Crossed product by actions of finite groups with the Rokhlin Property

0.27 MB·

by  Luis Santiago

CROSSED PRODUCT BY ACTIONS OF FINITE GROUPS WITH THE
ROKHLIN PROPERTY
LUIS SANTIAGO
4
1
0 Abstract. We introduce and study a Rokhlin-type property for actions of finite groups on (not
2 necessarily unital) C*-algebras. We show that the corresponding crossed product C*-algebras can
belocallyapproximatedbyC*-algebrasthatarestablyisomorphictoclosedtwo-sidedidealsofthe
n
givenalgebra. Wethenusethisresulttoprovethatseveralclasses ofC*-algebras areclosed under
a
J crossed products by finitegroup actions with this Rokhlin property.

7
2
Contents

]
A 1. Introduction 1
O 2. Preliminary definitions and results 2
. 3. The Rokhlin property 5
h
4. Local approximation of A⋊ G 13
t α
a
5. Permanence properties 16
m
References 22
[

1
v
2
1. Introduction
5
8
TheRokhlinpropertyisaformoffreenessforgroupactionsonnoncommutativeC*-algebras. For
6
. finite group actions on unital C*-algebras this property first appeared (under different names) in
1
[20] and [12] for actions of Z on UHF-algebras, and in [15] for actions of finite groups on arbitrary
0 p
4 unital C*-algebras. Let us recall the definition of the Rokhlin property:
1
: Definition 1. ([17, Definition 3.1]) Let A be a unital C*-algebra and let α: G Aut(A) be
v →
an action of a finite group G on A. The action α is said to have the Rokhlin property if there
i
X
exists a partition of unity (p ) A (see Subsection 2.1 for the definition of A ) consisting of
g g∈G ∞ ∞
r projections such that ⊂
a
α (p ) =p ,
g h gh
for all g,h G.
∈
When A is separable this property can be reformulated in terms of finite subsets and projections
of A ([28, Definition 1.1]). The Rokhlin property is quite restrictive and is not useful if there is
no abundance of projections. For instance, if either the K -group or the K -group of the algebra
0 1
is isomorphic to Z then the algebra does not have any action of a finite group with the Rokhlin
porperty (this follows by [17, Theorem 3.13]). On the contrast, if a C*-algebra absorbs tensorially
the UHF-algebra of infinite type M and G is a finite group of order n then the algebra has an
n∞
action of G with the Rokhlin property. In fact, in this case the actions of G with the Rokhlin
property are generic ([29, Theorem 5.17]).
Date: January 28, 2014.
1
For C*-algebras that are not unital there are at least two Rokhlin-type properties. One of them
is called the multiplier Rokhlin property ([27, Definition 2.15]), and is defined using projections of
the multiplier algebra of the given C*-algebra. The other one, which is simply called the Rokhlin
property ([22, Definition 3.1]), is defined for σ-unital C*-algebras using projections in the central
sequence algebra of the given C*-algebra. In this paper we introduce a Rokhlin-type property
(Definition 2) which somehow resembles [28, Definition 1.1]. It is defined using positive elements
of the algebra instead of projections. This new Rokhlin-type property is essentially that of [22,
Definition3.1](theyagreeforseparableC*-algebras), butithastheadvantagethatitcanbedefined
forgeneralC*-algebras(notjustforσ-unitalones). ActionswiththemultiplierRokhlinpropertyor
the Rokhlin property in the sense of [22, Definition 3.1] also have this new Rokhlin property. Also,
it can be seen that if the algebra is unital our definition agrees with the standard Rokhlin property
for unital C*-algebras. In this paper we also study the crossed product C*-algebras obtained by
actions of finite groups with the Rokhlin property on different classes of C*-algebras. Our main
motivation comesfromthestudyofcrossedproductsbyfinitegroupactions onstablyprojectionless
C*-algebrasandourobjectiveistoextendtheresultsof[23]tonon-unitalC*-algebras. Theproblem
of classifying actions with the Rokhlin property will be considered in [13].
The paper is organized as follows. In Section 3 we define and study a Rokhlin-type property for
actions of finite groups on arbitrary C*-algebras. Section 4 is dedicated to show that the crossed
product C*-algebra by an action of a finite group on a C*-algebra with this Rokhlin property can
belocally approximated by matrix algebras over hereditary C*-subalgebras of the given C*-algebra
(Theorem 2). This result may beconsidered as a nonunital version of [23, Theorem 3.2]. In Section
5 we use this approximation result to show that several classes of C*-algebras are closed under
crossed products by finite group actions with the Rokhlin property (Theorem 3). These classes of
C*-algebras include purely infinite C*-algebras, C*-algebras of stable rank one, C*-algebras of real
rank zero, C*-algebras of finite nuclear dimension, separable -stable C*-algebras, where is a
D D
strongly self-absorbing C*-algebra, simple C*-algebras, simple C*-algebras with strict comparison
of positive elements, simple stably projectionless C*-algebras, and C*-algebras that are stably
isomorphic to sequential direct limits of one-dimensional NCCW-complexes.
