Picard groups of certain stably projectionless C*-algebras

0.24 MB·English

by  Norio Nawata

PICARD GROUPS OF CERTAIN STABLY PROJECTIONLESS
C∗-ALGEBRAS
NORIONAWATA
Abstract. We compute Picard groups of several nuclear and non-nuclear
2 simple stably projectionless C∗-algebras. In particular, the Picard group of
1 Razak-Jacelon algebraW2 isisomorphictoasemidirectproduct of Out(W2)
0 with R×. Moreover, for any separable simple nuclear stably projectionless
+
2 C∗-algebrawithafinitedimensionallattice ofdenselydefined lowersemicon-
p tinuoustraces,weshowthatZ-stabilityandstrictcomparisonareequivalent.
e (This is essentially based on the result of Matui and Sato, and Kirchberg’s
S central sequence algebras.) Thisshows ifA isaseparablesimplenuclear sta-
bly projectionless C∗-algebra with a unique tracial state (and no unbounded
7 trace)andhasstrictcomparison,thefollowingsequenceisexact:
2
1 −−−−−→ Out(A) −−−−−→ Pic(A) −−−−−→ F(A) −−−−−→ 1

] whereF(A)isthefundamentalgroupofA.
A
O
.
h
1. Introduction
t
a
m Let A be a C∗-algebra. Brown, Green and Rieffel introduced the Picard group
Pic(A) of A in [5]. We say that an automorphism α of A is inner if there exists
[
a unitary element u in the multiplier algebra M(A) of A such that α(a) = uau∗

2 for any a ∈ A. Let Inn(A) denote the set of inner automorphisms of A, and
v
let Out(A) = Aut(A)/Inn(A). They showed that if A is σ-unital, then Pic(A)
0
is isomorphic to Out(A⊗K). Kodaka computed Picard groups of several unital
3
9 C∗-algebras in [20], [21] and [22]. In particular he computed the Picard groups of
1 the irrational rotation algebras A . If θ is not quadratic irrational number, then
θ
. Pic(A) is isomorphic to Out(A ) and if θ is a quadratic number, then Pic(A ) is
7 θ θ
0 isomorphic to Out(Aθ)⋊Z. Kodaka considered the following set
2 FP/∼={[p]|p is a full projection in A⊗K such that p(A⊗K)p∼=A}
1
: where [p] is the Murray-von Neumann equivalence class of p and showed that if
v
i Out(A) is a normal subgroup of Out(A⊗K), then FP/ ∼ has a suitable group
X
structure and the following sequence is exact:
r
a 1 −−−−→ Out(A) −−−−→ Pic(A) −−−−→ FP/∼ −−−−→ 1.
Note that there exists a simple unital AF algebra A with a unique tracial state
such that FP/ ∼ of A does not have any suitable group structure. K-theoretical
method enables us to show that Out(A) is a normal subgroup of Out(A⊗K) (see
[20, Proposition 1.5]).
The set of FP/ ∼ is similar to the fundamental group F(M) of a II factor
1
M introduced by Murray and von Neumann in [26]. Watatani and the author
introduced the fundamental group F(A) of a simple unital C∗-algebra A with a
unique tracial state τ based on Kodaka’s results. The fundamental group F(A)
2010 Mathematics Subject Classification. Primary46L05, Secondary46L08;46L35.
Key words and phrases. Picardgroup; Fundamental group; Stablyprojectionless C∗-algebra;
Cuntzsemigroup;Kirchberg’scentralsequence algebra.
TheauthorisaResearchFellowoftheJapanSocietyforthePromotionofScience.
1
2 NORIONAWATA
is defined as the set of the numbers τ ⊗ Tr(p) for some projection p ∈ M (A)
n
such that pM (A)p is isomorphic to A. We showed that F(A) is a multiplicative
n
subgroup of R× and computed fundamental groups of several C∗-algebras in [29].
+
Moreover we showed that any countable subgroup of R× can be realized as the
+
fundamental group of a separable simple unital C∗-algebra with a unique trace in
[30]. Note that the fundamental groups of separable simple unital C∗-algebras are
countable. Furthermore the author introduced the fundamental group of a simple
stablyprojectionlessC∗-algebrawithunique(uptoscalarmultiple)denselydefined
lower semicontinuous trace τ in [27]. If τ is normalizedand A is σ-unital, then the
fundamental group of F(A) of A is defined as the set of the numbers d (h) for
τ
some positive element h ∈ A⊗K such that h(A⊗K)h is isomorphic to A where
d is the dimension function defined by τ. Note that if A is unital, then this
τ
definition coincides with the previous definition and there exist separable simple
stably projectionless C∗-algebras such that their fundamental groups are equal to
R×. The fundamental group of a II factor M is equal to the set of trace-scaling
+ 1
constantsforautomorphismsofaII∞factorM⊗B(H). Thischaracterizationshows
that the fundamental groups of II factors are related to the structure theorem
1
for type III factors where 0 < λ ≤ 1 (see [41] and [42]). We have a similar
λ
characterization,that is, if A is σ-unital, then the fundamental groupof A is equal
to the set of trace scaling constants for automorphisms of A⊗K.
