On the classification of simple Z-stable C*-algebras with real rank zero and finite decomposition rank

0.22 MB·English

by  Wilhelm Winter

ON THE CLASSIFICATION OF SIMPLE Z-STABLE
C∗-ALGEBRAS WITH REAL RANK ZERO AND FINITE
6 DECOMPOSITION RANK
0
0
2
WILHELMWINTER

n
a
J

4

Abstract. Weshowthat,ifAisaseparablesimpleunitalC∗-algebrawhich
]
A absorbs the Jiang–Su algebra Z tensoriallyand whichhasreal rankzeroand
finite decomposition rank, then A is tracially AF in the sense of Lin, with-
O
out any restriction on the tracial state space. As a consequence, the Elliott
. conjecture is true for the class of C∗-algebras as above which, additionally,
h
satisfy the Universal Coefficients Theorem. In particular, such algebras are
t
a completely determined by their ordered K-theory. They are approximately
m homogeneous of topological dimensionless than orequal to3, approximately
subhomogeneous of topological dimensionat most2 and their decomposition
[
rankalsoisnogreaterthan2.

2
v
1 0. Introduction
8
1 It is the aim of the Elliott classification program to find complete K-theoretic
2 invariantsforseparablesimplenuclearC∗-algebras. ForpurelyinfiniteC∗-algebras,
0
this task was accomplished by Kirchberg and Phillips, and there are numerous
5
0 classification results for special inductive limits of (sub)homogeneous C∗-algebras,
/ cf. [15] for an overview. However, Toms (in the stably finite case) and Rørdam
h
(in the infinite case) have given examples which show that the Elliott invariant, at
t
a least in its current form (cf. [15], Section 2.2), is not complete for the class of all
m simplenuclearC∗-algebras. Theconstructionofthese counterexamplesis basedon
: techniquesintroducedbyVilladsen;theyinvolvelimitsofhomogeneousC∗-algebras
v
i with ‘fast dimension growth’, i.e., the covering dimension of the underlying base
X
spaces grows faster than the vector space dimension of the matrix algebras which
r occur as fibers.
a
ItisinterestingtoaskforasubclassofseparablesimplenuclearC∗-algebraswhich
canbedescribedinanaturalwayand,atthesametime,containstheclasseswhich
have been classified so far, but excludes the known counterexamples to the Elliott
conjecture. There is growing body of evidence that a very useful notion of ‘good’
behavior for simple C∗-algebras is the concept of Z-stability, i.e., the property of
absorbing the Jiang–Su algebra Z tensorially. This algebra was constructed by
Jiang andSu in [6]; it is a simple, stably finite and infinite dimensionalC∗-algebra
Date:January2006.
2000 Mathematics Subject Classification. 46L85,46L35.
Key words and phrases. NuclearC∗-algebras,coveringdimension,K-theory,classification.
Supported by: EU-NetworkQuantum Spaces -Noncommutative Geometry(Contract No.
HPRN-CT-2002-00280)andDeutscheForschungsgemeinschaft (SFB478).
1
2 WILHELMWINTER
which is KK-equivalent to the complex numbers. In some sense, the algebra Z
might be thought of as a stably finite analogue of the Cuntz algebra O∞ (cf. [19]
and[20]). None of the abovementioned counterexamplesis Z-stable,but virtually
all classes for which the Elliott conjecture has been confirmed so far consist of Z-
stable C∗-algebras ([20]). On the other hand, at the present stage it is not known
howZ-stabilityalonecanbeusedtoobtainclassificationresults,butinthesenotes
we show that, in combination with other natural conditions, it can indeed be used
to verify the Elliott conjecture for a large class of stably finite simple nuclear C∗-
algebras.
Z-stableC∗-algebrasbehaveverywellinmanyrespects(cf.[4],[5]and[16]). For
instance, a separable simple nuclear Z-stable C∗-algebra is either purely infinite
or stably finite (in which case it has weakly unperforated K -group and satisfies
0
Blackadar’ssecondfundamentalcomparabilityproperty). Moreover,astably finite
separable simple Z-stable C∗-algebra has real rank zero if and only if the positive
part of the ordered K -group has dense image in the positive continuous affine
0
functions on the tracial state space. If one thinks of the tracial state space T(A)
as an underlying space of a Z-stable and stably finite nuclear C∗-algebra A, one
might regard the elements of A as continuous affine functions on T(A). Then A
has real rank zero (i.e., elements with finite spectrum are dense in the subspace of
all self-adjoint elements of A, cf. [1]) precisely if K (A) is dense in A in the (in
0 + +
general non-Hausdorff) topology coming from T(A). In view of this observation it
is natural to look for classification results in the real rank zero case first.
