})$$\mathcal{H}_{B}$(R1) $\pi_{X}(ax)=\pi_{A}(a)\pi_{X}(x)$ ,(R2) $\pi_{X}(xb)=\pi_{X}(x)\pi_{B}(b)$ ,(R3)$\pi_{A}(_{A}\langle x, y\rangle)=\pi_{X}(x)\pi_{X}(y)^{*}$(R4)$\pi_{B}(\langle x, y)_{B})=\pi_{X}(x)^{*}\pi_{X}(y)$for$x,$ $y\in

X$

, and

for all $a\in A,$ $x,$ $y\in X$ , and $b\in B$ , where
denotes the set of all boundedlinear operators frominto.
Now we suppose that $X$ is a right Hilbert -module with an -inner product. We define a linear operator $_{x,y}$ on $X$ by$B(\mathcal{H}_{B},\mathcal{H}_{A})$$\mathcal{H}_{B}$$\mathcal{H}_{A}$

$A$

$A$

$\langle\cdot, \cdot\rangle$$\Theta_{x,y}(z)=x\cdot\langle y, z\rangle$

for all $x,$$y,$$z\in X$$\{_{x,y}|xy\in X\}$

. We denote by. Then$X$$\mathcal{K}(X)$is a full leftthe-algebra generated by the set-Hilbert module with respect to$C^{*}$$\mathcal{K}(X)$126
the natural left action defined by $t\cdot x=t(x)$ for
and $x\in X$ and the innerproduct. Thus $X$ is a
-Hilbert bimodule.
$t\in \mathcal{K}(X)$$\mathcal{K}(X)- A$$\mathcal{K}(X)\langle x , y\rangle\equiv_{x,y}$Deflnition 3.1. Let $(A, G, a)$ and $(B, G, \beta)$ be -dynamical systems. Let $X$ be aleft -Hilbert module and a right -Hilbert module with an-compatible action of . Suppose thatandare covariant representationsof and , respectively. Then we say that a representationof $X$
into
is covariant if$C^{*}$

$A$

$B$

$G$

$\eta$$(a, \beta)$$(\pi_{A}, u, ?t_{A})$$(\pi_{B}, v, \mathcal{H}_{B})$

$B$

$A$

$(\pi_{A}, \pi_{X}, \pi_{B}, u, v)$$B(\mathcal{H}_{B}, \mathcal{H}_{A})$’$\pi_{X}(\eta_{t}(x))=u_{t}\pi_{X}(x)v_{t}$Then we can define the representation

for all$\pi_{X}\cross v$

of$x\in X,$ $t\in G$$X\cross_{\eta}G$

into

.

$\mathcal{B}(\mathcal{H}_{B}, \mathcal{H}_{A})$by$( \pi_{X}\mathrm{x}v)(x)=\int_{G}\pi_{X}(x(s))v_{s}ds$

for $x\in K(X, G)$ . Thus we obtain the representation.
into$(\pi_{A}\cross u, \pi_{X}\cross v, \pi_{B}\cross v)$

of$B(\mathcal{H}_{B},\mathcal{H}_{A})$$X\cross_{\eta}G$

Define the representation$\overline{\pi_{X}}$of $X$ into

$B(L^{2}(\mathcal{H}_{B}, G),$

$L^{2}(?t_{A}, G))$by$(\overline{\pi_{X}}(x)\xi)(t)=\pi_{X}(\eta_{t^{-1}}(x))\xi(t)$for all $x\in X,$ $t\in G$ andrepresentation of $X$ into$\xi\in L^{2}(\mathcal{H}_{B}, G)$. Then$(\overline{\pi_{A}},\overline{\pi_{X}},\overline{\pi_{B}}, \lambda^{A}, \lambda^{B})$$B(L^{2}(\mathcal{H}_{B},G),$ $L^{2}(\mathcal{H}_{A}, G))$, that is, we haveis a covariant$(\overline{\pi_{X}}(\eta_{\epsilon}(x))\xi)(t)=((\lambda^{A}S\overline{\pi_{X}}(x)\lambda^{B}’)\epsilon\xi)(t)$and. The following definition of a reduced crossed
product of a Hilbert C’-module was introduced by the author [9].