I would like to thank Eusebio Gardella for useful discussions concerning the subject matter of
this paper.
2. Preliminary definitions and results
2.1. Central sequence algebras. Let A be a C*-algebra. Let us denote by A∞ the quotient of
ℓ (N,A) by the ideal c (N,A). That is,
∞ 0
A∞ := ℓ (N,A)/c (N,A).
∞ 0
The elements of ℓ (N,A) will be written as sequences of elements of A. If (a ) ℓ (N,A) then
∞ n n ∞
∈
its image in the quotient A∞ will be denoted by [(a ) ].
n n
Let B be a C*-subalgebra of A. Then we will identify B with its image in A∞ by the embedding
a B (a,a, ) A∞.
∈ 7−→ ··· ∈
Similarly, wewillidentify B∞ B′ with thecorrespondingC*-subalgebraof A∞ B′. Letusdenote
∩ ∩
by A and Ann(B,A∞) the C*-algebras
∞
A := A∞ A′, Ann(B,A∞) := a A∞ B′ : ab= 0, b B .
∞
∪ { ∈ ∩ ∀ ∈ }
Then Ann(B,A∞) is a closed two-sided ideal of A∞ B′. Denote the corresponding quotient by
∩
F(B,A) and the quotient map by π. We call F(A) := F(A,A) the central sequence algebra of A.
Note that if B is σ-unital then F(B,A) is unital. Let α: G Aut(A) be an action of a finite group
→
G on the C*-algebra A and let B be a G-invariant subalgebra of A. Then α induces natural actions
on A∞, A , F(A), and F(B,A). For simplicity we will denote all these actions again by α.
∞
2
Lemma 1. Let A be a C*-algebra and let α: G Aut(A) be an action of a finite group G on A.
→
LetJ beaG-invariant ideal. ThenJ has aquasi-central approximate unitconsistingof G-invariant
positive contractions.
Proof. By [24, Theorem 3.12.14] it is enough to show that J has a G-invariant approximate unit.
Let (u ) be an approximate unit for J. For each i I put v = 1 α (u ). Then it is clear
i i∈I ∈ i |G| g∈G g i
that v 1 for all i I, v v for all i,j I with i j, and α (v ) = v for all i I and g G.
k ik ≤ ∈ i ≤ j ∈ ≤ g iP i ∈ ∈
Let a J and let ǫ > 0. Then there exists i such that
0
∈
u α (a) α (a) < ǫ,
i g−1 g−1
k − k
for all i i and g G. Now using that *-automorphisms are norm preserving we get
0
≥ ∈
α (u )a a = α (u α (a) α (a)) = u α (a) α (a) < ǫ,
g i g i g−1 g−1 i g−1 g−1
k − k k − k k − k
for all i i and g G. Hence, by the triangle inequality
0
≥ ∈
1 1
v a a α (u )a a < (Gǫ) = ǫ,
i g i
k − k ≤ G k − k G | |
| | g∈G | |
X
for all i i . This shows that (v ) is an approximate unit for J. (cid:3)
0 i i∈I
≥
Lemma 2. Let A be a C*-algebra, let α: G A be an action of a finite group G on A, and let B
→
be a σ-unital C*-subalgebra of A. Let F be a finite subset of A∞ B′, let F be a finite subset of
1 2
∩
Ann(B,A∞), and let F be a finite subset of
3
a A∞ : ab= b for all b B .
{ ∈ ∈ }
Then there exists a positive contraction d in the fixed point algebra (B∞ B′)G (considered as a
∩
subalgebra of (A∞ B′)G) such that
∩
db= b, b B,
∀ ∈
da = ad, a F ,
1
∀ ∈
da = 0, a F ,
2
∀ ∈
da = a, a F .
3
∀ ∈
Proof. Let (un)n∈N BG be a countable approximate unit for B (this approximate unit exists by
⊂
the previous lemma). Let F , F , and F be as in the statement of the lemma. Without loss of
1 2 3
generality we may assume that their cardinalities is the same and that their elements are positive.
Write
F = a(i) = [(a(i)) ] :1 i m ,
1 n n
{ ≤ ≤ }
F = b(i) = [(b(i)) ] :1 i m ,
2 n n
{ ≤ ≤ }
F = c(i) = [(c(i)) ]: 1 i m .