The known classification theorems for approximately homogeneous (AH) or ap-
proximatelysubhomogeneous(ASH) C∗-algebrasalluse conditionsinvolvingtopo-
logicalcoveringdimensionoftheunderlyingbasespaces. Thispointofviewwasalso
pursued in [24], where we verified the Elliott conjecture for separable simple uni-
tal C∗-algebras with finite decomposition rank and compact and zero-dimensional
space of extremal tracial states. (The decomposition rank is a notion of covering
dimension for nuclear C∗-algebras introduced in [7] by E. Kirchberg and the au-
thor; cf. also [21] and [22].) Similar results were obtained in [2] and [11], under
the additional assumption that the algebras have only countably many extremal
tracial states. In the (locally) AH real rank zero case, the trace space conditions
are not necessary, as was shown in [12] and [14]. In the present article we show
that the condition of [24] on the tracial state space can be removed if the algebras
in question are Z-stable – in other words, we verify the Elliott conjecture for the
class of separable simple unital Z-stable C∗-algebraswith real rank zero and finite
decomposition rank (satisfying the UCT). Following the lines of [24], we do not
prove a classification result directly. Instead, we show that Z-stability, real rank
zero and finite decomposition rank together imply tracial rank zero; classification
willthenfollowfromatheoremofLin. Asaconsequence,ouralgebrashavedecom-
positionranknogreaterthan2,andtheyareAHoftopologicaldimensionlessthan
or equal to 3 and ASH of topologicaldimension at most 2. Moreover,we conclude
from [3] and [20] that, if A has finite decomposition rank and real rank zero, then
A is Z-stable if and only if it is approximately divisible.
I would like to thank Ping Wong Ng, Andrew Toms and, in particular, Nate
Brown for helpful comments and inspiring conversations.
CLASSIFICATION, Z-STABILITY AND DECOMPOSITION RANK 3
1. Real rank zero, decomposition rank and order zero maps
Belowwerecallthe definitionsofdecompositionrankandoforderzeromapsand
describe the particular situation for real rank zero algebras.
1.1 Definition: (cf. [7], Definitions 2.2 and 3.1) Let A and F be separable C∗-
algebras, F finite-dimensional.
(i) A completely positive map ϕ:F →A has order zero, ordϕ =0, if it preserves
orthogonality, i.e., ϕ(e)ϕ(f)=ϕ(f)ϕ(e)=0 for all e,f ∈F with ef =fe=0.
(ii) A completely positive map ϕ : F → A is n-decomposable, if there is a decom-
position F = F ⊕...⊕F such that the restriction of ϕ to F has order zero for
0 n i
each i∈{0,...,n}; we say ϕ is n-decomposable with respect to F =F ⊕...⊕F .
0 n
(iii) A has decomposition rank n, drA = n, if n is the least integer such that the
following holds: Given {b ,...,b } ⊂ A and ε > 0, there is a completely positive
1 m
approximation (F¯,ψ,ϕ) for b ,...,b within ε (i.e., F¯ is a finite-dimensional C∗-
1 m
algebra and ψ : A → F¯ and ϕ : F¯ → A are completely positive contractive and
kϕψ(b )−b k < ε) such that ϕ is n-decomposable. If no such n exists, we write
i i
drA=∞.
1.2 Let ϕ:F →A be an order zero map. We say a ∗-homomorphismσ :F →A′′
ϕ
is a supporting ∗-homomorphism for ϕ, if
(1) ϕ(x)=ϕ(1 )σ (x)=σ (x)ϕ(1 )∀x∈F .
F ϕ ϕ F
By [24], 1.2, any order zero map has a supporting ∗-homomorphism. Moreover,
each positive contractive q ∈ C∗(ϕ(1 )) defines a completely positive contractive
F
order zero map ϕ :F →A by
q
ϕ (.):=qσ (.).
q ϕ
It is clear from (1) that q commutes with ϕ(F). If q happens to be a projection,
then ϕ is a ∗-homomorphism by [21], Proposition 3.2.
q
1.3 Following [24], Definition 2.2, we say that a completely positive map
ϕ:F =M ⊕...⊕M →A
r1 rs
is a discrete order zero map, if ordϕ = 0 and each ϕ(1 ), i = 1,...,s, is a
Mri
multiple of a projection. The next lemma says that, if A has real rank zero, then
any order zero map into A can be approximated by discrete order zero maps.