Inthis paper weshallcompute Picardgroupsofseveralnuclearandnon-nuclear
simple stably projectionless C∗-algebras. In the case of stably projectionless C∗-
algebras, the theory of the Cuntz semigroup enables us to compute Picard groups
of several examples. We shall show that if A is a separable simple exact Z-stable
stably projectionless C∗-algebra with a unique tracial state τ and no unbounded
trace, then the following sequence is exact:
1 −−−−→ Out(A) −−−−→ Pic(A) −−−−→ F(A) −−−−→ 1.
Since there exists a unital simple Z-stable algebra A with a unique tracial state
suchthatOut(A)isnotanormalsubgroupofPic(A),Z-stablestablyprojectionless
C∗-algebras are more well-behaved than unital stably finite Z-stable C∗-algebras.
LetW betheRazak-Jacelonalgebrastudiedin[12],[35]and[36],whichhastrivial
2
K-groupsandauniquetracialstateandnounboundedtrace. ThenW isZ-stable,
2
andhencethesequenceaboveisexactinthiscase. Moreoverweshallshowthatthe
exact sequence above splits. Therefore Pic(W ) is isomorphic to Out(W )⋊R×.
2 2 +
BasedontheresultofMatuiandSato,andKirchberg’scentralsequencealgebras,
for any separable simple infinite-dimensional non-type I nuclear C∗-algebra with a
finite dimensional lattice of densely defined lower semicontinuous traces, we shall
show that Z-stability and strict comparison are equivalent. (It is important to
consider property (SI).) In particular, if A is a simple C∗-algebra with a finite
dimensional lattice of densely defined lower semicontinuous traces in the class of
Robert’sclassificationtheorem([35,Corollary6.2.4]),thenAisZ-stable. Moreover
we see that there are many examples that the sequence above are exact. But we
do not know whether the exact sequence above splits in this case. This question
is related to the existence of a one parameter trace scaling automorphism group
of A⊗K. In the final part of this paper we shall give some remarks and a reason
of the notation of W . Some results show every separable simple Z-stable stably
2
projectionless C∗-algebra A with a unique tracial state has similar properties of
(McDuff) II factors.
1
PICARD GROUPS OF CERTAIN STABLY PROJECTIONLESS C∗-ALGEBRAS 3
2. The Picard group
In this section we shall review basic facts on the Picard groups of C∗-algebras
introduced by Brown, Green and Rieffel in [5] and some results in [27].
LetAbeaC∗-algebraandX arightHilbertA-module. Forξ,η ∈X,a”rankone
operator” Θ is defined by Θ (ζ) = ξhη,ζi for ζ ∈ X. We denote by K (X)
ξ,η ξ,η A A
theclosureofthelinearspanof”rankoneoperators”Θ andbyKtheC∗-algebra
ξ,η
of compact operators on an infinite-dimensional separable Hilbert space. Let H
A
denote the standard Hilbert module {(xn)n∈N | xn ∈ A, x∗nxn converges in A}
with an A-valued inner product h(xn)n∈N,(yn)n∈Ni= xP∗nyn. Then there exists a
natural isomorphism of A⊗K to K (H ). P
A A
Let A and B be C∗-algebras. An A-B-equivalence bimodule is an A-B-bimodule
F which is simultaneously a full left Hilbert A-module under a left A-valued inner
product h·,·i and a full right Hilbert B-module under a right B-valued inner
A
product h·,·i , satisfying hξ,ηiζ =ξhη,ζi for any ξ,η,ζ ∈F. We say that A is
B A B
MoritaequivalenttoB ifthereexistsanA-B-equivalencebimodule. Itiseasytosee
thatK (F)isisomorphictoA. AdualmoduleF∗ ofanA-B-equivalencebimodule
B
F isaset{ξ∗;ξ ∈F}withtheoperationssuchthatξ∗+η∗ =(ξ+η)∗,λξ∗ =(λξ)∗,
bξ∗a = (a∗ξb∗)∗, hξ∗,η∗i = hη,ξi and hξ∗,η∗i = hη,ξi. The bimodule F∗ is
B B A A
aB-A-equivalencebimodule. Wereferthereaderto[33]and[34]forthebasicfacts
on equivalence bimodules and Morita equivalence. For A-A-equivalence bimodules
E and E , we say that E is isomorphic to E as an equivalence bimodule if there
1 2 1 2
exists a C-linear one-to-one map Φ of E onto E with the properties such that
1 2
Φ(aξb)=aΦ(ξ)b, hΦ(ξ),Φ(η)i= hξ,ηi andhΦ(ξ),Φ(η)i =hξ,ηi fora,b∈A,
A A A A
ξ,η ∈ E . The set of isomorphic classes [E] of the A-A-equivalence bimodules E
1
forms a group under the product defined by [E ][E ] = [E ⊗ E ]. We call it the
1 2 1 A 2
PicardgroupofAanddenoteitbyPic(A). TheidentityofPic(A)isgivenbytheA-
A-bimoduleE :=Awith ha ,a i=a a∗ andha ,a i =a∗a fora ,a ∈A. The
A 1 2 1 2 1 2 A 1 2 1 2
inverseelementof[E]inthePicardgroupofAisthedualmodule [E∗]. Letαbean
automorphismofA,andletEA =AwiththeobviousleftA-actionandtheobvious
α
A-valued inner product. We define the right A-action on EA by ξ·a = ξα(a) for
α
anyξ ∈EA anda∈A,andtherightA-valuedinnerproductbyhξ,ηi =α−1(ξ∗η)
α A
for any ξ,η ∈ EA. Then EA is an A-A-equivalence bimodule. For α,β ∈ Aut(A),
α α
EA is isomorphic to EA if and only if there exists a unitary u ∈ M(A) such that
α β
α = ad u ◦β. Moreover, EA ⊗EA is isomorphic to EA . Hence we obtain an
α β α◦β
homomorphism ρ of Out(A) to Pic(A). An A-B-equivalence bimodule F induces
A
an isomorphism Ψ of Pic(A) to Pic(B) by Ψ([E])=[F∗⊗E ⊗F] for [E]∈Pic(A).