3 n n
{ ≤ ≤ }
Then for each j N there exists k N such that
j
∈ ∈
1 1 1
(1) u a(i) a(i)u < , u b(i) < , u c(i) c(i) < ,
k j n − n jk j k j n k j k j n − n k j
for n kj and 1 i m. Moreover, we may choose the sequence (kj)j∈N satisfying kj < kj+1 for
all j ≥N. Define≤d=≤[(d ) ] B∞ A∞ by
n n
∈ ∈ ⊆
u for 1 n < k
1 2
d = ≤
n
(uj for kj+1 n < kj+2.
≤
3
Thend B∞ B′ anddb= bfor allb B,since(un)n∈N isanapproximate unitforB. Inaddition,
∈ ∩ ∈
by the inequalities in (1) we have
da(i) = a(i), db(i) = 0, dc(i) = c(i),
for 1 i m. This concludes the proof of the lemma. (cid:3)
≤ ≤
2.2. The Cuntz semigroup. Let us briefly recall the definition of the Cuntz semigroup of a C*-
algebra. Let A be a C*-algebra and let a,b A be positive elements. We say that a is Cuntz
∈
subequivalent to b, and denote this by a - b, if there exists a sequence (dn)n∈N A such that
⊂
d∗bd a as n . We say that a is Cuntz equivalent to b, and denote this by a b, if a - b
n n
→ → ∞ ∼
and b - a. It is easy to see that - is a preorder relation on A and that is an equivalence
+
∼
relation. We denote the Cuntz equivalence class of an element a A by [a]. The Cuntz semigroup
+
∈
of A, denoted by Cu(A), is the set of Cuntzequivalence classes of positive elements of A , where
⊗K
denotes the algebra of compact operators on a separable Hilbert space. The addition on Cu(A)
K
is given by [a]+[b] := [a′ +b′], where a′,b′ (A ) are such that a a′, b b′, and a′b′ = 0.
+
∈ ⊗K ∼ ∼
The preorder relation - induces an order relation on Cu(A); that is, [a] [b] if a - b. We say that
≤
[a] is compactly contained in [b], and denote this by [a] [b], if whenever [b] sup [b ], for some
n n
≪ ≤
increasing sequence ([bn])n∈N, one has [a] [bk] for some k. A sequence ([an])n∈N is said to be
≤
rapidly increasing if [a ] [a ] for all n. It was shown in [8] that Cu(A) is closed under taking
n n+1
≪
suprema of increasing sequences and that every element of Cu(A) is the supremum of a rapidly
increasing sequence. In particular, we have
[(a ǫ) ] [a], [a] =sup[(a ǫ) ],
+ +
− ≪ −
ǫ>0
for all a (A ) andall ǫ > 0. Here (a ǫ) denotes theelement obtained by evaluating—using
+ +
∈ ⊗K −
functional calculus—the continuous function f(t) = max(0,t ǫ), with t (0, ), at the element
− ∈ ∞
a.
The following lemma states two well known result regarding Cuntz subequivalence. The first
statement was proved in [19, Lemma 2.2] in the case δ = 0 and in [32, Lemma 1] for an arbitrary
δ. The second statement was shown in [19, Lemma 2.3].
Lemma 3. Let A be a C*-algebra and let a,b A . The following statements hold:
+
∈
(i) If a b < ǫ, then (a (ǫ+δ)) - (b δ) for all δ > 0;
+ +
k − k − −
(ii) If a - b then for every ǫ > 0 there exists δ > 0 such that (a ǫ) - (b δ) .
+ +
− −
We say that Cu(A) is almost unperforated if for any [a],[b] Cu(A) satisfying (n+1)[a] n[b],
for some n N, one has [a] [b]. Consider the set L(A) of all∈functionals λ: Cu(A) [0, ≤] that
∈ ≤ → ∞
are additive, order-preserving, and that preserve suprema of increasing sequences. We say that a
simpleC*-algebra Ahasstrict comparison of positive elements if for [a],[b] Cu(A), λ([a]) < λ([b])
∈
for all λ L(A) satisfying λ([b]) < , implies [a] [b]. The following is [39, Lemma 6.1].
∈ ∞ ≤
Lemma 4. Let A be a simple C*-algebra. Then Cu(A) is almost unperforated if and only if A has
strict comparison of positive elements.
Lemma 5. Let A be a C*-algebra and let I be a σ-unital closed two-sided ideal of A. Suppose
that Cu(A) is almost unperforated, then Cu(A/I) is almost unperforated.
Proof. Let π: A A/I denote the quotient map. Let n N and x,y Cu(A/I) be such that
→ ∈ ∈
(n+1)x ny.
≤
Choose a,b (A ) such that [π(a)] = x and [π(b)] = y. Then it follows by [6, Theorem 1.1]
+
∈ ⊗K
that,
(n+1)[a]+z n[b]+z,
≤
4
where z is the largest element of Cu(I) (this element exists because I is σ-unital). Since 2z = z we
have
(n+1)([a]+z) n([b]+[z]).
