数理解析研究所講究録 1418 巻 2005 年 41-50

Acharacterization of coactions which fix
Cartan subalgebras(Joint work with Takehiko YAMANOUCHI)

北海道大学大学院理学研究科青井久 (Hisashi AOI)

Department of Mathematics Faculty of Science,
Hokkaido University

1Preparation
In this section, we summarize the basic facts about measured groupoids andvon Neumann algebras associated to them. Further details regarding theseobjects can be found in [3], [8], [9]. We also briefly discuss actions of locallycompact quantum groups on von Neumann algebras.
We assume that all von Neumann algebras in this paper have separable
preduals, and: standard Borel space,
7% :discrete measured equivalence relation on $(X, \mu))$: left counting measure on 7%,: normalized 2-cocycle on 72,$\mathcal{R}(x):=\{y\in X : (x, y)\in \mathcal{R}\}$ ,[72]: bimeasurable nonsingular transformations
such thatfor . in $X$ },is in.

$(X, \mu)$$\nu$$\sigma$$:=\{\varphi$$\varphi(x)$$\Gamma(\varphi):=\{(x, \varphi(x)) :$\mathcal{R}(x)$x\in \mathrm{D}\mathrm{o}\mathrm{m}(\varphi)\}$$\mathrm{a}.\mathrm{e}$$x$$(\varphi\in[7\%])$

42

and a vonDefinition 1. (1) We define a von Neumann algebra
following:theby
Neumann subalgebra $W^{*}(X)$ which act on$W^{*}(\mathcal{R}, \sigma)$$L^{2}(\mathcal{R}, \nu)$is a left finite function on$W^{*}(X).--\{L^{\sigma}(d) : d\in L^{\infty}(X, \mu)\}$ ,$W^{*}(\mathcal{R}, \sigma):=$where we regard
defined by{La f) :$\{$$L^{\infty}(X, \mu)$$f$

as functions on the diagonal of$\mathcal{R}\}’$$\mathcal{R}$$\{L^{\sigma}(f)\xi\}(x, z):=,\sum_{y:(/,x)\in \mathcal{R}}f(x, y)\xi(y, z)\sigma(x, y, z)$, and$L^{\sigma}(f)$is.

algebra and $D$ be a subalgebra of
be a von(2) Let$D$is a Cartan subalgebra of if $D$ satisfies the following:$A$.

$\backslash ^{-}|arrow\backslash \mathrm{e}\mathrm{u}\mathrm{n}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{n}$$A$. We call

$A$

(i)

is maximal abelian in$D$(ii) $D$ is regular in

$A$

$A$

.

, i.e., the normalizer $N_{A}(D)$ generates$N_{A}(D1$ $:=$

, where{ $u\in A$ : is unitary and $uDu^{*}=D$ }.
$u$(iii) there exists a faithful normal conditional expectation
$D$

$A$

$E_{D}$

from

$A$

onto

.

Theorem 2 ([3, Theorem 1]). For each inclusion of a von Neumann algebra and a Cartan subalgebra $D$ of , there exists a standard Borel space
on $X$ with a normal$(X, \mu)$ and a discrete measured equivalence relation.
ized 2-cocycle such that $(D\subseteq A)$ is isomor $hic$ to
$A$

$A$

$\mathcal{R}$$(W^{*}(X)\subseteq W^{*}(\mathcal{R}, \sigma))$$\sigma$

Theorem 3 ([1, Corollary 3.5) . Suppose is a von Neumann algebra withand $D=W^{*}(X)$ . Thena Cartan subalgebra $D$ of such that
correspondence between the set of Borel subrelations
there exists aof on $(X, \mu)$ and the set of von Neumann subalgebras of A whichcontain $D$ :
$)$

$A$

$A$

$A=W^{*}(\mathcal{R}, \sigma)$$b_{\dot{l}}jective$

$S$

$B$

$\mathcal{R}$$B\mapsto S_{B}\subseteq \mathcal{R}$$S\mapsto W^{*}(S, \sigma):=\{L^{\sigma}(f)\in A : \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f)\subseteq S\}\subseteq A$

.