Lemma: ([24], Lemma2.4) Let Aand F beC∗-algebras, Awith realrankzero and
F finite-dimensional. Suppose ϕ : F → A is completely positive contractive with
orderzeroandletδ >0begiven. Then thereareaunitalembedding ι:F →F˜ ofF
intosomefinite-dimensionalC∗-algebraF˜ andadiscreteorderzeromapϕ˜:F˜ →A
such that ϕ˜(1 )≤ϕ(1 ) and kϕ(x)−ϕ˜◦ι(x)k<δ·kxk for all 06=x∈F.
F˜ F
1.4 Proposition: ([24], Proposition 2.5) Let A be a C∗-algebra with real rank
zero and decomposition rank n. For any finite subset {b ,...,b } ∈ A and ε > 0,
1 k
there is a c.p. approximation (F,ψ,ϕ) such that ϕ is discretely n-decomposable and
kϕψ(b )−b k, kψ(b )ψ(b )−ψ(b b )k<ε for j, l =1,...,k.
j j j l j l
4 WILHELMWINTER
2. The Jiang–Su algebra and approximate tracial division
BelowwerecallsomefactsabouttheJiang–SualgebraZ andestablishoneofthe
technical tools for the proof of our main theorem.
2.1Forrelativelyprimenaturalnumberspandq definetheirprimedimensiondrop
interval to be the C∗-algebra
I[p,q]:={f ∈C([0,1],M ⊗M )|f(0)∈1 ⊗M , f(1)∈M ⊗1 }.
p q Mp q p Mq
In [6], Jiang and Su constructed an infinite-dimensional simple unital C∗-algebra
Z, which is KK-equivalent to C and has a unique normalized trace τ¯. It can be
written as an inductive limit of prime dimension drop intervals; moreover,there is
a unital embedding of any prime dimension drop interval into Z.
2.2TheJiang–Sualgebraisstronglyself-absorbinginthesenseof[19]. Inparticular
this means that, if A is unital and Z-stable, there exists a sequence
(θn :A⊗Z →A)n∈N
of unital ∗-homomorphisms which satisfies
kθn(a⊗1Z)−akn−→→∞0∀a∈A
(cf. [19], Remark 2.7).
2.3 Let A be simple and unital. By [4], A and A ⊗ Z have isomorphic Elliott
invariants iff the ordered group K (A) is weakly unperforated. In [18], Toms has
0
givenanexampleofasimpleunitalC∗-algebrawithweaklyunperforatedK -group
0
which is not Z-stable; this is a counterexample to the Elliott conjecture. It is
not known, although well possible, if real rank zero and finite decomposition rank
together do imply Z-stability. At least we have the following
Theorem: Let A be a separable simple unital C∗-algebra with real rank zero and
finite decomposition rank. Then, A and A⊗Z have isomorphic Elliott invariants;
furthermore, A⊗Z has real rank zero and finite decomposition rank.
Proof: By[24],Theorem3.9,K (A)isweaklyunperforated,soAandA⊗Z have
0
isomorphicElliottinvariants. By[24],Corollary5.2,K (A) isdenseinAff(T(A))
0 + +
(the positive continuous affine functions on the Choquet simplex T(A)), whence
K (A⊗Z) is dense in Aff(T(A⊗Z)) . But now it follows from [16], Theorem
0 + +
7.2, that A⊗Z has real rank zero; dr(A⊗Z) is finite by [7], 3.2.
2.4 The next two lemmas will provide a method of dividing elements of a Z-stable
C∗-algebra approximately with respect to traces. They mark a key step in the
proof of our main result, Theorem 4.1.
Lemma: For any n ∈ N and 0 < β < 1/2(n+1) there is a completely positive
contractiveorderzeromap̺:Cn+1 →Z suchthatτ¯(̺(e ))>β fori=1,...,n+1,
i
where the e denote the canonical generators of Cn+1.
i
Proof: Choose k ∈N such that
1 1 1
< −β .
k 2(cid:18)2(n+1) (cid:19)
Setp:=k·(n+1)andq :=p+1. Letκ:C([0,1])→I[p,q]be the canonicalunital
embedding and ι : I[p,q] → Z a unital ∗-homomorphism; such an ι always exists
(cf. 2.1).