≤
Using now that Cu(A) is almost unperforated we get [a]+z [b]+z. Hence,
≤
x = Cu(π)([a]+z) Cu(π)([b]+z) = y.
≤
This implies that Cu(A/I) is almost unperforated. (cid:3)
2.3. Strongly self-absorbing C*-algebras. Recall that a unital nuclear C*-algebra is said
D
to be strongly self-absorbing if there exists a *-isomorphism φ: that is approximately
D → D⊗D
unitarily equivalent to the map i: given by i(a) = a 1 for all a . A C*-algebra
D
D → D⊗D ⊗ ∈ D
A is -stable if A =A.
D ⊗D ∼
Thefollowing result was proved in [16, Proposition 4.1] undertheassumption that the algebra
D
is K -injective. This assumption is in fact redundantsince every strongly self-absorbing C*-algebra
1
is K -injective ([42, Theorem 3.1 and Remark 3.3]).
1
Lemma 6. Let A and be separable C*-algebras such that is strongly self-absorbing. Then
D D
A is -stable if and only if for any ǫ > 0 and any finite subsets F A and G there exists a
D ⊂ ⊂ D
completely positive contractive map φ: A such that
D →
(i) bφ(1 ) b < ǫ,
D
k − k
(ii) bφ(d) φ(d)b < ǫ,
k − k
(iii) b(φ(dd′) φ(d)φ(d′)) < ǫ,
k − k
for all b F and d,d′ G.
∈ ∈
3. The Rokhlin property
InthissectionweintroduceaRokhlin-typepropertyforactionsoffinitegroupsonnotnecessarily
unital C*-algebras and study several permanence properties of it. We also study its relation with
the Multiplier Rokhlin property ([27, Definition 2.15]) and with the Rokhlin property as defined in
[22, Definition 3.1].
Definition 2. Let A be a C*-algebra and let α: G Aut(A) be an action of a finite group G on
→
A. We say that α has the Rokhlin property if for any ǫ > 0 and any finite subset F A there exist
⊂
mutually orthogonal positive contractions (r ) A such that
g g∈G
⊂
(i) α (r ) r < ǫ, for all g,h G;
g h gh
k − k ∈
(ii) r a ar < ǫ, for all a F and g G;
g g
k − k ∈ ∈
(iii) ( r )a a < ǫ, for all a F.
k g∈G g − k ∈
The elements (r ) will be called Rokhlin elements for α.
P g g∈G
Proposition 1. Let A be a C*-algebra and let α: G Aut(A) be an action of a finite group G
→
on A. Then the following are equivalent:
(i) α has the Rokhlin property.
(ii) For any finitesubsetF Athereexist mutuallyorthogonal positive contractions (r )
g g∈G
⊂ ⊂
A∞ F′ such that
∩
(a) α (r ) = r , for all g,h G;
g h gh
∈
(b) ( r )b = b, for all b F.
g∈G g ∈
(iii) ForanyseparableC*-subalgebraB Athereareorthogonalpositivecontractions(r )
g g∈G
P ⊆ ⊂
A∞ B′ such that
∩
(a) α (r ) = r , for all g,h G;
g h gh
∈
(b) ( r )b = b, for all b B.
g∈G g ∈
P 5
(iv) For any separable G-invariant C*-subalgebra B A there are projections (p )
g g∈G
⊆ ⊂
F(B,A) such that
(a) α (p ) =p , for all g,h G;
g h gh
∈
(b) p = 1 .
g∈G g F(B,A)
Proof. (i) (Piii). Let B beaseparable C*-subalgebra of A. Choosefinitesubsets (Fn)n∈N of Asuch
⇒
(n)
that F F for all n, and F = B. For each n choose Rokhlin elements (r ) such
n ⊂ n+1 n∈N n g g∈G
that
S
1 1 1
α (r(n)) r(n) < , r(n)b br(n) < , ( r(n))b b < ,
k g h − gh k n k g − g k n k g − k n
g∈G
X
for all g,h G and all b F . Let r = r(n) A∞. Then α (r )= r for all g,h G, and
n g g g h gh
∈ ∈ n ∈ ∈
(cid:16) (cid:17)
r b = br , ( r )b = b,
g g g
g∈G
X
for all b F . Using that F is dense in B we get the previous equalities for all b B.
∈ n∈N n n∈N n ∈
This shows that the elements r , with g G, belong to A∞ B′ and that the satisfy (a) and (b)
g
S S ∈ ∩
of (iii).
(iii) (ii). This is clear by taking B to be the C*-subalgebra generated by the finite set F.