Therefore if A is Morita equivalent to B, then Pic(A) is isomorphic to Pic(B).
Brown, Green and Rieffel showed that if A is σ-unital, then Pic(A) is isomorphic
to Out(A⊗K) (see [5, Theorem 3.4 and Corollary 3.5]).
IfA is σ-unital, then forany A-A-equivalencebimodule E there existsa positive
element h in A⊗K such that E is isomorphic to hH as a rightHilbert A-module.
A
Note that h(A⊗K)h is isomorphic to A and hH has a suitable structure as an
A
A-A-equivalence bimodule in this case. (See, for example, [27, Proposition 2.3].)
LetAbeaC∗-algebra,andletτ beadenselydefinedlowersemicontinuoustrace
on A. Put dτ(h)=limn→∞τ ⊗Tr(hn1) for h∈(A⊗K)+. Then dτ is a dimension
function. The following proposition is a key proposition in this paper.
Proposition2.1. LetAbeasimpleσ-unitalC∗-algebrawithauniquetracialstate
τ and no unbounded trace. Define a map T of Pic(A) to R× by T([hH ])=d (h).
+ A τ
Then T is a well-defined multiplicative map and T([EA])=1.
id
Proof. This is an immediate consequence of [27, Proposition 3.1 and Proposition
3.4 ] (put [X]=[EA] ). (cid:3)
id
4 NORIONAWATA
Remark 2.2. [27, Proposition 3.1 and Proposition 3.4] are shown by using the
countable bases of (right) Hilbert modules. See [14], [15] and [46] for bases of
Hilbert modules.
Put F(A)=Im(T). Then we see that F(A) is equal to the set
{d (h)∈R× | h is a positive element in A⊗K such that A∼=h(A⊗K)h}
τ +
by the results in [27]. We call F(A) the fundamental group of A, which is a mul-
tiplicative subgroup of R×. We refer the reader to [29], [30] and [27] for details of
+
the fundamental groups of C∗-algebras. If A is σ-unital, then F(A) is equal to the
set of trace-scaling constants for automorphisms:
S(A):={λ∈R× | τ ⊗Tr◦α=λτ ⊗Tr for some α∈Aut(A⊗K) }.
+
3. The Cuntz semigroup
In this section we shall review basic facts of the Cuntz semigroups and some
results in [6], [9], [36] and [38]. See, for example, [2] for details of the Cuntz
semigroups. Let A be a C∗-algebra. For positive elements a,b∈A we say that a is
Cuntz smaller than b, written a - b, if there exists a sequence {xn}n∈N of A such
that kx∗bx −ak → 0. Positive elements a and b are said to be Cuntz equivalent,
n n
written a ∼ b, if a - b and b - a. Define the Cuntz semigroup Cu(A) as the set
of Cuntz equivalence classes of positive elements in A⊗K endowed with the order
[a] ≤ [b] if a is Cuntz smaller than b, and the addition [a]+[b] = [a′ +b′] where
a ∼ a′, b ∼ b′ and a′b′ = 0. Note that this definition is different from the original
definition W(A) in [7]. (We have Cu(A) = W(A ⊗K).) The Cuntz semigroup
Cu(A)isalsodefinedusingHilbertrightA-modules(see[6]). Forpositiveelements
a,b ∈ A⊗K we say that a is compactly contained in b, written a ≪b if whenever
[b] ≤ supn∈N[bn] for an increasing sequence {[bn]}n∈N, then there exists a natural
numbernsuchthat[a]≤[b ]. Coward,ElliottandIvanescu[6]showedthatCu(A)
n
has the following properties:
(1) every increasing sequence in Cu(A) has a supremum,
(2) for any element [a] in Cu(A) there exists an increasing sequence {[an]}n∈N of
Cu(A) such that [a ]≪[a ] for any n∈N and [a]=sup[a ],
n n+1 n
(3) the operation of passing to the supremum of an increasing sequence and the
relation ≪ are compatible with addition.