for

$s,$

$t\in G$$\xi\in L^{2}(\mathcal{H}_{B}, G)$

Deflnition 3.2. Let $(A, G, a)$ and $(B, G,\beta)$ be -dynamical systems. Let be an-compatible action of on a left A- and right -Hilbert module $X$ . Consider
a representationof $X$ , whereandare faithful representations of and , respectively. Thenis automatically faithful. Consider
the representation
intoof. Then we say
thatis the reduced crossed product of $X$ by , and we denote it. It is easy to verify thatbyis a left-Hilbert moduleand a right $(B\cross\rho’,G)$ -Hilbert module. We remark that
does not dependon the choice of a pair of faithful representations
andof and .
$C^{*}$$\eta$$G$$(\alpha, \beta)$$B$$(\pi_{A}, \pi_{X}, \pi_{B})$$A$$(\pi_{A}, \mathcal{H}_{A})$$B$$\overline{\pi_{X}}\cross\lambda^{B}$$(\pi_{B}, \mathcal{H}_{B})$$\pi_{X}$$X\cross_{\eta}G$$B(L^{2}(\mathcal{H}_{B}, G),$ $L^{2}(\mathcal{H}_{A}, G))$$G$$(\overline{\pi_{X}}\cross\lambda^{B})(X\cross_{\eta}G)$$X\cross_{\eta,t}G$$X\cross_{\eta,\mathrm{r}}G$$(A\cross_{\alpha,\mathrm{r}}G)$$X\cross_{\eta,r}G$$\pi_{A}$$\pi_{B}$

$B$

$A$

Proposition 3.3([9, Proposition 2.13]). Let $(A, G,\alpha)$ be a C’-dynamical system
and let $X$ be a right -Hilbert module. Suppose that there exists an -compatible
action of on X. If is amenable, thenis isomorphic to.
$A$$\eta$$G$$\alpha$$G$$X\cross_{\eta}G$

The reader is referred to [9] for crossed products of Hilbert
their representations mentioned in the above.

$X\cross_{\eta,\mathrm{r}}G$$C^{*}$-modules and127$\hat{A}$Given a C’-dynamical system $(A, G, a),$which is defined byinduces the natural action of$a$$(t, [\pi])\in G\cross\hat{A}arrow[\pi 0\alpha_{t^{-1}}]\in\hat{A}$

$G$

on

.

This map makes
into a topological transformation group acting on . Let, which is defined by $S_{[\pi]}=\{t\in G|[\pi\circ a_{t^{-1}}]=[\pi]\}$ . If
be the stability group at
all stability groups are trivial, i.e.,
consists only of the identity of at every, it is said that acts freely on .
$\hat{A}$$G$$S_{[\pi]}$$[\pi]$$G$$S_{[\pi]}$$[\pi]\in\hat{A}$$\hat{A}$$G$

Theorem 3.4. Let $(A, G, \alpha)$ be a-dynamical system and let $X$ be a $hll$ right-Hilbert module with an -compatible action of G. Suppose that acts ffeely
on \^A. Then the following condtions (1) $-(3)$ are equivalent:$C^{*}$$A$$\alpha$(1) For any closed right
(2)

$G$

$A$

$G$

$\eta$-submodule

is discrete, and for any closedof , we have$X$$\mathrm{Y}$$(A\cross_{\alpha}G)$$X=\mathrm{Y}\oplus \mathrm{Y}^{\perp}$-submodule$\tilde{\mathrm{Y}}$of.

$X\cross_{\eta}G$, we have$X\mathrm{x}_{\eta}G=\tilde{\mathrm{Y}}\oplus(\tilde{\mathrm{Y}})^{\perp}$(3)
is discrete, and for any closed right
have
$G$

$(A\cross_{\alpha,r}G)$-submodule$\tilde{\mathrm{Y}}$of$X\cross_{\eta,\mathrm{r}}G$, we$X\cross_{\eta,\tau}G=\overline{\mathrm{Y}}\oplus(\overline{\mathrm{Y}})^{\perp}$. Sinceis a full right -Hilbert module, it is a $\mathcal{K}(X)-A-$
imprimitivity bimodule. Hence it folllows from Theorem 2.2 that is a dual
is a dual -algebra
algebra. Since acts freely on
is discrete andis a([2, Theorem], or [5]). This shows codition (2) by Theorem 2.2, since
full right-Hilbert bmodule.
(2). This is trivial.
is a dual C’-algebra. Since(3). Condition (3) implies that
discrete,isis embedded into
as a -algebra. Since every -subalgebraof a dual -algebra is dual, so is also . Hence condition (1) follows from Thoerem
2.2.