⇒
(ii) (i). Let F be a finite subset of A. Let (r ) A∞ F′ be elements satisfying conditions
g g∈G
⇒ ⊂ ∩
(a) and (b) of (ii). By [21, Lemma 10.1.12] these elements can be lifted to mutually orthogonal
positive contractions ((r ) ) ℓ (N,A). Let ǫ > 0. Then using the definition of A∞ and that
g,n n g∈G ∞
∈
the elements (r ) satisfy conditions (a) and (b) of (ii) we get
g g∈G
α (r ) r < ǫ, g,h G,
g h,n gh,n
k − k ∀ ∈
r b br < ǫ, g G, b F,
g,n g,n
k − k ∀ ∈ ∀ ∈
( r )b b < ǫ, b F,
g,n
k − k ∀ ∈
g∈G
X
for some n N. This shows that α has the Rokhlin property.
∈
(iii) (iv). Let B be a separable G-invariant C*-subalgebra of A. By (iii) there exist positive
⇒
orthogonal contractions (r ) A∞ B′ satisfying (a) and (b) of (iii). Let π: A∞ B′ F(A,B)
g
⊂ ∩ ∩ →
bethe quotient map. Set p = π(r ). Then by (b) of (iii) we have p = 1 . Also, by (a) of
g g g∈G g FB,A
(iii) we get α (p ) = p for all g,h G. Since the elements (r ) are orthogonal, the elements
g h g,h g g∈G
∈ P
(p ) are orthogonal. It follows that p2 = p p = p for all h G. Hence, (p ) are
g g∈G h h g∈G g h ∈ g g∈G
projections.
P
(iv) (ii). Let F be a finite subset of A and let B be the C*-subalgebra of A generated by the
⇒
finite set α (F). Then B is G-invariant and separable. Let (p ) F(B,A) be projections
g∈G g g g∈G ⊂
satisfyingconditions(a)and(b)of(iv). Thenby[21,Lemma10.1.12]thereexitmutuallyorthogonal
S
positive contractions (f ) A∞ B′ such that π(f )= p . Note that these elements satisfy
g g∈G g g
⊂ ∩
α (f )b = f b, ( f )b = b,
g h gh g
g∈G
X
for all g,h G and all a B. By Lemma 2 applied to the finite sets F = f : g G , F =
1 g 2
∈ ∈ { ∈ }
α (f ) f :g,h G , and F = f , there exists a positive contraction d (A∞ B′)G
{ g h − gh ∈ } 3 { g∈G g} ∈ ∩
such that
P
db= b, df = f d, α (f )d = f d, ( f )d = d,
g g g h gh g
g∈G
X
6
for all b B, and g,h G. For each g G set r = df d. Then (r ) A∞ B′. Also, by the
g g g g∈G
∈ ∈ ∈ ⊂ ∩
previous equations and that d (A∞ B′)G we get
∈ ∩
r = d2,
g
g∈G
(2) X
α (r ) = r g,h G,
g h gh
∀ ∈
r r = 0 g,h G,g = h.
g h
∀ ∈ 6
Therefore, since (r ) A∞ F′ and d2b = b for all b B, it follows that the elements (r )
g g∈G g g∈G
⊂ ∩ ∈
satisfy conditions (a) and (b) of (ii). (cid:3)
Corollary 1. Let A be a unital C*-algebra and let α: G Aut(A) be an action of a finite group
→
G on A. Then α has the standard Rokhlin as in [28, Definition 1.1] (without the assumption of
separability) if and only if it has the Rokhlin property as defined in Definition 2.
Proof. The forward implication is clear. Let us show the opposite implication. Let A and α be
as in the statement of the corollary. Let F be a finite subset of A and let B be the unital C*-
subalgebra of A generated by α (F). It is clear that B is separable and G-invariant. Also,
g∈G g
F(B,A) = A∞ B′ since B is unital. By (iv) of Proposition 1 there exists a partition of unity
∩ S
(p ) A∞ B′ consisting of projections such that
g g∈G
⊂ ∩
α (p ) =p ,
g h gh
for all g,h G. Let φ: C[G] A∞ B′ be the *-homomorphism defined by φ( z g) =
∈ → ∩ g∈G g
z p , where z C for all g G. Using that the C*-algebra C[G] = C|G| is semiprojective by
g∈G g g g ∈ ∈ ∼ P
[21, Lemma 14.1.5 and Theorem 14.2.1], we can lift φ to a unital map from C[G] to ℓ (N,A). In
∞
P
other words, there exists a partition of unity (qg)g∈G ℓ∞(N,A), with qg = (qg(n))n∈N, consisting
⊂
of projections such that
q(n) = 1, n N,
g
∀ ∈
g∈G
X
(n) (n)
lim (α (q ) q ) = 0, g,h G,
n→∞ g h − g,h ∀ ∈
lim (q(n)a aq(n))= 0, a F.
g g
n→∞ − ∀ ∈
This implies that α has the Rokhlin property as in [28, Definition 1.1]. (cid:3)
Let A be a σ-unital C*-algebra and let α: G Aut(A) be an action of a finite group G on A.