Moreoverthey showedthat Cu(A) is a functor which is continuous with respect
toinductivelimits([6,Theorem2]). Forapositiveelementa∈A⊗Kandǫ>0we
denote by (a−ǫ) the element f(a) in A⊗K where f(t)=max{0,t−ǫ},t∈σ(a).
+
Then we have (a−ǫ) ≪a.
+
Followingthe definition in [38], the Cuntz semigroupCu(A) is said to be almost
unperforated if (k+1)[a] ≤ k[b] for some k ∈ N, then [a] ≤ [b]. Rørdam showed
that if A is Z-stable, then Cu(A) is almost unperforated (see [38, Theorem 4.5]).
We denote by T(A) the set of densely defined lower semicontinuous traces on A
and T (A) the set of tracial states on A. If A is simple exact C∗-algebra with
1
traces, then Cu(A) is almost unperforated if and only if A has strict comparison,
that is, if a,b ∈ (A⊗K) with d (a) < d (b) for any τ ∈ T(A), then [a] ≤ [b].
+ τ τ
(See [9, Proposition 4.2, Remark 4.3 and Proposition 6.2] and [38, Proposition 3.2
and Corollary 4.6].) The following proposition is an immediate corollary of [9,
Theorem 6.6]. (Note that they considered the more general case.) But we shall
give a self-containedproofbased ontheir arguments (see also [9, Proposition6.4]).
PICARD GROUPS OF CERTAIN STABLY PROJECTIONLESS C∗-ALGEBRAS 5
Proposition 3.1. Let A be a simple exact C∗-algebra,and let a andb be positive
elements in A⊗K. Assume that Cu(A) is almost unperforated and 0 is an accu-
mulation point of the spectrum σ(a) of a. Then if d (a)≤d (b) for any τ ∈T(A),
τ τ
then a is Cuntz smaller than b.
Proof. Let a and b be positive elements in A⊗K such that d (a) ≤ d (b) for any
τ τ
τ ∈T(A). We may assume that kak=kbk=1. For any k ∈N we have
k k+1
d (diag(a,..,a))=kd (a)≤kd (b)<(k+1)d (b)=d (diag(b,..,b)).
τ z }| { τ τ τ τ z}|{
Hence k[a]≤(k+1)[b] for any k ∈N because A has strict comparison. Let ǫ>0,
and choose a positive function c on σ(a) such that c (t) > 0 on t ∈ (0,ǫ) and
ǫ ǫ
c (t) = 0 on σ(a)\(0,ǫ). Then we have [c (a)]+[(a−ǫ) ] ≤ [a]. Note that for
ǫ ǫ +
any ǫ > 0, c (a) is a nonzero positive element because 0 is an accumulation point
ǫ
of σ(a). Hence we have 2[a] ≤ sup n[c ] by the simplicity of A and Brown’s
n∈N ǫ
theorem in [4]. There exists a natural number m such that 2[(a−ǫ) ] ≤ m[c (a)]
+ ǫ
since 2[(a−ǫ) ]≪2[a]. Therefore we have
+
(m+2)[(a−ǫ) ]≤m[(a−ǫ) ]+m[c (a)]≤m[a]≤(m+1)[b].
+ + ǫ
By the assumption that Cu(A) is almost unperforated, we see that [(a−ǫ) ]≤[b]
+
for any ǫ>0, and hence we have [a]≤[b]. (cid:3)
Corollary 3.2. Let A be a simple exact stably projectionless C∗-algebra, and let
a andb be positive elements inA⊗K. Assume thatCu(A) is almostunperforated.
Then if d (a)=d (b) for any τ ∈T(A), then a is Cuntz equivalent to b.
τ τ
Proof. For any nonzero positive element a in A⊗K, 0 is an accumulation point of
σ(a) because A is a stably projectionless C∗-algebra. Hence we obtain the conclu-
sion by Proposition 3.1. (cid:3)
Based on the result in [36], we say that a C∗-algebra A has almost stable rank
one if for every σ-unital hereditary subalgebra B ⊆ A⊗K we have B ⊆ GL(B).
Robert showedthat if A is a Z-stable stably projectionlessC∗-algebra,then A heas
almoststable rank one (see [36, Corollary4.5]and[38]). The following proposition
is [36, Proposition 4.7]. See [6, Theorem 3] for the proof.
Proposition 3.3. Let A be a σ-unital C∗-algebra such that A has almost stable
rank one and a and b positive elements in A⊗K. Then a is Cuntz smaller than
b if and only if there exists a right Hilbert A-module X ⊆ bH such that X is
A
isomorphic to aH as a right Hilbert A-module, and a is Cuntz equivalent to b if
A
and only if aH is isomorphic to bH as a right Hilbert A-module.
