数理解析研究所講究録 1250 巻 2002 年 9-15

9Crossed products of Cuntz algebras by quasi-freeactions of abelian groups
Takeshi KATSURA (勝良健史)
Department of Mathematical Sciences
University of Tokyo, Komaba, Tokyo, 153-8914, JAPAN-mail: katsu@ms.u-tokyo.ac.jp$\mathrm{e}$1Introduction
The crossed products of C’-algebras give us plenty of interesting examples and the structures of them have been examined by several authors. In [KK1] and [KK2], A. Kishimotoand A. Kumjian dealt with, among others, the crossed products of Cuntz algebras byquasi-ffee actions of the real group R. In [Kal] and [Ka2], we examined the crossed
products of Cuntz algebras by quasi-free actions of arbitrary locally compact, second
countable, abelian groups. In this note, we summarize the results of [Kal] and [Ka2], anddiscuss several examples.

2Preliminaries
In this section, we review some basic objects and fix the notation.
, the Cuntz algebrais the universal C’-algebra generated by
For $n=2,3$ ,, , satisfying $\sum_{i=1}^{n}ShSi2=1$ [C1]. In this note, we only consider
isometries , ,, see [Ka3]. For
the case $n<\infty$ . For similar results on the crossed products of$k\in \mathrm{N}=\{0,1, \ldots\}$
and, we define the setof -tuples by$S_{2}$$S_{1}$$\ldots$$n$$O_{n}$$\ldots$$S_{n}$$O_{\infty}$$\mathcal{W}_{n}^{(k)}$$\mathcal{W}_{n}^{(k)}=$$\mathcal{W}_{n}^{(0)}=\{\emptyset\}$$k$$\{(i_{1}, i_{2}, \ldots, i_{k})|i_{j}\in\{1,2, \ldots, n\}\}$.

,, we denote its length by, ). For $\mu=$ ( , i2,, ), $\nu=$. For $\mu=$ ( , i2,,. Note that
).
,
,i2,
by $\mu\nu=$ ( , i2,, we define their productof
axiom
countthesecondsatisfies
group
which
compact
abelian
Let be alocallymultiplicative operations ofability and be the dual group of . We always
abelian groups except for , which is the group of the unit circle in the complex plane C.
.
is denoted by
The pairing of $t\in G$ and,and fix it. Since the isometries
Let us take
$t\in
G$
,
a
there
is
any
,
,
also satisfy the relation above for, . One canfor $i=1,2$ ,such that
automorphism:
is astrongly continuous group homomorphism:
see that

We setand set$\mathcal{W}_{n}=\bigcup_{k=0}^{\infty}\mathcal{W}_{n}^{(k)}$$i_{1}$$i_{k}$$\ldots$$|\emptyset|=0$$S_{\mu}=S_{i_{1}}S_{i_{2}}\cdots$ $S_{i_{k}}\in O_{n}$$(j_{1},j_{2}, \ldots,j_{l})\in \mathcal{W}_{n}$$\mu\nu$$k$$\in \mathcal{W}_{n}$$s_{\emptyset}=1$$\in \mathcal{W}_{n}$$i_{1}$$i_{1}$$\ldots$$\ldots$$i_{k},j_{1}$$|\mu|$$i_{k}$$\ldots,j_{l}$

$G$

$G$

$\Gamma$$\mathrm{u}\mathrm{s}\mathrm{e}+\mathrm{f}\mathrm{o}\mathrm{r}$$\mathrm{T}$$\langle t|\gamma\rangle\in \mathrm{T}$$\gamma\in\Gamma$$\omega$$\langle t|\omega_{2}\rangle S_{2}$$\ldots$$\langle t|\omega_{1}\rangle S_{1}$