→
We say that α has the Rokhlin property in the sense of [22, Definition 3.1] if there exist projections
(p ) F(A) such that
g g∈G
⊂
p = 1 , α (p )= p g,h G.
g F(A) g h g,h
∀ ∈
g∈G
X
Corollary 2. Let Abeaσ-unital C*-algebra andlet α: G Aut(A) bean action of afinitegroup
→
G on A. Suppose that α has the Rokhlin property in the sense of [22, Definition 3.1], then α has
the Rokhlin property. Moreover, if A is separable then α has the Rokhlin property in the sense of
[22, Definition 3.1] if and only if it has the Rokhlin property.
Proof. This is a consequence of the equivalence of (i) and (iii) of Proposition 1. (cid:3)
Let A be a C*-algebra and let α: G Aut(A) be an action of a finite group G on A. Let M(α)
→
denote the extension of α to the multiplier algebra M(A) of A. Recall that α has the multiplier
Rokhlin property ([27, Definition 2.15]) if for every finite set F A and every ǫ > 0, there are
⊂
mutually orthogonal projections (p ) M(A) such that:
g g∈G
⊂
7
(i) M(α) (p ) p < ǫ for all g,h G;
g h gh
k − k ∈
(ii) p a ap < ǫ for all g G and all a F;
g g
k − k ∈ ∈
(iii) p = 1 .
g∈G g M(A)
CorollaPry 3. Let A be a C*-algebra and let α: G Aut(A) be an action of a finite group G on
→
A. If α has the multiplier Rokhlin property then α has the Rokhlin property.
Proof. Let A and α be as in the statement of the corollary. Let us show that (A,α) satisfies (iv) of
Proposition 1. LetB beaseparable G-invariant C*-subalgebra of A. Choosefinitesubsets (Fn)n∈N
of B consisting of contractions such that F is dense in the unit ball of B. For each n N let
n∈N n ∈
(n)
(p ) M(A) be mutually orthogonal projections satisfying properties (i), (ii), and (iii) given
g g∈G ⊂ S
above for F and ǫ = 1. Select a G-invariant approximate unit (u ) for A that is quasi-central
n n λ λ∈Λ
for M(A) (this approximate unit exists by Lemma 1). Let n N be fixed. Then for each a F
n
∈ ∈
we have
1
lim (u p(n)u )(u p(n)u )a = lim p(n)p(n)au4 = 0< , g,h G, g = h
λ k λ g λ λ h λ k λ k g h λk n ∀ ∈ 6
1
lim α (u p(n)u )a u p(n)u a = lim (M(α) (p(n)) p(n))au2 < , g,h G,
λ k g λ h λ − λ gh λ k λ k g h − gh λk n ∀ ∈
1
lim (u p(n)u )a a(u p(n)u ) = lim (p(n)a ap(n))u2 < , g G,
λ k λ g λ − λ g λ k λ k g − g λk n ∀ ∈
1
lim ( u p(n)u )a a = ( p(n))a a = 0< ,
λ k λ g λ − k k g − k n
g∈G g∈G
X X
(n) (n)
Therefore, there exists a positive contraction u A such that for r := up u A the following
g g
∈ ∈
hold:
1
r(n)r(n)a < , g,h G, g = h, a F
k g h k n ∀ ∈ 6 ∀ ∈ n
1
(n) (n)
(α (r ) r )a < , g,h G, a F
k g h − gh k n ∀ ∈ ∀ ∈ n
1
r(n)a ar(n) < , g G, a F
k g − g k n ∀ ∈ ∀ ∈ n
1
( r(n))a a < , a F .
k g − k n ∀ ∈ n
g∈G
X
For each g G set r := (r(n)) ℓ (N,A). Then by the inequalities above, for each a F
∈ g g n ∈ ∞ ∈ n∈N n
we have r r a = 0 for all g,h G with g = h, (α (r ) r )a = 0 for all g,h G, ar = r a for all
g h g h gh g g
∈ 6 − ∈ S
g G, and ( r )a = a. Since F is dense in the unit ball of B, the same equalities hold
∈ g∈G g n∈N n
for every a B. Let π: A∞ B′ F(B,A) be quotient map. For each g G put p = π([r ]).
g g
∈P ∩ →S ∈
Then (p ) are mutually orthogonal projections that satisfy conditions (a) and (b) of (iv) of
g g∈G
Proposition 1. Therefore, by the same proposition α has the Rokhlin property. (cid:3)
The following is an example of a Z -action with the Rokhlin property that does not have the
2
multiplier Rokhlin property:
Example 1. Consider the space X = 1 : n N with the topology induced from R. Let
{n ∈ }
φ: X X be the map defined by φ(1/(2n 1)) = 1/(2n), and φ(1/(2n)) = 1/(2n 1), for all
n N.→Itisclearthatφisbijective, continuou−s,proper,itsinverseiscontinuous,andtha−tφ2 = id .