43

Let
be a locally compact quantum group ( $M$ is a von
Neumann algebra, : $M\mapsto M\mathrm{O}M$ is a coproduct, (resp. ) is a left (resp.
right) invariant weight on ). A normal unital inje-homomorphismfrom onto
is called an action of on if satisfies the following:$\mathrm{G}=(M, \triangle, \varphi, \psi)$$\triangle$$\psi$$\varphi$$\mathrm{J}/I$$A$$\mathrm{c}.\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}*$$M\otimes A$$\mathrm{G}$

$A$

$(y$

$\alpha$$(\triangle\otimes id_{A})\alpha=(id_{M}\otimes\alpha)\alpha$.

In particular, if is cocommutative, i.e., $M$ is equal to the group von Neumann algebra $W^{*}(K)$ which is generated by the left regular representationof a locally compact group $K$ , and is equal to,:$K$then the action is called a coaction of$\mathrm{G}$$\lambda_{K}$$\triangle_{K}\wedge$$\triangle$$\lambda_{K}(k)\mapsto\lambda_{K}(k)8\lambda_{K}(k)$$\alpha$2

A reduction to coaction caseIn the discussion that follows, we fix a von Neumann algebra and a Cart
subalgebra $D$ of with an equivalence relation
and a normalized
2-cocycle of such that the pair $(D\subseteq A)$ is equal to.
$D$
We assume that the action fixes , i.e.,
is equal tofor each$d\in D$ . It follows that the fixed-point algebra $A^{\alpha}:=\{a\in A:\alpha(a)=1\otimes a\}$is an intermediate subalgebra for.
We will prove that each such a action should be a coaction.
$A$

$A$

$\mathcal{R}_{011}(X, \mu)$$(W^{*}(X)\subseteq W^{*}(\mathcal{R}, \sigma))$$\mathcal{R}$$\sigma$$1\otimes d$$\alpha(d)$$\alpha$$D\subseteq A$

Proposition 4. Under the situation as above, the von Neumann subalgebra$\{(id_{lM}\otimes\omega)(\alpha(a)):a\in A, \omega \in A_{*}\}’$, whereof is contained in$\Lambda f${ $u\in M:u\iota s$ unitary and$IG(\mathrm{G}):=$

is the intrinsic group of G.
In particular, if is faithful, thengroup.
$\alpha$$\alpha$$IG(\mathrm{G})’$$\triangle(u)=u\otimes u$}

is a coaction of some locally

Proof For each $u\in N_{A}(D)$ , set $w:=\alpha(u)(1\otimes u^{*})\in M\otimes A$ .

normalizes$D$, for any$d\in D$, we have$w(1\otimes d)=\alpha(u)(1\otimes u^{*})d=\alpha(u)(1\otimes u^{*}du)(1 \copyright u^{*})$$=\alpha(u)\alpha(u^{*}du)(1\otimes u^{*})=\alpha(du)(1\otimes u^{*})$$=(1\otimes d)w$.

$CO\mathit{7}mpact$

Since

$u$

44

Hence belongs to $(M\otimes A)\cap(\mathrm{C}\otimes D)’=M\otimes D$ . So we nlay and do assume
that is an $M$ -valued function. Moreover, we have
$w$

$w$

$(\triangle\otimes id_{A})(w)=(\triangle\otimes id_{A})(\alpha(u)(1\otimes u^{*}))$$=(\triangle\otimes id_{A})(\alpha(u))(1\otimes 1\otimes u^{*})$$=(id_{M}\otimes\alpha)(\alpha(u))(1\otimes 1\otimes u^{*})$$=(id_{M}\otimes\alpha)(\alpha(u)(1\otimes u^{*}))(1\otimes\alpha(u))(1\otimes 1\otimes u^{*})$$=w_{12}w_{23}$

is an
Hence
belongs to$IG(\mathrm{G})$$w$$IG(\mathrm{G})’\otimes A$

3-valued function. So we have that $\alpha(u)=w(1\otimes u)$. Since $N_{A}(D)$ generates , we get the conclusion.
$A$$\square$Coactions derived from l-cocycles

Let $K$ be a locally compact group. A Borel map
following:
1-cocycle if satisfies$c$:$\mathcal{R}arrow K$

is called a$\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{e}$$c$$c(x, x)=1_{K}$$c(x, y)c(y, z)=c(x, z)$Each 1-cocycle

$c$

for
for

$\mathrm{a}.\mathrm{e}$$\mathrm{a}.\mathrm{e}$. $x\in X$ ,.