Proof. (1)$\Rightarrow(2)$

$X$

$A$

$A$

$G$

$\hat{A},$$G$$C^{*}-$$C^{*}$$A\cross_{\alpha}G$$X\cross_{\eta}G$$(A\cross_{\alpha}G)$$\Rightarrow(3)$$\Rightarrow(1)$$G$$A\cross_{a,r}G$$A$$C^{*}$$C^{*}$$A\cross_{\alpha,r}G$$A$$C^{*}$-dynamical system $(A, G, a)$ , we say that is pointwise unitary if
For afor every irreducible representationof , there exists a strongly continuous
unitary representation of on the Hilbert spacesuch that$C^{*}$$\alpha$$A$$(\pi, H_{\pi})$

$G$

$u$

$H_{\pi}$’$\pi(a_{t}(x))=u_{t}\pi(x)u_{t}$for all $x\in A$ and $t\in G$ .
-dynamical system and let $X$ be a full right
Theorem 3.5. Let $(A, G, a)$ be a
$A- Hilbe\hslash$ module with an a-compatible actionof G. Suppose that is a compact
group. Consider the following condtions.
$C^{*}$$G$$\eta$(1) For any closed right(2) For any closed$A$of , we have
, we have-submodule of-submodule$\mathrm{Y}$$X$$\tilde{\mathrm{Y}}$$(A\cross_{\alpha}G)$$X\cross_{\eta}G$$X=\mathrm{Y}\oplus \mathrm{Y}^{\perp}$.

$X\cross_{\eta}G=\tilde{\mathrm{Y}}\oplus(\tilde{\mathrm{Y}})^{\perp}$.

128
Then we have (1)(2). Furthermore we suppose thatabelian. If is of type I and is pointwise unitary, we have (2)$\Rightarrow$

$A$

$a$

Proof. (1)$\Rightarrow(2)$is (compact)$G$$\Rightarrow(1)$.

. It follows from the proof of Theorem 3.4 above that

$A$

is

a dual -algebra. On the other hand, by Imai-Takai’s duality we see that there
exists a coaction 6 of onsuch that(see$C^{*}$$G$$(A\cross_{\alpha}G)\cross_{\delta}G\cong A\otimes C(L^{2}(G))$$A\cross_{\alpha}G$

the paragrapf following Remark 3.6 for the detail of crossed products by coactions).
Since is compact,
is embedded into
as a C’-algebra. Since$A\otimes C(L^{2}(G))$ is a dual C’-algebra, so is. This shows condition (2) by
Theorem 2.2.
(2)
. Suppose that is abelian. Since is of type I and is pointwise
unitary, it follows from the proof ofin [5, Theorem 3.2] that the actionof induced by the dual action of acts freely on the spectrum. Sinceis discrete,is a dual -algebra. Hence $A\otimes C(L^{2}(G))$ is also a dual
C’-algebra, which shows that is dual. Thus condition (1) follows from Thoerem$G$$A\cross_{\alpha}G$$(A\cross_{\alpha}G)\cross_{\delta}G$$A\cross_{\alpha}G$$G$$\Rightarrow(1)$

$A$

$a$

$(\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i})$$\hat{G}$$\hat{G}$$\hat{\alpha}$$\hat{G}$$(A\cross_{\alpha}G)^{\wedge}$$(A\cross_{\alpha}G)\cross_{\hat{\alpha}}\hat{G}$$C^{*}$$A$

2.2.

$\square$Remark 3.6. Let $(A, G, a)$ be a C’-dynamical system. Suppose thatis an-subalgebra of . Thena-invariant
is not necessarily embedded into
as a -algebra (see [1]). Let $(A, G, a)$ and $(B, G, \beta)$ be C’-dynamicalsystems and let $X$ be an $A-B$ -imprimitivity bimodule with an-compatible
action of . If is an -invariant closed $A-B$-submodule of $X$ , thenisembedded into
as an $(A\cross_{\alpha}G)-(B\cross\rho G)$ -bimodule. This is not suprising.
For, the closed $A-B$ -submodules in the case of Hilbert C’-modules correspondto the closed ideals in the case of C’-algebras. The author will discuss the further
details of this subject elsewhere.
$B$

$A$

$C^{*}$$A\cross_{\alpha}G$$B\cross_{\alpha}G$$C^{*}$$(\alpha, \beta)$$G$$\eta$$\mathrm{Y}$$\mathrm{Y}\cross_{\eta}G$$\eta$$X\cross_{\eta}G$