$n$

$*-$

$\langle t|\omega_{n}\rangle S_{n}$$\alpha_{t}^{\omega}$$\alpha^{\omega}$$=$$(\omega_{1},\omega_{2}, \ldots,\omega_{n})\in\Gamma^{n}$$O_{n}arrow O_{n}$$G\ni t\mapsto*\alpha_{t}^{\omega}\in \mathrm{A}\mathrm{u}\mathrm{t}(O_{n})$$\alpha_{t}^{\omega}(S_{i})=\langle t |\omega_{i}\rangle S_{i}$$\ldots$$n$

10Definition 2.1 Let$O_{n}$$\omega=$$(\omega_{1},\omega_{2}, \ldots,\omega_{n})\in\Gamma^{n}$be given. We define the action$\alpha^{\omega}$: $G\cap$by(i $=1,$ 2, \ldots ,n, t $\in G)$ .

$\alpha_{t}^{\omega}(S\dot{.})=\langle t|\omega:\rangle S_{\dot{1}}$:The action
becomes quasi-free (for adefinition of quasi-ffee actions on
Cuntz algebras, see [E] . Conversely, any quasi-ffee action of the abelian group on
is conjugate to”.
for some
Since the abelian groupis amenable, the reduced crossed product of the action:
coincides with the full crossed product of it. We denote it byandcall it the crossed product. The crossed producthas a -subalgebra
which is isomorphic to. Throughout this paper, we always consider
as a, and use $f,g$ ,C’-subalgebra offor denoting elements of.
The Cuntz algebra
is naturally embedded into the multiplier algebra, we define an elementof. For each $\mu=$inof by, we define a(reverse) shift automorphism
. For
:
by
. Once noting that
,
for
for
one can easily verify thatfor any
and any.
, where spam
From this fact, we havespan.
means the closure of the linear$Gr[searrow] O_{n}$$\alpha^{\omega}$

$G$

$)$

$\alpha^{\omega}$$\omega$$O_{n}$$\in \mathrm{I}$$G$$\alpha^{\omega}$$Gr\backslash$$O_{n}$$O_{n}\mathrm{x}_{\alpha^{\omega}}G$$C^{*}$$O_{n}\mathrm{x}_{\alpha^{\omega}}G$$\mathbb{C}1\aleph_{\alpha}\cdot G$$C_{0}(\Gamma)$$C_{0}(\Gamma)$$C_{0}(\Gamma)\subset Onx\text{\^{a}} GG$$O_{n}\mathrm{x}_{\alpha^{\omega}}G$$\ldots$$O_{n}$$M(O_{n}x_{\alpha^{\omega}}G)$$(i_{1},i_{2}, \ldots,i_{k})$$O_{n}\mathrm{x}_{\alpha^{\omega}}G$$\Gamma$$\mathcal{W}_{n}$$\gamma_{0}\in\Gamma$$\sum_{j=1}^{k}\omega_{i_{j}}$$C_{0}(\Gamma)arrow C_{0}(\Gamma)$$\sigma_{\gamma 0}$$f\in C_{0}(\Gamma)$$(\sigma_{\gamma 0}f)(\gamma)=f(\gamma+\gamma_{0})$$\omega_{\mu}=$$\omega_{\mu}$$\mu\in \mathcal{W}_{n}$$\alpha_{t}^{\omega}(S_{\mu})=\langle t|\omega_{\mu}\rangle S_{\mu}$$f\in C_{0}(\Gamma)\subset O_{n}\mathrm{x}_{\alpha^{\omega}}G$$fS_{\mu}=S_{\mu}\sigma_{\omega_{\mu}}f$$\mu\in \mathcal{W}_{n}$$O_{n}\mathrm{x}_{\alpha}.G=\overline{\mathrm{s}\mathrm{p}\mathrm{m}}\{S_{\mu}fS_{\nu}^{*}|\mu, \nu\in \mathcal{W}_{n}, f\in C_{0}(\Gamma)\}$3The ideal structure of$\mathcal{O}_{n}\mathrm{x}_{\alpha^{\omega}}G$