$(x, y, z)\in \mathcal{R}^{2}$oninto $K$ deter mines a unitary$y)$, . Since is a 1-cocycle, the map$L^{2}(K)\otimes L^{2}(\mathcal{R})$$U_{c}$$\{U_{c}\xi\}(k_{\mathrm{t}}x, y):=\xi(c(x, y)^{-1}k\backslash$ $x$

is a coaction of $K$ . In fact,$\alpha_{c}$$(a\in A)$

is defined by the following:$\{\alpha_{c}(L^{\sigma}(f))\xi\}(k, x, z).--\sum_{y:(y,x\rangle\in R}f(x, y)\xi(c(x, y)^{-1}k,$

By the definition of$\alpha_{C}$

by

$c$

$\alpha_{c}(a):=U_{c}(1\otimes a)U_{c}^{*}$$W^{*}(\mathrm{K}\mathrm{e}\mathrm{r}(c), \sigma)$.

$y$, $z)\sigma(x, y, z)$ .

, we have that the fixed-point algebra$A^{\alpha_{\mathrm{c}}}$is equal to.

We claim that the converse also holds.
Theorem 5. For each coaction
there exists a Borel 1-cocycle$\alpha$of If on A which satisfies$c:\mathcal{R}arrow K$

such that$\alpha$$D\subseteq A^{\alpha}\subseteq A$is equal to$\alpha_{c}$

.

,

45

Proof. Suppose is in $N_{A}(D)$ . By the definition, Ad determines an aut0morphism. Set $w:=\alpha(u)(1\otimes u^{*})$ . By using the same argument as
in the proof of Proposition 4, is a(If)-valued function. Moreover, for$x\inX$$w(x)$,
almost allis equal tofor some $k(x)\in K$ . We notethat the map depends only on . Now, we define a map from the graph
to $K$ by the following:
$u$

$u$

$\rho\in[\mathcal{R}]$$W^{*}$$w$$\lambda_{K}(k(x))$

$k$

$c$

$\rho$$\Gamma(\rho^{-1})$$c(\rho(x), x):=k(x)$$(x\in \mathrm{D}\mathrm{o}\mathrm{m}(\rho))$By using this construction, we can define a map from
to $K$ . We notethat the map is well-defined, i.e., if there exists
and
in
and a
measurable subset
such that $\rho_{1}(x)=\rho_{2}(x)$ for all $x\in E$ , then there. It
exists null set
such that $c(\rho_{1}(x))x)=c(\rho_{2}(x).x)$ for all
is easy to check that is a 1-cocycle. Moreover, wethatis equal
tofor all $u\in N_{A}(D)$ . Hen ce we conclude that is equal to .
$\mathcal{R}$

$c$

$c$

$\rho_{1}$$[\mathcal{R}]$$\rho_{2}$$E\subseteq X$$F\subseteq X$$x\in E\backslash F$$\alpha(u)$$1_{1}\mathrm{a}\mathrm{v}\mathrm{e}$$c$$\alpha_{c}(u)$$\square$$\alpha$$\alpha_{c}$By using the above characterization, we will develop a theory of coactionsin terms of l-cocycles.
In the rest of this paper, we fix a coaction of $K$ on and a l-cocycle$arrow K$ which. We denote bythesatisfies
crossed product of by , ,$A$$\alpha$$c:\mathcal{R}$$\hat{\mathrm{G}}(K)_{\alpha_{c}}\ltimes W^{*}(\mathcal{R}, \sigma)$$\alpha_{c}=\alpha$$A$$\mathrm{i}.\mathrm{e}$$\alpha$$\hat{\mathrm{G}}(K)_{\alpha_{c}}\ltimes W^{*}(\mathcal{R}, \sigma):=(L^{\infty}(K^{\backslash })\otimes \mathrm{C}\vee\alpha_{c}(W^{*}(\mathcal{R}, \sigma))’$.