A A
Corollary3.2 and Proposition3.3 are important in the proofof our main result.
These propositions show that every separable simple Z-stable stably projection-
less C∗-algebra A with a unique tracial state has similar properties of II factors
1
(Murray-vonNeumanncomparisontheory). Moreoverwehavethefollowingpropo-
sition.
Proposition 3.4. Let A be a simple exact separable stably projectionless C∗-
algebra with unique (up to scalar multiple) densely defined lower semicontinuous
trace τ. Assume that τ is normalized,Cu(A) is almostunperforated, A has almost
stable rank one and F(A) = R×. Then every nonzero hereditary subalgebra of A
+
is isomorphic to A.
Proof. Corollary 3.2 and Proposition 3.3 imply if a and b are positive elements in
A⊗K such that d (a)= d (b), then aH is isomorphic to bH as a right Hilbert
τ τ A A
A-module. HenceK (aH )isisomorphictoK (bH ). Thereforeifd (a)=d (b),
A A A A τ τ
6 NORIONAWATA
then a(A⊗K)a is isomorphic to b(A⊗K)b because K (aH ) is isomorphic to
A A
a(A⊗K)a. Since F(A) = R×, for any t ∈ (0,1] there exists a positive element h
+
such that h(A⊗K)h is isomorphic to A and d (h)=t. Note that for any nonzero
τ
hereditarysubalgebraB ofaseparableC∗-algebraAthereexistsanonzeropositive
elementh inAsuchthatBisisomorphictoh Ah . Sinced (h )∈(0,1],weobtain
0 0 0 τ 0
the conclusion. (cid:3)
4. Main result
The following theorem is the main result in this paper. See [20, Corollary 4.8]
and [29, Proposition 3.26] for the unital case.
Theorem 4.1. Let A be a simple exact σ-unital stably projectionless C∗-algebra
withauniquetraicalstateτ andnounboundedtrace. AssumethatCu(A)isalmost
unperforated and A has almost stable rank one. Then the following sequence is
exact:
1 −−−−→ Out(A) −−−ρA−→ Pic(A) −−−T−→ F(A) −−−−→ 1.
Proof. It is clear that ρ is one-to-one, T is onto and Im(ρ ) ⊆Ker(T). We shall
A A
showthatKer(T)⊆Im(ρ ). Let[E]∈Ker(T). ThenCorollary3.2andProposition
A
3.3 imply E is isomorphic to (h⊗e )H as a right Hilbert A-module where h is a
11 A
strict positive element in A and e is a rank one projection in K because we have
11
d (h⊗e )=1bykτk=1. Itcaneasilybecheckedthat(h⊗e )H isisomorphic
τ 11 11 A
to a right Hilbert A-module A with the obvious right A-action and ha,bi = a∗b
A
fora,b∈A. Consequentlyweseethatthereexistssomeautomorphismαsuchthat
[E]=[EA], and hence [E]∈Im(ρ ). (cid:3)
α A
Corollary 4.2. Let A be a simple exact separable Z-stable stably projectionless
C∗-algebrawithaunique tracialstate τ andnounboundedtrace. Thenthe follow-
ing sequence is exact:
1 −−−−→ Out(A) −−−ρA−→ Pic(A) −−−T−→ F(A) −−−−→ 1.
Proof. This is an immediate consequence of [38, Theorem 4.5], [36, Corollary 4.5]
and Theorem 4.1. (cid:3)
Remark 4.3. There exists a unital simple AF algebra A with a unique tracial
statesuchthatOut(A)isnotanormalsubgroupofPic(A). (See[28].) OfcourseA
is a unital stably finite Z-stable C∗-algebra. Therefore the corollary above shows
that Z-stable stably projectionless C∗-algebras are more well-behaved than unital
stably finite Z-stable C∗-algebras.
We shall show some examples.
LetW betheRazak-Jacelonalgebrastudiedin[12],[35]and[36],whichhastriv-
2
ialK-groupsandauniquetracialstateandnounboundedtrace. TheRazak-Jacelon
algebraW is constructedasaninductive limitC∗-algebraofRazak’buildingblock
2
in [32], that is,
k k+1
A(n,m)=f ∈C([0,1])⊗M (C) | f(0)=diag(c,..,c,0 ),f(1)=diag(c,..,c), 
 m n 
z}|c{∈M (C) z}|{
n
 
where n and m are natural numbers with n|m and k := m −1. Let O denote the
n 2
Cuntz algebra generated by 2 isometries S and S . There exists by universality a
1 2
one-parameterautomorphism groupα of O2 given by αt(Sj)=eitλjSj. Kishimoto
andKumjian showedthat if λ and λ areall nonzeroofthe same signand λ and
1 2 1
λ generate R as a closed subgroup, then O ⋊ R is a simple stable projectionless
2 2 α
C∗-algebrawithunique(uptoscalarmultiple)denselydefinedlowersemicontinuous
PICARD GROUPS OF CERTAIN STABLY PROJECTIONLESS C∗-ALGEBRAS 7
trace in [18] and [19]. Moreover Robert [35] showed that W ⊗K is isomorphic to
2
O ⋊ R for some λ and λ . (See also [8].) In particular, W ⊗ K has a one
2 α 1 2 2
parameter trace scaling automorphism group σ (see [18]).