In [Kal], we completely determined the ideal structures of the crossed product, we define the closed subset
For an ideal I of the crossed product. The closed subset
satisfies$O_{n}\nu_{\alpha}.G$$\gamma\in X_{I}$(ii) For any$\gamma\in X_{I}$$X_{I}$

of$\Gamma$.

by$X_{I}$$I\cap \mathrm{C}\mathrm{o}(\mathrm{r})=\mathrm{C}0(\mathrm{F}\backslash X_{I})$(i) For any$O_{n}x_{\alpha}.G${1,2, \ldots ,n}, we have
, there exists i {1,2,n} such thatand any i$\gamma+\omega:\in X_{I}$

$\in$

$\in$

\ldots ,.

$\gamma^{-\omega}:\in X_{I}$.

The closed subset of satisfying two conditions above is said to be invariant A. For aclosed ;-invariant
closed set $X$ is -invariant if and only if$X$,bysubsetof we define$\Gamma$$\omega$$X= \bigcup_{=1}^{n}.\cdot(X+\omega.\cdot)$$\omega$$0$$I_{X}\subset O_{n}\aleph_{\alpha}\cdot G$$\Gamma$$I_{X}=\overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}}\{S_{\mu}fS_{\nu}^{*}|\mu, \nu\in \mathcal{W}_{n}, f\in C_{0}(\Gamma\backslash X)\}$.

and invariant under the gauge action A of
One can see thatis an ideal of,, which is defined by
for ,
and
on
. With atechnique using conditional expectations, we can prove the following.
$I_{X}$$O_{n}\mathrm{x}_{\alpha}.G$$O_{n}\nu_{\alpha^{w}}G$$\mathrm{T}$$\beta_{t}(S_{\mu}fS_{\nu}^{*})=t^{|\mu|-|\nu|}S_{\mu}fS_{\nu}^{*}$$\mu$$\nu\in \mathcal{W}_{n}$$f\in C_{0}(\Gamma)$$t\in \mathrm{T}$and $X\succ*I$ be tween
Proposition 3.1 ([Kal, Theorem 3.14]) The two maps
and the set of closed -invariant subsets ofthe set of gauge invariant ideals ofare the inverses of each other.
$I\mapsto X_{I}$$O_{n}\mathrm{x}_{\alpha}.G$$\omega$$\Gamma$

The ideal structure of
conditions$O_{n}\mathrm{x}_{\alpha^{\omega}}G$

depends on whether$\omega$$\in\Gamma^{n}$satisfies the folowing

11
Condition 3.2 For each i

$\in$

{1,2,

(i) For any positive integer ,$k$(ii) There exists $j\neq i$ such$\mathrm{a}\mathrm{n}\mathrm{d}-\omega_{i}$\ldots ,n}, one of the following two conditions is satisfied:$k\omega_{i}\neq 0$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}-\omega_{j}$.

is in the closed semigroup generated by$\omega_{1},\omega_{2}$,,$\ldots$$\omega_{n}$.

This condition is an analogue of Condition (II) in the case of Cuntz-Krieger algebras[C2] or Condition (K) in the case of graph algebras [KPRR].