We recall that a unitary $V\in W^{*}(K)\otimes A$ is called an a-l-cocycle ifsatisfies the following:$(\triangle_{K}\wedge\otimes id_{A})(V)=V_{23}(id_{M}\otimes\alpha)(V)$

Another coaction of $K$ on-cocycle andexists an$\alpha’$$\alpha- 1$

$V$

$A$

.

is said to be cocycle conjugate toof such that$\mathrm{a}*\mathrm{a}\mathrm{u}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{m}$$V$$\alpha$if there$A$$\theta$$(id_{M}\otimes\theta)\circ\alpha’\circ\theta^{-1}=\mathrm{A}\mathrm{d}V\circ\alpha$.

For each Borel map $\phi:Xarrow K$ , a unitary $(V_{l}\xi)(k, x, y):=\xi(\phi(x)^{-1}k, x, y)$-cocycle. So we get the following
is an$\alpha- 1$

is cohomologous to
Proposition 6. . Suppose a Borel 1-cocycle$i.e.$, there exists a Borel map : $Xarrow K$ suchanother Borel 1-cocycle ,. Then the coactionisthat $d(x, y)=\phi(x\grave{)}c(x, y)\phi(y)^{-1}$ for $a.e$ .
$hic$
is isomor. Hence the crossed productcocycle conjugate to.
to$c:\mathcal{R}arrow K$$d$$\phi$$(x, y)\in \mathcal{R}$$\hat{\mathrm{G}}(K)_{\alpha_{\mathrm{c}}}\ltimes A$$\alpha_{c’}$$\hat{\mathrm{G}}(K)_{\alpha_{c’}}\ltimes A$$\alpha_{c}$464Connes spectrum and asymptotic range

into aLetbe a Borel 1-cocycle from an equivalence relation$K$ on the von
$K$
of
the
coaction
weconsiderAgainlocally compact group .
. We will show that the Connes spectrum
Neumann algebra
can be described in terms of the 1-cocycle .
of the coactionis the, the essential range
For each such a 1-cocycle :$K$$c^{-1}(F)$ofmeasurehas complement
smallest closed subset of such that
if and only if, for any
zero. It is easy to check that $k\in K$ belongs to$\nu(c^{-1}(U))>0$. The asymptotic
of , one has(compact) neighborhood,of the 1-cocycle is by definition
range
by .
stands for the restriction of to the reductionwhere$\mathcal{R}$$c:\mathcal{R}arrow K$$\alpha_{c}$$A:=W^{*}(\mathcal{R}, \sigma)$$c$$\alpha_{c}$$\mathcal{R}arrow K$$c$$\sigma(c)$

$F$

$\nu$

$\sigma(c)$$U$$r^{*}(c)$$k$$\cap\{\sigma(c_{B}) :$c$$c_{B}$

Theorem 7. The Connes spectrum.
range

B\underline{\subseteq}X, \mu(B)>0\}$$B$$\mathcal{R}_{B}$$c$$\Gamma(\alpha_{c})$of$c\iota_{c}$is equal to the asymptotic$r^{*}(c)$To prove this theorem, we use the following

Lemma 8. Let$W^{*}(K)_{*}$

of

$K$

$L^{\sigma}(f)\in A$

Then

and

$\omega$$\in A(K)$ ,. For any$L^{\sigma}((\omega\circ c)f)$$(\alpha_{c})_{\omega}(L^{\sigma}(f)):=(\omega\otimes id)(\alpha_{c}(L^{\sigma}(f))$