Theorem 4.4. The Picard group of Razak-Jacelonalgebra W is isomorphic to a
2
semidirectproductofOut(W )with R×. MoreoverifA isa simple exactseparable
2 +
W -stable C∗-algebra with a unique tracial state τ and no unbounded trace, then
2
the Picard group of A is isomorphic to a semidirect product of Out(A) with R×.
+
Proof. NotethatweseethatAisstablyprojectionlessC∗-algebrabecauseA⊗K∼=
A⊗W ⊗K has a one parameter trace scaling automorphism group id⊗σ. Since
2
W is Z-stable, we have the following exact sequence:
2
1 −−−−→ Out(A) −−−ρA−→ Pic(A) −−−T−→ F(A) −−−−→ 1
by Corollary 4.2. We have Pic(A)∼=Out(A⊗K) and F(A)=S(A) (see Section 2
). Therefore we see that F(A) = R× and the exact sequence above splits because
+
A⊗Khasaoneparametertracescalingautomorphismgroup. ConsequentlyPic(A)
is isomorphic to Out(A)⋊R×. (cid:3)
+
Remark4.5. (i)BytheresultofBrown,RieffelandGreenandthetheoremabove,
we have
Out(W ⊗K)∼=Out(W )⋊R×.
2 2 +
(ii) We do not assume that A is nuclear in the theorem above. Hence we have
Pic(W ⊗C∗(F ))∼=Out(W ⊗C∗(F ))⋊R×
2 r n 2 r n +
where F is a non-amenable free group with n generators. Moreover Proposition
n
3.4 shows that every nonzero hereditary subalgebra of W ⊗C∗(F ) is isomorphic
2 r n
to W ⊗C∗(F ).
2 r n
(iii) Let A be a simple unital AF algebra with two extremal tracial states. Then
W ⊗AisasimplestablyprojectionlessC∗-algebrawithtwoextremaltracialstates
2
and in the class of Robert’s classification theorem [35]. It can be checked that
Out(W ⊗A) is not a normal subgroup of Pic(W ⊗A) by Robert’s classification
2 2
theoremanda similarpropositionas[20,Proposition1.5]. (We needto replacethe
K -groups with the trace spaces.)
0
5. Z-stability of stably projectionless C∗-algebras
In this section we shall generalize the result of Matui and Sato in [24] to stably
projectionless C∗-algebras. Note that our arguments are essentially based on their
arguments.
We shallreviewsomeresultsofKirchberg’scentralsequence algebrain[16]. For
a separable C∗-algebra A, set
c0(A):={(an)n∈N ∈ℓ∞(N,A)| lim kank=0}, A∞ :=ℓ∞(N,A)/c0(A).
n→∞
Let B be a C∗-subalgebra of A. We identify A and B with the C∗-subalgebras of
A∞ consisting of equivalence classes of constant sequences. Put
A∞ :=A∞∩A′, Ann(B,A∞):={(an)n ∈A∞∩B′ |(an)nb=0foranyb∈B}.
Then Ann(B,A∞) is an closed ideal of A∞∩B′, and define
F(A):=A∞/Ann(A,A∞).
We call F(A) the central sequence algebra of A. A sequence (a ) is said to be
n n
central if limn→∞kana−aank = 0 for all a ∈ A. A central sequence is a rep-
resentative of an element in A∞. Since A is separable, A has a countable ap-
proximate unit {hn}n∈N. It is easy to see that [(hn)n] is a unit in F(A). If
A is unital, then F(A) = A∞. Moreover we see that F(A) is isomorphic to
8 NORIONAWATA
M(A)∞ ∩A′/Ann(A,M(A)∞) since for any (y ) ∈ M(A)∞ ∩A′, (y h ) is a
n n n n n
central sequence in A and [(y ) ] = [(y h ) ] in M(A)∞ ∩A′/Ann(A,M(A)∞).
n n n n n
Let {eij}i,j∈N be the standard matrix units of K. Define a map ϕ of F(A) to
F(A⊗K) by ϕ([(x ) ]) = [(x ⊗ n e ) ]. Then it is easily seen that ϕ is a
n n n Pi=1 ii n
well-defined injective homomorphism. It can be checked that ϕ is surjective by
using matrix units and the centrality of sequence because a similar argument as
above shows any element in F(A⊗K) is equal to [( n x ⊗e ) ] for some
sequence {xn,i,j}n∈N in A. Hence F(A) is isomorphiPc tio,j=F1(An,⊗i,jK). iW,jenshall show
the followingproposition(whichis basedon[45, Proposition2.2])by a similarway
as in [37, Theorem 7.2.2]. See [16, Proposition 4.11] for more general cases.