Theorem 3.3 ([Kal, Theorem 5.2]) When satisfies Condition 3.2, any ideal is gauge
invariant. Hence there is one-tO-One correspondence between the set of ideals ofand the set of closed -invariant subsets of .
$\omega$$O_{n}\mathrm{x}_{\alpha^{\omega}}G$$a$$\Gamma$$\omega$does not satisfy Condition 3.2, there exists $i_{0}\in\{1,2, \ldots, n\}$ such thatWhen
is not in the closed semigroup generated
for some positive integer , andbe the. Note that such is unique. Let
for any,
and,by
the image in ofand denote byquotient group of by the subgroup generated by.
by. Define a -subalgebraA of
The C’-algebra is isomorphic to the Toeplitz algebra of the Hilbert module coming from.
, hence there is asurjective map :the automorphism : of
It is not hard to see that there is aone-t0-0ne correspondence between the set of ideals of, we define. For an ideal I of
and the set of closed subset ofwhich corresponds to the ideal $\pi(I\cap A)$ . The closed setofthe closed subset. Conversely,
and any
any
for anysatisfies thatsatisfying the condition above, we can construct the ideal
for any closed set $Y$ of(see Definition 5.17 and Proposition 5.23 of [Kal]).
so thatof$\omega$$k$$k\omega_{i_{0}}=0$$\omega_{1},\omega_{2}$$\omega_{n}$$\ldots$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}-\omega_{i}$$i\neq i_{0}$$-\omega_{i_{0}}$$\Gamma’$$[\gamma]$$\Gamma$$\omega_{1}$$C^{*}$$\gamma\in \mathrm{F}$$\Gamma’$$i_{0}$$A=\overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}}\{S_{i_{0}}^{k}fS_{i_{\mathrm{O}}}^{l}’|f\in C_{0}(\Gamma), k, l\in \mathrm{N}\}$$O_{n}\mathrm{x}_{\alpha^{\omega}}G$$A$$\mathbb{Z}$$\sigma_{\omega}$$\pi$$C_{0}(\Gamma)$$\Gamma’\cross \mathrm{T}$$Aarrow C_{0}(\Gamma)\mathrm{x}_{\sigma_{\omega}:_{0}}$$O_{n}\mathrm{x}_{\alpha^{\omega}}G$$C_{0}(\Gamma)\mathrm{x}_{\sigma_{\omega}:_{0}}\mathbb{Z}$$\mathrm{Y}_{I}$$\mathrm{Y}_{I}$$\Gamma’\cross \mathrm{T}$$\mathit{0}’\in \mathrm{T}$$i\neq i_{0}$$([\gamma+\omega_{i}], \theta’)\in Y_{I}$$([\gamma], \theta)\in \mathrm{Y}_{I}$$\Gamma’\cross \mathrm{T}$$\mathrm{Y}_{I_{Y}}=Y$$O_{n}\mathrm{x}_{\alpha^{\omega}}G$$I_{Y}$

Theorem 3.4 ([Kal, Theorem 5.49]) In the above setting, we have $I_{Y_{I}}=I$ for any
respondence between the set of ideals
G. Thus there is one-tO-One
ideal I ofsatisfying the condition above.
and the set of closed subsets of
of
$co$

$a$

$O_{n}\mathrm{x}_{\alpha^{\omega}}$$\Gamma’\cross \mathrm{T}$$O_{n}\mathrm{x}_{\alpha}.G$

On the way to prove the two theorems above, we get another proofs of the following
known facts (see [Ki] and [OP]):$\bullet$$O_{n}\mathrm{x}_{\alpha^{\omega}}G$$\mathrm{a}\mathrm{n}\mathrm{d}-\omega_{i}$$\bullet$is simple if and only if the closed semigroup generated by, [Kal, Theorem 4.8].
is equal to for any $i=1,2$ ,$\Gamma$$\ldots$,$\ldots$,$\omega_{n}$$n$is primitive if and only if the closed group generated byequal to [Kal, Theorem 4.12],$O_{n}\mathrm{x}_{\alpha^{\omega}}G$$\omega_{1},\omega_{2}$$\omega_{1},\omega_{2}$,,$\ldots$is$\omega_{n}$$\Gamma$

By Theorem 3.3 and Theorem 3.4, we can show that the strong Connes spectrum,
is the intersection of the closed semigroups generated byof the action, [Kal, Proposition 6.2]. The crossed productwhere $i=1,2$ ,and$n$$\alpha^{\omega}$$-\omega_{i}$$\ldots$$n$$\omega_{1},\omega_{2}$$\tilde{\Gamma}(\alpha^{\omega})$$\ldots$,$\omega_{n}$$O_{n}\nu_{\alpha^{\omega}}G$is12
isomorphic to the Cuntz Pimsner algebra of acertain Hilbert bimodule. Prom this fact,
we have the following exact sequence.