X
∈
Thisimpliesthatthemapα: C (X) C (X)definedbyα(f)(x) = f(φ−1(x))isa*-automorphism
0 0
satisfying α2 = id . Consider th→e action of Z on C (X) defined by this automorphism. Let
C0(X) 2 0
us see that this action has the Rokhlin property. For each n N, let r(1),r(2) C (X) denote the
n n 0
∈ ∈
8
characteristic functions of the open subsets 1/(2k 1) : 1 k n and 1/(2k) : 1 k n ,
{ − ≤ ≤ } { ≤ ≤ }
(1)
respectively (note that these functions are continuous since the given sets are open). Then r and
n
(2) (1) (2) (1) (2)
rn are orthogonal, α(rn ) = rn , and (rn +rn )n∈N is an approximate unit for C0(X). Since
C (X) is commutative it follows that the given Z -action has the Rokhlin property. It is clear that
0 2
this action does not have the multiplier Rokhlin property since X can not be written as the union
of two disjoint open sets X and X satisfying φ(X ) = X . (If X = X X where X and X
1 2 1 2 1 2 1 2
∪
are disjoint and open, then either X or X contains a subset of X of the form 1/n : n > k . In
1 2
{ }
particular, it follows that φ(X )= X .)
1 2
6
Let A be a C*-algebra and let α: G Aut(A) be an action of a finite group G on A. Recall
→
that a map ω: G U(M(A)), where U(M(A)) denotes the unitary group of M(A), is said to be an
→
α-cocycle if it satisfies ω = ω α (ω ) for all g,h G. Let β: G Aut(A) be an action of G on
gh g g h
∈ →
A. The actions α and β are said to be exterior equivalent if there exists an α-cocycle ω such that
β (a) = ω α (a)ω∗ for all g G and all a A.
g g g g
∈ ∈
Theorem 1. Let A and B be C*-algebras and let α: G Aut(A) be an action of a finite group
→
G on A. The following statements hold:
(i) If β: G Aut(B) is an action with the Rokhlin property, then the tensor action
→
α β: G Aut(A B)
min min
⊗ → ⊗
has the Rokhlin property. Here denotes the minimal tensor product;
min
⊗
(ii) If α has the Rokhlin property and B is a G-invariant hereditary C*-subalgebra of A then
the restriction of α to B has the Rokhlin property;
(iii) If α has the Rokhlin property and I is a G-invariant closed two-sided ideal of A then the
action induced by α on A/I has the Rokhlin property;
(iv) If for any finite subset F A and any ǫ > 0 there exist a C*-algebra B with an action
⊂
β: G Aut(B)withtheRokhlinproperty,andanequivariant*-homomorphismφ: B A
→ →
such that dist(a,φ(B)) < ǫ for all a F. Then α has the Rokhlin property;
∈
(v) If (A,α) is a direct limit—with equivariant connecting *-homomorphisms—of a direct sys-
tem (A ,α ) , and every action α : G Aut(A ) has the Rokhlin property, then α has
λ λ λ∈Λ λ λ
→
the Rokhlin property;
(vi) If α has the Rokhlin property and H is a subgroup of G then the restriction of α to H has
the Rokhlin property.
(vii) Ifβ: G Aut(A)isanactionwiththeRokhlinpropertyandαandβ areexteriorequivalent
→
then α has the Rokhlin property.
Proof. (i) Given that G is finite the tensor action α β is well defined. Let us show that α β
min
⊗ ⊗
has the Rokhlin property. Since A B is the closure of the span of simple tensors, it is enough
min
⊗
to show that for any ǫ > 0 and any finite set F A B consisting of simple tensors, there exist
min
⊂ ⊗
orthogonal positive contractions (f ) in A B satisfying (i), (ii), and (iii) of Definition 2.
g g∈G min
⊗
Let ǫ > 0 and
F = a b :a A, b B, 1 k n .