Proof. We may and do assume that$\eta_{2}\in L^{2}(K)$where $A(K)$ is the Fourier algebra
equals$\zeta_{1}$,$\zeta_{2}\in L^{2}(\mathcal{R})$has the form
, we have$\mathrm{t}\iota’$’$\omega=\omega_{\eta_{1},\eta \mathrm{z}}$for solne$\eta_{1}$,$((\alpha_{c})_{\omega}(L^{\sigma}(f))(_{1}|(_{2})$$=(\alpha_{c}(L^{\sigma}(f))(\eta_{1}8 \zeta_{1})|\eta_{2}\otimes(_{2})$$= \int\int\sum_{y.(y,x)\in \mathcal{R}}\eta_{1}(c(x, y)^{-1}k)\overline{\eta_{2}(k^{\wedge})}f(x, y)\zeta_{1}(y, z)\sigma(x, y, z)\overline{(_{2}(x,z)}d\nu(x_{i}\approx\grave{)}dk$$= \int\sum_{y:(y,x)\in \mathcal{R}}\omega(c(x, y))f(x, y)\zeta_{1}(y, z)\sigma(x, y, z)\overline{(_{2}(x,z)}d\nu(x, z)$$=(L^{\sigma}((\omega\circ c)f)(_{1}|\zeta_{2})$.

Thus we are done.
$Pr\cdot oof$$\square$of Theore $m7$. Since the center$\mathrm{F}(\mathrm{a}\mathrm{c})=\cap${$\mathrm{S}\mathrm{p}((\alpha_{c})^{e})$:Hence, it suffices to show that Sp

$e$

$Z$

(

is contained in$A^{\alpha}\grave{)}$$D$, we have: non-zero projection in $D$ }.

$(\alpha_{c})=\sigma(c)$.

47

Let $k\in\sigma(c)$ . Take alny compact neighborhood of . Since $\nu(c^{-1}(U))>$
that $\nu(E)>0$ and, there exists a measurable subset. If $\in A(K)$ vanishes on. Then define. Fromsome neighborhood of , then, by Lemma 8, we have.
belongsHence, Lemma 1.2 (ii) , it follows that
Chapter.
. By [6, Chapter , Lemma 1.2 (iv)], lies into. We will show that, for eac.ll open
Conversely suppose thatneighborhood of , $c^{-1}(V)$ is not a -null set. Indeed, if $\nu(c^{-1}(V))$ is equal. So, forfor eachto 0 for some , we have, by Lemma we haveeach $\omega\in A(K)$ such that
$U$

$k$

$E\subseteq c^{-1}(U)\mathrm{s}\mathrm{u}\mathrm{c}.\mathrm{f}_{1}$$0$$a:=L^{\sigma}(\chi_{E})\in A\backslash \{0\}$$L^{\sigma}(\chi_{E})\in A$$\omega$$U$$(\alpha_{c})_{\omega}(a)=0$$\mathrm{I}\mathrm{V}$$0$$\mathrm{S}\mathrm{p}_{\alpha_{\mathrm{c}}}(a)\subseteq U$

$]$

$k$

$\mathrm{I}\mathrm{V}$$A^{\alpha_{\mathrm{c}}}(U)$$[6_{\backslash }$$\mathrm{S}\mathrm{p}(\alpha_{c})$$k\in \mathrm{S}\mathrm{p}(\alpha_{c})$

$V$

$k$

$\nu$

$V$

$L^{\sigma}(f)\in A$$L^{\sigma}(f)=L^{\sigma}(f\chi_{c^{-1}(V)^{c}})$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\omega$$\subseteq U$$8_{j}$$(\alpha_{c})_{\omega}(L^{\sigma}(f))=L^{\sigma}(f\chi_{c^{-1}(V)^{c}}(\omega\circ c))=0$.

for
So we conclude thatch $a\in A$ and $\omega\in A(K)$ such that
. In the meantime, Sinceis open for each $h\in V$ , there$\omega\inA(K)$$\omega(h)=1$. This shows that forandsuch thatexists].
, Lemma. This contradic.ts [6, Chapter
each $a\in A$ ,.
Therefore belongs to$(\alpha_{c})_{\backslash d}(a)=0$$\mathrm{e}\mathrm{a}$$V$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\omega\subseteq U$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\omega$$\subseteq V$$\mathrm{I}\mathrm{V}$$h\not\in \mathrm{S}\mathrm{p}_{\alpha_{c}}(a)$$k$$1.2(\mathrm{i}\mathrm{v})$$\square$$\sigma(c)$By using the above theorem and [4, Lemma 1.13], we get the following
be an $AFD$ type II factor. Suppose that
Corollary 9 (cf. [5]). Let
and are coactions of a locally compact group $K$ on such that each ofcontains a Cartan subalgebra of A. If $\Gamma(\alpha)=\Gamma(\alpha’)=K$ . then isandcocycle conjugate to $a’$ .
$A$$\alpha$$A$$\alpha’$$A^{\alpha}$$A^{\alpha’}$$\alpha$Proof. Suppose that

(resp.
. By [2], there exists$A^{ty}$$A^{\alpha’}$) contains a Cartan subalgebra$D_{1}$(resp.