$K_{0}(C_{0}(\Gamma))\uparrow$$K_{0}(O_{n_{\mathrm{I}}^{*}\alpha^{\omega}}G)$$K_{1}(O_{n^{*}\alpha^{w}}G)$$K_{1}(C_{0}(\Gamma))$where is the embedding :$\iota$4$\mathrm{A}\mathrm{F}$$\iota$$C_{0}(\Gamma)\epsilonarrow O_{n}\aleph_{\alpha^{\omega}}G$[Kal, Proposition 6.5].

-embeddability and pure infiniteness of$\mathcal{O}_{n}\mathrm{x}_{\alpha^{\omega}}G$

In [Ka2], we gave asufficient condition for the crossed products
to be AFembeddable. To the best of the author’s knowledge, this is the first case to have succeededin embedding crossed products of purely infinite C’-algebra into-algebras except
trivial cases.
$O_{n}\mathrm{x}_{\alpha}.G$$\mathrm{A}\mathrm{F}$

Theorem 4.1 ([Ka2, Theorem 3.8])
then the crossed productis AF-embeddable.

$If-\omega:\not\in\overline{\{\omega_{\mu}|\mu\in \mathcal{W}_{n}\}}$

for any $i=1,2$ ,$\ldots$, ,$n$$O_{n}*_{\alpha}\cdot G$

In [KK1], Kishimoto and Kumjian proved that
becomes stable and projec. Hencetionless whensatisfies
is stably finite inthis case. Theorem 4.1 gives another proof of this fact.
In [KK2], they gave anecessary and sufficient condition that
becomes simple
and purely infinite. Here, we generalze their result.
$O_{n}\mathrm{x}_{\alpha}.\mathrm{R}$$\omega$$-\omega:\not\in\overline{\{\omega_{\mu}|\mu\in \mathcal{W}_{n}\}}$$\in \mathrm{R}^{n}$$O_{n}\mathrm{n}_{\alpha^{\omega}}\mathrm{R}$$O_{n^{\aleph}\alpha^{w}}\mathrm{R}$Theorem 4.2 ([Ka2, Coroll
infinite if and only if$\mathrm{y}$$4.9]$ )$\Gamma=\{\omega_{\mu}|\mu\in \mathcal{W}_{n}\}$$\mathfrak{M}e$.

crossed product$O_{n}\mathrm{x}_{\alpha^{w}}G$

is simple and purely

By the two theorems above and the characterization of simplicity, we have the following
dichotomy.

Corollary 4.3 ([Ka2, Corollary 4.8]) The crossed productinfinite or $AF$-ernbeddable when it is simple.

$O_{n}\mathrm{x}_{\alpha}.G$

is either purely

5Examples
5.1

When G is compact

When is compact, its dual group becomes discrete. In this case, for any
thecrossed productis agraph algebra of some skew product graph which is rowfinite (see [KP]) and apart of our results here has been already proved in, for example,[BPRS]. Particularly, we have the following.
$G$$\Gamma$$\omega$$\in\Gamma^{n}$$O_{n}x_{\alpha^{\omega}}G$

Proposition 5.1 ([Ka2, Proposition 3.9]) When$G$

is compact, the follovnng are equiv

13
(i)

$-\omega_{i}\not\in\overline{\{\omega_{\mu}|\mu\in \mathcal{W}_{n}\}}$

for any $i=1,2$ ,$\ldots$, .
$n$(ii) The crossed product$O_{n}\succ 1_{\alpha^{\omega}}G$

is stably finite.

(iii) The crossed product$O_{n}\mathrm{x}_{\alpha^{\omega}}G$

is AF-embeddable.

(iv) The crossed product$O_{n}\mathrm{x}_{\alpha^{\omega}}G$

itself is an AF-algebra.