k k k k
{ ⊗ ∈ ∈ ≤ ≤ }
Choosean approximate unit(u ) for A consisting of Ginvariant elements (this approximate unit
i i∈I
exists by Lemma 1). Then there exists i I such that
∈
ǫ ǫ
(3) u a a u < , u a a < ,
i k k i i k k
k − k 2 k − k 2
for 1 k n. Choose Rokhlin elements (r ) B for β corresponding to the finite set
g g∈G
b ,b ,≤ ,b≤ and to the number ǫ. For each g G p⊂ut f = u r . Then the elements (f )
{ 1 2 ··· n} 2 ∈ g i⊗ g g g∈G
9
are mutually orthogonal since the elements (r ) are orthogonal. Now by the triangle inequality,
g g∈G
the choice of (r ) , and the inequalities in (3) we get
g g∈G
ǫ
(α β) (f ) f u (β (r ) r ) < ǫ,
g h gh i g h gh
k ⊗ − k ≤ k ⊗ − k ≤ 2
ǫ ǫ
f (a b ) (a b )f (u a a u ) r b + a u (r b b r ) < + = ǫ,
g k k k k g i k k i g k k i g k k g
k ⊗ − ⊗ k≤ k − ⊗ k k ⊗ − k 2 2
ǫ ǫ
( f )(a b ) (a b ) (u a a ) r b + a ( r b b ) < + = ǫ,
g k k k k i k k g k k g k k
k ⊗ − ⊗ k ≤k − ⊗ k k ⊗ − k 2 2
g∈G g∈G g∈G
X X X
for 1 k n. We have shown that (f ) are Rokhlin elements for β corresponding to the finite
g g∈G
≤ ≤
set F and the number ǫ. This implies that β has the Rokhlin property.
(ii) Let A, B, and α be as in the statement of (ii). Let F be a finite subset of B. Then by (ii) of
Proposition1thereexistorthogonalpositivecontractions(r ) A∞ F′ satisfyingα (r )= r
g g∈G g h gh
⊂ ∩
for all g,h G, and ( r )a = a for all a F. By Lemma 2 applied to the separable C*-
∈ g∈G g ∈
subalgebra of B generated by the elements in F and to the finite sets F = r :g G , F = 0 ,
1 g 2
P { ∈ } { }
and F = r , there exists a positive contraction d B∞ B′ A∞ B′ such that da = a
3 { g∈G g} ∈ ∩ ⊂ ∩
for all a F, dr = r d for all g G, and ( r )d = d. For each g G set f = dr d. Then
∈ P g g ∈ g∈G g ∈ g g
f B∞ B′ since B is a hereditary C*-subalgebra of A. Also, it is straightforward to check that
g
∈ ∩ P
the elements (f ) satisfy (a) and (b) of (ii) of Proposition 1. Therefore, by the equivalence of
g g∈G
(i) and (ii) in the same proposition, the restriction of α to B has the Rokhlin property.
(iii) Let ǫ > 0 and let F A/I be a finite set. Let π: A A/I denote the quotient map.
⊂ →
Choose a finite set F A such that π(F)= F. Choose Rokhlin elements (r ) for F and ǫ > 0.
g g∈G
⊂
Then it is straightforward that (φ(r )) are Rokhlin elements for F and ǫ.
g g∈G
(iv) Let A and αebe as in the statemeent of the proposition. Let F = a1,a2, ane be a finite
{ ··· }
subset of A. Then by assumption there exist a C*-algebra B with an action β: G Aut(B)
→
with the Rokhlin property, an equivariant *-homomorphism φ: B A, and a finite subset F =
b ,b , b B such that a φ(b ) < ǫ for 1 i n. Sinc→e β has the Rokhlin property,
{ 1 2 ··· n} ⊂ k i − i k 3 ≤ ≤
there exist orthogonal positive contractions (rg)g∈G B such that e
⊂
ǫ ǫ
α (r ) r < ǫ, r b b r < , ( r )b b < ,
g h gh g i i g g i i
k − k k − k 3 k − k 3
g∈G
X
for g,h G and 1 i n. For each g G set f := φ(r ). Then (f ) are orthogonal positive
g g g g∈G
∈ ≤ ≤ ∈
contractions. Using that φ is equivariant we get
β (f ) f = φ(α (r ) r ) α (r ) r < ǫ.
g h gh g h gh g h gh
k − k k − k ≤ k − k
Also, using the triangle inequality we get
ǫ ǫ ǫ
f a a f φ(r b b r ) + f (a φ(b )) + (a φ(b ))f < + + = ǫ,
g i i g g i i g g i i i i g
k − k≤ k − k k − k k − k 3 3 3
ǫ ǫ ǫ
( f )a a φ(( r )b b ) + ( f )(a φ(b )) + a φ(b ) < + + = ǫ,
g i i g i i g i i i i
k − k ≤k − k k − k k − k 3 3 3
g∈G g∈G g∈G
X X X
for g G and 1 i n. Hence, (f ) are Rokhlin elements for α, F and ǫ. It follows that β
g g∈G
∈ ≤ ≤
has the Rokhlin property.
(v) This is a particular case of (iv).
(vi) Let H be a subgroup of G and let n denote the number of elements of G/H. Let F A and
⊂
let ǫ > 0. Choose a family K of representatives of H in G and Rokhlin elements (r ) for α, F,
g g∈G
and ǫ/n. Then it is easy to check using the triangle inequality that (f ) , with f = f
h h∈H h g∈K hg
for all h H, are Rokhlin elements for the restriction of α to H, the finite set F, and the number
∈ P
ǫ > 0.
10