Here we briefly review the definition of the crossed products by coactions. Let
be a locally compact group with left invariant Haar measure $ds$ . We denote by
the left regular representation of on $L^{2}(G)$ . We define the representation of$L^{1}(G)$ on $L^{2}(G)$ by
$G$

$G$

$\lambda$$\tilde{\lambda}$$\tilde{\lambda}(f)=\int_{G}f(s)\lambda_{\theta}ds$

for $f\in L^{1}(G)$ . Then the reduced group-algebraof $G$ is defined as thenorm closure ofin the set of all bounded linear operators on $L^{2}(G)$ . We
$C^{*}$$C_{r}^{*}(G)$$\tilde{\lambda}(L^{1}(G))$

write

for
above.
Let be a C’-algebra and denote byof the injective -tensor productof $M(A\otimes_{\min}C_{f}’(G))$ by$\tilde{\lambda}(f)$$\lambda_{f}$$A$$M(A\otimes_{\min} C_{f}^{*}(G))$$C^{*}$$A\otimes_{\min}C_{f}’(G)$the multiplier algebra. We then define the C’-subalgebra$\overline{M}(A\otimes_{\min}C_{f}’(G))$$\overline{M}(A\otimes_{\min}C_{f}^{*}(G))=${ $m\in M(A\otimes_{\min}C_{f}’(G))|m(1\otimes x),$We denote by$W_{G}$$(1\otimes x)m\in A\otimes_{\min}C_{f}^{*}(G)$for all $x\in C_{f}^{*}(G)$ }.

the unitary operator on $L^{2}(G\cross G)$ defined by$(W_{G}\xi)(s, t)=\xi(s, s^{-1}t)$

for $\xi\in L^{2}(G\cross G)$ and$s,$ $t\in

G$

.

129
Define the homomorphism

from$\delta_{G}$

into$C_{r}^{*}(G)$$\delta_{G}(\lambda_{f})=W_{G}(\lambda_{f}\otimes 1)W_{G^{*}}$$\overline{M}(C_{f}^{*}(G)\otimes_{\min}C_{f}^{*}(G))$

for$f\in L^{1}(G)$We say that an injective homomorphism 6 frominto
coaction of a locally compact group on if 6 satisfies:
$A$

$G$

$\underline{(\mathrm{C}}1)$

there is an approximate identity$M(A\otimes_{\min}C_{f}^{*}(G))$

by.

$\overline{M}(A\otimes_{\min} C_{f}^{*}(G))$

is a

$A$

$\{e_{i}\}$

for

$A$

such that$\delta(e_{i})arrow 1$

strictly in;(C2)(id CE9 )for all $a\in A$ , where we always denote by $id$
the identity map on each considered set.
Furthermore, the coaction 6 is said to be nondegenerate if it satisfies the additional condition:$(\delta\otimes \mathrm{i}\mathrm{d})(\delta(a))=$$\delta_{G}$(C3) for every nonzero$\varphi\in A^{*}$$(\delta(a))$, there exists$\psi\in C_{f}^{*}(G)^{*}$

such that$(\varphi\otimes\psi)\circ\delta\neq 0$.

This is equivalent to the condition that the closed linear span of, wherebe equal tois the identity of the multiplier algebra $M(A)$for A. (In (C2) and (C3), we implicitly extended 6 to $M(A\otimes_{\min}C_{f}^{*}(G))$ , which isensured by condition (C1).) We always denote by the same symbol 6 the extensionof 6 to $M(A\otimes_{\min}C_{f}’(G))$ .
Let be a coaction of a locally compact group on and let $C_{0}(G)$ be theset of all continuous functions on vanishing at infinity. We denote bythemultiplication operator on $L^{2}(G)$ given by $f\in C_{0}(G)$ which is defined by$\delta(A)(1_{A}\otimes C_{f}’(G))$$A\otimes_{\min}C_{f}^{*}(G)$$1_{A}$$A$$G$$\delta$$G$$M_{f}$$(M_{f}\xi)(t)=f(t)\xi(t)$for all $\xi\in L^{2}(G)$ . Then the crossed productof by 6 is the C’-subalgebra$M(A\otimes
C(L^{2}(G)))$ofgenerated by the set $\{\delta(a)(1\otimes M_{f})|a\in A, f\in C_{0}(G)\}$ , where$C(L^{2}(G))$ always denotes the-algebra of all compact linear operators on $L^{2}(G)$ .
$X$
Let be a right -Hilbert module. We define a linear operator
on $X$ by$A\cross_{\delta}G$$A$$C^{*}$$A$$\Theta_{x,y}$$\mathrm{O}-_{x,y}(z)=x\cdot\langle y, z\rangle$

for all$z\in X$ .