5.2

When G is discrete

aclosedbecomes compact. Let us denote by, .
,$i=1,2$for. One can see thatsatisfies Condition 3.2. Since the closed set $X$ is -invariant if and only
Hence any
if $X+\Lambda_{\omega}=X$ , the set of all closed -invariant subsets of is one-t0-0ne correspondentis aclosed subgroup of .
. Here note thatto the set of all closed subset ofbijectively to the set of all
corresponds
3.3,
By Theorem
the set of all ideals of.
closed subset of
directly as well as other structures
We can examine the ideal structures of$G’$
be the quotient of by the closed subgroupof it. Let

is discrete, its dual groupWhen,,
semigroup generated by$G$$\omega_{1},\omega_{2}$$\omega$$\ldots$$\Lambda_{\omega}$$\Gamma$$-\omega_{i}\in\Lambda_{\omega}$$\omega_{n}$$\in\Gamma^{n}$$\ldots$$n$$\omega$$\Gamma$$\omega$$\Gamma$$\Lambda_{\omega}$$\Gamma/\Lambda_{\omega}$$O_{n}\mathrm{n}_{\alpha^{\omega}}G$$\Gamma/\Lambda_{\omega}$$O_{n}\mathrm{x}_{\alpha^{w}}G$$G$$\{t\in G|\alpha_{t}^{\omega}=\mathrm{i}\mathrm{d}\}=$$=${{ $t\in G|$$t$$\in G|$$\langle$$\langle$$t|\omega_{i}\rangle=1$$t|\gamma\rangle=1$

for $i=1,2$ ,for any$\ldots$$\in\Lambda_{\omega}$$\gamma$

,

$n$

}

}.

, we can define. Since
The dual group of $G’$ is naturally isomorphic to$G’\cap$
is simple and purely infinite by. The crossed product:an action
becomes acontinuous field over the compact
Theorem 4.2. The crossed product. Prom this observation,
space
whose fiber of any point is isomorphic tocorresponds bijectively to the set
we can easily see that the set of all ideals of.
of all closed subset of
has an infinite projection, hence isWhen is discrete, the crossed productnever AF-embeddable.
$\omega\in\Lambda_{\omega}^{n}\subset\Gamma^{n}$$\Lambda_{\omega}$$O_{n}\mathrm{n}_{\alpha^{\omega}}G’$$O_{n}$$\alpha^{\omega}$$O_{n}\mathrm{x}_{\alpha^{\omega}}G$$O_{n}*_{\alpha^{\omega}}G’$$\Gamma/\Lambda_{\omega}$$O_{n}\aleph_{\alpha^{\omega}}G$$\Gamma/\Lambda_{\omega}$$G$$O_{n}\mathrm{x}_{\alpha^{\omega}}G$

5.3

When GWhen$G=\mathbb{R}^{m}$$=\mathbb{R}^{m}$, its dual group

Definition 5.2 Let,,by$\omega_{1},\omega_{2}$$\omega$$=$$\omega_{n}\in \mathbb{R}^{m}$$\ldots$$\Gamma$

is also$L_{\omega}$

. For

.

,

, we define the following.

. We denote the affine space generated

is aclosed subset ofDefinition 5.3 Let$(+)$ if
04 , to be of type (0) if$=$, $\sum_{i=1}^{n}t:=1\}$ ,$C_{\omega}= \{\sum_{i=1}^{n}t_{i}\omega_{i}\in \mathbb{R}^{m}|t_{i}\geq 0$

We define the three types for elements of$C_{\omega}$$\in(\mathbb{R}^{m})^{n}$and their convex hull by$C_{\omega}$$\omega$$\omega$$(\omega_{1},\omega_{2}, \ldots, \omega_{n})\in(\mathbb{R}^{m})^{n}$$L_{\omega}= \{\sum_{i=1}^{n}t:\omega_{i}\in \mathbb{R}^{m}|\sum_{i=1}^{n}t_{i}=1\}$

respectively. The set$\mathbb{R}^{m}$$L_{\omega}$. We denote by$(\mathbb{R}^{m})^{n}$the interior of$C_{\omega}$in.