Proposition 5.1. Let A be a separable C∗-algebra. If there exist a unital homo-
morphism of the prime dimension drop algebra I(k,k+1) to F(A) for any k ∈N,
then A is Z-stable.
Proof. By a similar argumentas in [45, Proposition2.2]and the construction of Z
in [13], we see that there exists a unital homomorphism α of Z to F(A).
Let ϕ be an injective homomorphism of A to A⊗Z such that ϕ(a) = a⊗1Z,
and put C :=M(A⊗Z)∞∩ϕ(A)′/Ann(ϕ(A),M(A⊗Z)∞). Then we can regard
α as a unital homomorphism of Z to C since F(A) is isomorphic to M(A)∞ ∩
A′/Ann(A,M(A)∞). Define a unital homomorphism of β of Z to M(A⊗Z)∞∩
ϕ(A)′ byβ(x)=(1 ⊗x) , andlet[β]:Z →C be the quotienthomomorphism
M(A) n
of β. Then we see that C∗(α(Z),[β](Z)) in C is isomorphic to Z ⊗Z. By the
propertyofZ,thereexistsasequence{wm}m∈N ofunitaryelementsinC suchthat
limm→∞wm∗ [β](x)wm =α(x)foranyx∈Z andwm isintheconnectedcomponent
of 1 in U(C) for any m ∈ N. Since w is in the connected component of 1 in
C m C
U(C),thereexistsaunitaryelementu inM(A⊗Z)∞∩ϕ(A)′ suchthat[u ]=w
m m m
for any m ∈ N. For any a ∈ A,x ∈ Z and all y ∈ M(A⊗Z)∞∩ϕ(A)′ such that
[y]=α(x), we have
yϕ(a)= lim u∗ β(x)u ϕ(a)= lim u∗ β(x)ϕ(a)u = lim u∗ (a⊗x)u
m→∞ m m m→∞ m m m→∞ m m
by[y]=limm→∞[u∗mβ(x)um]andthe definitionofAnn(ϕ(A),M(A⊗Z)∞). Hence
we see that limm→∞u∗m(a ⊗ x)um is an element in ϕ(A)∞. Therefore for any
z ∈A⊗Z, limm→∞d(u∗mzum,ϕ(A)∞)=0. We obtain the conclusion by a similar
argument as in [37, Proposition 2.3.5 and Proposition 7.2.1]. (cid:3)
IfAisunital,everydenselydefinedlowersemicontinuoustraceonAisbounded.
Hence if A is simple and A⊗K has a nonzero projection, then there exists a full
hereditarysubalgebraB ofAsuchthateverydenselydefinedlowersemicontinuous
trace on B is bounded. In general, we have the following proposition.
Proposition 5.2. Let A be a σ-unital simple C∗-algebra. Then there exists a full
hereditarysubalgebraB ofAsuchthateverydenselydefinedlowersemicontinuous
trace on B is bounded.
Proof. Let Ped(A) be the Pedersen ideal of A, and let h be a nonzero positive
elementin Ped(A). Then hAh is containedinPed(A), andhence τ(b)<∞ for any
b ∈ hAh and any τ ∈ T(A) because Ped(A) is a minimal dense ideal. We refer
+
the reader to [3] and [31] for properties of the Pedersen ideal. Define a map Φ of
T(A) to T(hAh) by Φ(τ) = τ| . It can easily be checked that Φ(τ) is equal to
hAh
TrX constructed in [27, Proposition 2.4] where X = hA. Therefore we see that
τ
Φ is a bijective map since A is simple (or h is full). (See also references in [27].)
Consequentlyeverydensely definedlowersemicontinuoustrace onhAh is bounded
because every positive linear functional is automatically bounded. (cid:3)
PICARD GROUPS OF CERTAIN STABLY PROJECTIONLESS C∗-ALGEBRAS 9
IfAisseparable,thenAisZ-stableifandonlyifsomefullhereditarysubalgebra
isZ-stablebyProposition5.1andBrown’stheoremin[4]sinceF(A) isisomorphic
to F(A⊗K). (See also [44].) Therefore we may assume that A has no unbounded
trace by the proposition above. Note that if A has strict comparison and no un-
bounded trace, then for any a,b ∈ A satisfying d (a) < d (b) for all τ ∈ T (A),
+ τ τ 1
we have a-b.
Proposition5.3. LetAbeaseparableC∗-algebrasuchthatT (A)isanon-empty
1
compact set, and let {hm}m∈N be a countable approximate unit for A and ǫ > 0.