-algebra generated by the set
We denote by
the$\{\Theta_{x,y}|x, y\in X\}$ . Then $X$ is a full left
-Hilbert module with respect tothe natural left action defined by $t\cdot x=t(x)$ for
and $x\in X$ and the innerproduct. Thus $X$ is a
-Hilbert module.
)
We denote by $M(X)$ the set of all multipliers of a right Hilbert A-module
X. We refer to $M(X)$ as the multiplier bimodule of $X$ , and note that $M(X)$ is an$M(\mathcal{K}(X))-M(A)$ -Hilbert module, where
and $M(A)$ are the multiplieralgebras for,andrespectively.
Let:
be a coaction of a locally compact groupon the -algebra and let
be a coaction of:
on the C’-algebra . Suppose that $X$ is a $B-A$ -Hilbert bimodule. We saythat a linear map: $Xarrow M(X\otimes C_{f}^{*}(G))$ is a -compatible coaction (resp. a$x,$ $y,$$\mathcal{K}(X)$$C^{*}$$\mathcal{K}(X)$$t\in \mathcal{K}(X)$$\mathcal{K}\langle X$$\langle x, y\rangle\equiv_{x,y}$$\mathcal{K}(X)- A$$M(\mathcal{K}(X))$$A$$\mathcal{K}(X)$$\delta_{A}$$G$$Aarrow\overline{M}(A\otimes_{\min}C_{\mathrm{r}}^{*}(G))$$A$$C^{*}$$G$$\delta_{B}$$Barrow\overline{M}(B\otimes_{\min} C_{f}’(G))$$B$$\delta_{X}$$\delta_{A}$130-compatible coaction) of the locally compact group
following conditions:$\delta_{B}$(D1)(resp. (Dl)
(D2)

(D4)

$\delta_{X}$satisfies thefor all $x\in X$ and $z\in C_{f}’(G)$ ;lies in $X\otimes C_{f}’(G)$ for all $x\in X$ and $z\in C_{f}’(G);$ )for all $x\in X$ and $b\in B$ ;for all $x\in X$ and $a\in A;$ )$\delta_{X}(x\cdot a)=\delta_{x}(x)\cdot\delta_{A}(a)$$\delta_{B}(_{B}\langle x, y\rangle)=_{M(B\oplus_{\min}G_{f}^{*}(G))}\langle\delta_{X}(x), \delta_{X}(y)\rangle$(resp. (D3)

if

$X$

$X\otimes C_{f}^{*}(G)$$\delta_{X}(x)(1_{A}\otimes z)$$\delta_{X}(b\cdot x)=\delta_{B}(b)\cdot\delta_{X}(x)$(resp. (D2)
(D3)

lies in$(1_{B}\otimes z)\delta_{X}(x)$

on

$G$

,

$y\rangle_{4})=\langle\delta_{X}(x)$$\delta_{A}(\langle x,$$(\delta_{X}\otimes \mathrm{i}\mathrm{d})\circ\delta_{X}=(\mathrm{i}\mathrm{d}\otimes\delta_{G})\circ\delta_{\mathrm{x}}$;$\delta_{X}(y)\rangle_{M(A\mathrm{e}_{\min}\circ i(G))};$).

(In (D1) and (D2) (resp. in (Dl) and (D2)), we implicitly extended themodule action on the $(B\otimes_{\min}C_{f}^{l}(G))-(A\otimes_{\min} C_{f}^{*}(G))$ -Hilbert module
to actions of the multiplier algebras on the multiplier bimodule; in (D3) (resp. (D3))
we extended the inner products to $M(X\otimes C_{f}^{*}(G))$ ; and in (D4), we used the strictly
continuous extensions ofandto make sense of the compositions.)
Furthermore, we say thatis nondegenerate if
satisfies the following
additional conditions:$X\otimes C_{r}^{*}(G)$$\delta_{X}\otimes \mathrm{i}\mathrm{d}$$\mathrm{i}\mathrm{d}\otimes\delta_{G}$$\delta_{X}$(D5) the closed linear span of(D5) the closed linear span of$\delta_{X}$$(1_{B}\otimes C_{f}^{*}(G))\delta_{X}(X)$$\delta_{X}(X)$($1_{A}$

is equal to$X\otimes C_{f}’(G)$;C8) $C_{f}^{*}(G))$ is equal to $X$ C8) $C_{f}^{*}(G)$.