. The element is said to be of type.
, and to be of type (-) if$(\omega_{1},\omega_{2}, \ldots,\omega_{n})\in(\mathbb{R}^{m})^{n}$$\mathrm{O}\in C_{\omega}\backslash O_{\omega}$$O_{\omega}$$\omega$$0\in O_{\omega}$

14
On this type, we can prove the following. We omit proofs.
Lemma 5.4
is of type $(+)$ , then there eists:of :and is non-negative for any $i=1,2$ ,suchso that there
with
.
find
$If\omega$$\omega$$v$$\cdot$$e$$\dot{m}tsi_{0}$$v$$m\geq 2_{f}$$n$$e$$\dot{m}tsv\in \mathrm{R}^{m}\backslash \{0\}$$\omega$$n$$\ldots$$\omega_{\dot{w}}\cdot v=0$

Lemma 5.5 If is of type (0), then there
any $i=1,2$ ,, , and there exists
with$\ldots$such that the inner product, . Moreover when
we can$v\in \mathrm{R}^{m}\backslash \{0\}$$v$$\omega$$i_{0}$$\omega_{\dot{\mathrm{W}}}\cdot v=0$.

such that$\omega:\cdot v\geq 0$

forProm these two lemmas, we get the following characterizations of type (-) and type
$(+)$

.

Proposition 5.6 An element is of type (-) if and only if the closed semigroup gen,erated by
is a group. An element
is of type $(+)$ if and only
,
,
any
.
$i=1,2$for$\omega$$\omega_{1},\omega_{2}$$\ldots,\omega_{n}$$\{\omega_{\mu}|\mu\in \mathcal{W}_{n}\}$$if-\omega.\cdot\not\in$$\omega$$\ldots$$n$

Combining this proposition with Theorem 4.1 and Theorem 4.2, we have the following.
An element is called aperiodic if the closed group generated by,.
is$\omega$$\omega_{1},\omega_{2}$$\mathrm{R}^{m}$\ldots ,$\omega_{n}$Proposition 5.7 The crossed product
is $AF$-embeddable if
is of type $(+)$ .
The crossed productis simple and purely infinite if and only if is of type (-)
and aperiodic.
$O_{n}\mathrm{n}_{\alpha}.\mathrm{R}^{m}$$\omega$$O_{n}\aleph_{\alpha^{\omega}}\mathrm{R}^{m}$$\omega$

It is easy to see that an element does not satisfy Condition 3.2 if and only if 0is anextreme point ofand there is only one $i\in\{1,2, \ldots,n\}$ with $\omega:=0$ . In this case,is of type (0). The folowing is aconsequence of Lemma 5.4 and Lemma 5.5.
$\omega$$C_{\omega}$$\omega$

Proposition 5.8 If is of type (0) or if is of type $(+)$ and $m\geq 2$ , then there eists$i_{0}\in\{1,2, \ldots, n\}$ such that the closed semigroup generated by
,
is
case,
product
simple.
.
notHence in thisthe crossed
is not$\omega$$\omega$$\omega_{1},\omega_{2}$$\mathrm{R}^{m}$$\ldots,\omega_{n}and-\omega_{\dot{\eta}}$$O_{n}x_{\alpha^{[] d}}\mathrm{R}^{m}$

The condition for simplicity follows ffom the proposition above.

Proposition 5.9 When $m=1$ , the crossed productis simple if and only if
is of type $(+)$ or (-) and aperiodic.
hen $m\geq 2$ , the crossed productis simple if and only if is of type (-)
and aperiodic.
$O_{n}\mathrm{n}_{\alpha^{\omega}}\mathrm{R}^{m}$$O_{n^{\aleph}\alpha^{\omega}}\mathrm{R}^{m}$

When $m\geq 2$ , the crossed product$O_{n}\mathrm{n}_{\alpha^{\omega}}\mathrm{R}^{m}$$\omega$$\omega$is purely infinite if it is simple.

15
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