Then there exists a natural number N such that
max |τ(f )−τ(h f )|<ǫ
n m n
τ∈T1(A)
for any m ≥ N and for any sequence (fn)n∈N of positive contractions in A. In
particular, we have
lim max |τ(h f )−τ(f )|=0.
n n n
n→∞τ∈T1(A)
Proof. Foranyτ ∈T (A),wehaveτ(h )≤τ(h )andlimτ(h )=1. ByDini’s
1 m m+1 m
theorem, there exists a natural number N such that
max |1−τ(h )|<ǫ
m
τ∈T1(A)
for any m≥N. For any sequence (fn)n∈N of positive contractions in A,
max |τ(f )−τ(h f )|= max |τ((1−h )1/2f (1−h )1/2)|
n m n m n m
τ∈T1(A) τ∈T1(A)
≤ max |1−τ(h )|<ǫ.
m
τ∈T1(A)
(cid:3)
We denote by A˜ the unitization algebra of A. Note that we consider A = A˜
when A is unital. We recall some definitions.
Definition5.4. LetAbeaseparableC∗-algebrawithnounboundedtrace. Assume
that T (A) is a non-empty compactset. We saythat A has property (SI) if for any
1
central sequences (e ) and (f ) of positive contractions in A satisfying
n n n n
lim max τ(e )=0, lim liminf min τ(fm)>0,
n→∞τ∈T1(A) n m→∞ n→∞ τ∈T1(A) n
there exists a central sequence (s ) in A such that
n n
lim ks∗s −e k=0, lim kf s −s k=0.
n→∞ n n n n→∞ n n n
For a completely positive map ϕ of A˜ to A˜, we say that ϕ can be excised in small
central sequences in A if for any central sequences (e ) and (f ) of positive
n n n n
contractions in A satisfying the property above, there exists a sequence (sn)n∈N in
A such that
lim ks∗as −ϕ(a)e k=0for any a∈A˜, lim kf s −s k=0.
n→∞ n n n n→∞ n n n
Remark 5.5. In the definition above,it is important that e and f are elements
n n
in A. We see thatif id canbe excisedin smallcentralsequences in A, then Ahas
A˜
property (SI) (see [24, Proof of (iii)⇒(iv) of Theorem1.1]).
We shall generalize [23, Lemma 4.6] and [24, Lemma 2.4] to non-unital C∗-
algebras.
10 NORIONAWATA
Lemma 5.6. Let c be a positive element in a separable C∗-algebra A such that
T (A) is a non-empty compact set, and let θ ∈ R. For any central sequence (f )
1 n n
of positive contractions in A, we have
limsup max |τ(cf )−θτ(f )|≤2 max |τ(c)−θ|.
n n
n→∞ τ∈T1(A) τ∈T1(A)
Proof. Let {hm}m∈N be a countable approximate unit for A. A similar argument
as in the proof of [23, Lemma 4.6] shows that
limsup max |τ(cf )−θτ(h f )|≤2 max |τ(c)−θτ(h )|
n m n m
n→∞ τ∈T1(A) τ∈T1(A)
for any m∈N. By Proposition 5.3, we have
limsup max |τ(cf )−θτ(f )|≤2 max |τ(c)−θ|.
n n
n→∞ τ∈T1(A) τ∈T1(A)
(cid:3)
Lemma 5.7. Let A be a separable simple C∗-algebra such that T (A) is a non-
1
empty compact set, and let a be a nonzero positive element in A˜. If (f ) is a
n n
central sequence of positive contractions in A such that
lim liminf min τ(fm)>0,
m→∞ n→∞ τ∈T1(A) n
then
lim liminf min τ(fm/2afm/2)>0.
m→∞ n→∞ τ∈T1(A) n n
Proof. Put R := a1/2A. Since A is simple, R is a right ideal of A such that
R∗R =AaA is a dense ideal of A. Therefore there exists a sequence {vj}j∈N in A
such that {Pnj=1vj∗avj}n∈N is an approximate unit for A by a similar argumentas
in [4, Lemma 2.3]. By Proposition 5.3, there exists a natural number N such that
N
lim liminf min τ( v∗av fm)>0.
m→∞ n→∞ τ∈T1(A) X j j n
j=1
We have
N N
lim liminfminτ( v∗av fm)= lim liminfmin τ(v∗a1/2fma1/2v )
m→∞ n→∞ τ X j j n m→∞ n→∞ τ X j n j
j=1 j=1
N
= lim liminfmin τ(fm/2a1/2v v∗a1/2fm/2)
m→∞ n→∞ τ X n j j n
j=1
N
≤ kv k2 lim liminfminτ(fm/2afm/2).
X j m→∞ n→∞ τ n n
j=1
Hence we obtain the conclusion. (cid:3)
Let A be a separable simple C∗-algebra, and let τ be a tracial state on A.
Consider the GNS representation (π ,H ,ξ ) associated with τ. Then π (A)′′ is
τ τ τ τ
a finite von Neumann algebra and π (A) is strongly dense subalgebra of π (A)′′
τ τ
in general. Indeed, every approximate unit for π (A) is strongly convergent to
τ
1 . We can identify C∗(π (A),1 ) in B(H ) with its unitization algebra A˜.
Hτ τ Hτ τ
Therefore we obtain the following lemma (which is based on Haagerup’s theorem
([11, Theorem 3.1])) by the proof of [40, Lemma 2.1]. See also [25, Proposition 3.5
and Theorem 4.3].