CLASSIFICATION, Z-STABILITY AND DECOMPOSITION RANK 5
Sinceτ¯◦ι◦κisastateonC([0,1]),itisstraightforwardtofindorthogonalpositive
normalized functions f and f in C([0,1]) such that
0 1
1 n+1 1
τ¯◦ι◦κ(f +f )> − −β
0 1
2 2 (cid:18)2(n+1) (cid:19)
and f (1)=f (0)=0.
0 1
Next, define a completely positive contractive map ϕ:M ⊕M →I[p,q] by
q p
ϕ:=f ·(1 ⊗id )+f ·(id ⊗1 );
0 Mp Mq 1 Mp Mq
it is straightforwardto check that ϕ has order zero.
Let ̺ : Cn+1 → M and ̺ : Cn+1 → M be ∗-homomorphisms such that the
0 q 1 p
projections̺ (e )haverankk forj =0,1andi=1,...,n+1. Defineacompletely
j i
positive contractive map ̺:Cn+1 →I[p,q] by
̺:=ϕ◦(̺ ⊕̺ ).
0 1
Being the composition of an order zero map and a ∗-homomorphism, ̺ again has
order zero.
If g , g ∈ M and h , h ∈ M are projections such that rank(g ) = rank(g )
1 2 q 1 2 p 1 2
andrank(h )=rank(h ),therearepartialisometriesv ∈M ands∈M suchthat
1 2 q p
g =vv∗,g =v∗v,h =ss∗ andh =s∗s. Itisthenstraightforwardtocheckthat
1 2 1 2
τ¯◦ϕ(g ,h ) = τ¯◦ϕ(g ,h )
1 1 2 2
rank(g ) rank(h )
j j
= ·τ¯◦ι◦κ(f )+ ·τ¯◦ι◦κ(f ).
0 1
q p
As a consequence, we see that
k 1
τ¯◦̺(e ) = ·τ¯◦ι◦κ(f )+ ·τ¯◦ι◦κ(f )
i 0 1
k(n+1)+1 n+1
1 1 1
≥ ·τ¯◦ι◦κ(f )+ ·τ¯◦ι◦κ(f )−
0 1
n+1 n+1 k
1 1 1 1
> − −β −
2(n+1) 2(cid:18)2(n+1) (cid:19) k
> β
for i=1,...,n+1.
2.5 Lemma: Let ϕ : F →A and ̺ : Cn+1 →B be completely positive contractive
n
maps; suppose ϕ is n-decomposable with respect to the decomposition F = F
i=0 i
and ̺ has order zero. L
Then the completely positive contractive map ϕ¯:F →A⊗B, given by
n
ϕ¯(x):= ϕ(x1 )⊗̺(e )∀x∈F ,
Fi i+1
Xi=0
has order zero.
Proof: Obvious.
6 WILHELMWINTER
3. Tracial rank zero
In this section we recall the notion of tracially AF C∗-algebras and provide an
alternative characterization of when a simple unital C∗-algebra with Blackadar’s
second fundamental comparability property is tracially AF.
3.1 In [8], Lin introduced the notion of tracially AF algebras (or, equivalently, C∗-
algebrasoftracialrankzero). In the simple and unitalcase,the definition readsas
follows:
Definition: A simple unital C∗-algebra A is said to be tracially AF, if, for
any finite subset F ⊂ A, ε > 0 and 0 6= a ∈ A , there is a finite-dimensional
+
C∗-subalgebra B ⊂A with the following properties:
(i) kb1 −1 bk<ε∀b∈F
B B
(ii) dist(1 b1 ,B)<ε∀b∈F
B B
(iii) 1 −1 isMurray–vonNeumannequivalenttoaprojectioninthehereditary
A B
subalgebra aAa of A.
3.2RecallthataunitalC∗-algebraAissaidtohaveBlackadar’ssecondfundamental
comparabilityproperty, if, whenever p and q are projections in A such that τ(p)<
τ(q) ∀τ ∈ T(A), then p is Murray–von Neumann equivalent to a subprojection of
q. In [10], Corollary 6.15, Lin provided a characterization of when a C∗-algebra
withrealrankzeroandcomparabilityistraciallyAF.Forourpurposesthefollowing
slightlydifferentversionofLin’scharacterizationwillbemoreuseful. Itisprobably
wellknown;butsincewecouldn’tfindanexplicitproofintheliterature,weinclude
one here.