-automorphism of such thatof
. So
. Then we have
. Set
.
Hence
it
. Clearly,
is cocycle conjugate to
suffices to assume from the outset that $D_{1}=D_{2}=:D$ .
We may assume that the inclusion $(D\subseteq A)$ is of the form
on a stanfor an amenable ergodic type II equivalence relation
$(X,
\mu)$
with an invariant measure . By Theorem 5 there
dard Borel space
. By
and
and
from to $K$ such thatexist Borel 1-cocycles
Theorem 7, we have $r^{*}(c)=r^{*}(d)=K$ . So we may apply [4, Lemma 1.13],
and obtain that there exist cocycles and cohomologous to and re,for some
spectively as 1-cocycles on such that is equal to) is cocycle conjugate(resp.
the normalizer of . By Proposition 6,$D_{2})$$A$$D_{2}$$A$$\theta$$\mathrm{a}*$$A^{\alpha_{\theta}}=\theta(A^{\alpha})$$\alpha_{\theta}:=(id_{\mathrm{I}V^{*}(K)}\otimes\theta^{-1})\circ\alpha\circ\theta$$D_{2}=\theta(D_{1})\subseteq\theta(A^{\alpha})=A^{\alpha_{\theta}}$$\theta(D_{1})=$$\alpha$$\alpha_{\theta}$$(L^{\infty}(X)\subseteq$$\mathcal{R}$$W^{*}(\mathcal{R}))$

$\mu$

$c$

$d$

$\mathcal{R}$$\alpha=\alpha_{c}$$\overline{d}$$c$$\overline{c}$$\mathcal{R}$$\mathcal{R}$$\overline{d}\circ\rho$$\overline{c}$$\alpha’$$\alpha$$\alpha’=\alpha_{c’}$$d$$\rho\in N[\mathcal{R}]$

48

to

$\alpha_{\overline{c}}$(resp.

$X\in W^{*}(\mathcal{R})$$\alpha_{\overline{d}}$). Furthermore, a direct computation shows that for each,$\alpha_{\overline{c}0\rho}(X)=(1\otimes\Phi_{\rho}^{-1})(\alpha_{\overline{c}}(\Phi_{\rho}(X)))$

where$\Phi_{\rho}$

is an automorphism on$W^{*}(\mathcal{R})$which is defined by$\Phi_{\rho}(L(f)):=L(f\circ\rho)$

So we conclude that
Hence is cocycle conjugate to$(1\otimes\Phi_{\rho})\alpha_{\overline{c}\circ\rho}=\alpha_{\overline{c}}\circ\Phi_{\rho}$$\alpha’$$\alpha$5,.

, i.e.,$\alpha_{\overline{c}0\rho}$

is conjugate to.

$\alpha_{\overline{c}}$.

$\square$Exchangeability for a 1-cocycle with a smaller
range within the cohomology class

Suppose that there exists a closed subgroup $H$ of $K$ which coh omologous toand the range is contained in $H$ . By regarding as a 1-cocycle into $H$ ,of $H$ .
and the dual action
we obtain the crossed product. Namely, there
of $K$ is induced from
It follows that the dual action
onto
exists an isomorphism from
, where the action of $K$ is the induced action
such that II.
of
We will show that the converse also holds.
$d$