Suppose that a -algebra is concretely represented on some Hilbert space. Let
on
be a coaction ofand let $X$ be a right -Hilbert module,
and we suppose thatis concretely represented on some Hilbert space.
Given a -compatible coactionofon $X$ , the crossed productof $X$ byis the right-Hilbert closed submodule of(Hrc, ) generated by the set $\{\delta_{X}(x)(1_{A}\otimes M_{f})|x\in X, f\in C_{0}(G)\}$ .
Then the inner product on
is given in terms of the usual operator adjoint’$)arrow
B$$G)$ ) by(Hrc, )( (Hrc, $G),$:$A$$C^{*}$$A$$G$$\delta_{A}$$\mathcal{H}_{A}$$A$$\mathcal{K}(X)$$\mathcal{H}_{\mathcal{K}}$$G$$\delta_{X}$$\delta_{A}$$X\cross_{\delta_{X}}G$$M(X\otimes C_{f}^{*}(G))\subset$$(A\cross_{\delta_{A}}G)$$\delta_{X}$$B(L^{2}(\mathcal{H}_{A}, G),$ $L^{2}$

$G$

$)$

$X\cross_{\delta_{X}}G$$G$$\mathcal{B}(L^{2}(\mathcal{H}_{A}, G),$ $L^{2}$$\langle x :$L^{2}$$L^{2}(\mathcal{H}_{A},$y\rangle_{A\mathrm{x}_{\delta_{A^{G}}}}=x^{*}y$for$x,$ $y\in X\cross_{\delta_{X}}G$.

Suppose thatis nondegenerate. If, for every irreducible representation, there exists a unitary $W\in M(\pi(A)\otimes_{\min}C_{f}^{*}(G))$ such that$\delta_{A}$$A$$(\pi\otimes id)(\delta_{A}(a))=W(\pi(a)\otimes 1)W$for all$a$$\in A$, then

$\pi$

of

’

is called pointwise unitary.

$\delta_{A}$Theorem 3.7. Let be a C’-algebra with a nondegenerate coactionof and-compatible coaction
let $X$ be a full right -Hilbert module with anof .
Consider the following condtions.
$A$

(1)

$G$

$\delta_{A}$is discrete and for any closed submodule(2) For any closed$(A\cross_{\delta_{A}}G)$$G$$\delta_{A}$$A$-submodule$\mathrm{Y}$of , we have$\tilde{\mathrm{Y}}ofX\cross_{\delta_{X}}G$$G$$\delta_{X}$$X$, we have$X=\mathrm{Y}\oplus \mathrm{Y}^{\perp}$.

$X\cross_{\delta_{X}}G=\tilde{\mathrm{Y}}\oplus(\tilde{\mathrm{Y}})^{\perp}$

.

131

Then we have (1)sapce, we have (2)$\Rightarrow(2)$$\Rightarrow(1)$Proof. (1)$\Rightarrow(2)$

.

. If

$a$

is pointwise unitary and if$\hat{A}$is a Hausdorff. It follows from the proof of Theorem 3.4 thatis a dual-algebra (see [2, Theorem 3.1]).

is a dual-algebra. Then we see that
Thus we obtain condition (2) by Theorem 2.2.
. Condition (2) and Theorem 2.2 imply that(2)-algebra. Then condition (1) follows from [2, Thereom 3.1].
$C^{*}$$C^{*}$$A\cross_{\delta_{A}}G$$\Rightarrow(1)$$C^{*}$$A$$A\cross_{\delta_{A}}G$

is a dual$\square$of a compact group andRemark 3.8. Let be a -algebra with a coaction$X$of . If
let
be a full right -Hilbert module with an -compatible coaction
every closed
is orthogonally complemented,-submodule ofthen so is every closed submodule of $X$ in $X$ (see [2, Proposition 3.3]).
$A$$C^{*}$$(A\cross_{\delta_{A}}G)$$G$$\delta_{A}$$A$$\delta_{A}$$\delta_{X}$$G$$\overline{\mathrm{Y}}$$X\cross_{\delta_{X}}G$$\mathrm{Y}$REFERENCES
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$C^{*}-$