$c$

$\hat{\mathrm{G}}(H)_{\alpha_{c}},$$\ltimes A$$\overline{\alpha_{d}}$$\overline{\alpha_{d}}$$\hat{\alpha}_{c}$$L^{\infty}(K/H)\otimes(\hat{\mathrm{G}}(H)_{\alpha_{\mathrm{c}’}}\ltimes A)$$\hat{\mathrm{G}}(K)_{\alpha_{c}}\ltimes A$$\Pi$$\delta$$\circ(\alpha_{c})_{k}=\delta_{k}\circ\Pi$$\overline{\alpha_{d}}([7])$be a Borel l-cocycle
Theorem 10 (cf. [9, Theorem 3.5]). Let
and $H$ be a closed subgroup of K. Then the following are equivalent:$c:\mathcal{R}arrow K$:(1) There exists a Borel 1cocyclethat the range of is contained in $H$ .
$c_{0}$$\mathcal{R}arrow K$. cohomologous to , such$c$$c_{0}$(2) There exists an $injective*$ -homomorphism$\Theta$from$L^{\infty}(K/H)\underline{in}to$thesuch that
center of the crossed product
$k\in
K$
the
comes
,
where
left translation by on $K/H$ .
from
for all
Equivalently, if is the measure-theoretic spectrum of the center of thecrossed product ($i.e.$ , the measure space on which the Mackey action(the Poincar\’e flow) of $K$ is considered), then it is an extension of the$K$ -space $K/H$ .
$\hat{\mathrm{G}}(K)_{\alpha_{c}}\ltimes A$$\ell_{k}$$\Theta\circ\ell_{k}=(\alpha_{c})_{k}\circ \mathrm{O}-$

$k$

$Y$

(3) The covariant system$tem\{P, H, \beta\}$ .

$\{\hat{\mathrm{G}}(K)_{\alpha_{c}}\ltimes A, K,\hat{\alpha}_{c}\}$is induced from some sys-43

If one of
$A$

,

$H$

$(\mathit{1})\sim(\mathit{3})$, where-valued l-cocycle.

$d$$H,\overline{\alpha_{c’}}\}$

occurs, then one can take $\{P, H, \beta\}$ to be:is the -cocycle obtained by regarding$\{\hat{\mathrm{G}}(H)_{\alpha_{c}},$$\mathcal{R}arrow H$$l$$\ltimes$as an$c_{0}$

Proof. It is easy to check hat the condition (2) follows (1). By using the

Imprimitivity Theorem of [7], (2) is equivalent to (3). So we will prove.
If suchexists, then by using [7], the dual action
is induced from an action of $H$ on a von Neumann algebra . We
denote the induced action of by . By the assumption, there exists a
isomorphism II fromonto $L^{\infty}(K/H)\otimes P$ such that II$k\in
K$II for all
A direct computation shows thatis equal to. Moreover,since is defined bythere exists a dual action
on $H$ which is conjugate to . So there exist a von Neumann algebra anda coaction of $H$ on such that the dual action
is conjugate to the
induced action by . In particular, we have$(2)\Rightarrow(1)$$\mathrm{a}*- \mathrm{h}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{m}\Theta$$(\alpha_{c})_{k}$$P$$\beta$$\beta$$\delta$$*-$$\hat{\mathrm{G}}(K)_{\alpha_{c}}\ltimes A$$\circ\overline{(\alpha_{c})}_{k}=$$\delta_{k}\circ$$\Pi(\alpha_{c}(A))$$\mathrm{C}\otimes P^{\beta}$$\beta_{h}:=\mathrm{A}\mathrm{d}(\lambda_{H}(h)\otimes 1)|_{P\backslash }$$\beta$$\beta’$$B$$\beta$$B$$\tau$$\overline{(\alpha_{c})}$$\hat{\tau}$$\hat{\mathrm{G}}(K)_{\alpha_{c}}\ltimes A\cong L^{\infty}(K/H)\otimes\hat{\mathrm{G}}(H)_{\tau}\ltimes B$

Under the above isomorphism, we have that there exists amorphismfrom onto
such that the fixed-point subalgebra
contains a Cart
. So comes from a 1-cocycle
subalgebra:. By the construction, we conclude thatis cohomologous to as a cocycle into $K$ .
Therefore we complete the proof.
$\mathrm{i}\mathrm{s}\mathrm{o}$$\eta$$A$$B$$\eta(D)$$B^{\tau}$$\tau$$c_{0}arrow$$\mathcal{R}arrow H$$c$$\infty$$\square$

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