Lemma: Let A be a simple unital C∗-algebra which has the second fundamental
comparability property; suppose that K (A) has dense image in the positive con-
0 +
tinuousaffine functions on T(A), Aff(T(A)) , and that every hereditary subalgebra
+
of A contains a nonzero projection. Then A is tracially AF if and only if there is
n∈Nsuchthat,foranyfinitesubsetF ⊂Aandε>0,thereisafinite-dimensional
C∗-subalgebra B ⊂A with the following properties:
(i) kb1 −1 bk<ε∀b∈F
B B
(ii) dist(1 b1 ,B)<ε∀b∈F
B B
(iii) τ(1 )> 1 ∀τ ∈T(A).
B n
As for the hypotheses of the preceding lemma, note that if a C∗-algebra A has
comparability and real rank zero, then automatically K (A) has dense image in
0 +
Aff(T(A)) and every hereditary subalgebra of A contains a nonzero projection.
+
Before we turn to the proof of the lemma, we need two more technical results.
3.3 Recall that any finite-dimensional C∗-algebra can be written as the universal
C∗-algebra generated by a set of matrix units with the respective relations. By
the results of Chapter 14 in [13], finite-dimensional C∗-algebras are semiprojec-
tive. Equivalently, the defining relations are stable (cf. [13], Definition 14.1.1 and
Theorem 14.1.4). As a direct consequence we obtain the following:
Proposition: Let B be a finite-dimensional C∗-algebra. For every γ >0 there is
ϑ>0 such that the following holds:
Suppose A is another C∗-algebra and ϕ : B → A is a ∗-homomorphism. If p ∈ A
CLASSIFICATION, Z-STABILITY AND DECOMPOSITION RANK 7
is a projection satisfying
kpϕ(b)−ϕ(b)pk<ϑ·kbk∀06=b∈B,
then there is a ∗-homomorphism ϕ˜:B →pAp such that
kϕ˜(b)−pϕ(b)pk<γ·kbk∀06=b∈B.
3.4Lemma: LetAbeasimpleunitalC∗-algebrawhich has thesecondfundamental
comparability property; suppose that K (A) has dense image in the positive con-
0 +
tinuous affine functions on T(A). Let α,β > 0 and a nontrivial projection p ∈ A
with τ(p)>α+2β ∀τ ∈T(A) be given.
Then, there are s,t∈N and ∗-homomorphisms ι :M ⊕M →M ⊕M ,
0 t t+1 t+s t+1+s
ι : M ⊕M → M ⊕M and κ : M ⊕M → A such that the map
1 s s t+s t+1+s t+s t+1+s
ι +ι is a unital ∗-homomorphism,
0 1
κι (1 +1 )=1 −p,
0 Mt Mt+1 A
s α
> +β
t+1 1−α
and
t·θι1(1Ms⊕Ms)<s·θι0(1Mt⊕Mt+1)<(t+1)·θι1(1Ms⊕Ms)
for any tracial state θ on M ⊕M .
t+s t+1+s
Proof: Let f ∈ Aff(T(A)) be the image of 1 −p under the canonical map
+ A
r : K (A) → Aff(T(A)). Note that f is nowhere zero since A is simple and any
0
tracial state on A is faithful.
We first show that, for any t∈N, there is a unital ∗-homomorphism
ν :M ⊕M →(1 −p)A(1 −p).
t t+1 A A
By our hypotheses on A, there is a projection e∈(1 −p)A(1 −p) such that
A A
1 1
·f <r(e)< ·f.
t+1 t
From comparison, we now obtain t pairwise orthogonal subprojections of 1 −p
A
whichareallMurray–vonNeumann equivalentto e; this yields a∗-homomorphism
ν¯:M →(1 −p)A(1 −p).
t A A
We have
r(1 −p−ν¯(1 ))=f −t·r(e)<r(e),
A Mt
whence there is a partial isometry s∈A such that
s∗s≤ν¯(e )
11
and
ss∗ =1 −p−ν¯(1 ).
A Mt
But now it is straightforwardto construct ∗-homomorphisms
ν :M →(1 −p)A(1 −p)
1 t+1 A A
and
ν :M →(1 −p)A(1 −p)
0 t A A
such that
ν (e )=1 −p−ν¯(1 )
1 t+1,t+1 A Mt
8 WILHELMWINTER
and
ν (e )+ν (e )=ν¯(e )∀j ∈{1,...,t}.
0 jj 1 jj jj
Themapν isdeterminedbysettingν (e ):=(1 −p−ν¯(1 ))sandν (e ):=
1 1 t+1,1 A Mt 1 j,1
ν¯(e )s∗s for j =1,...,t. ν is determined by ν (e ):=ν¯(e )(ν¯(e )−s∗s) for
j,1 0 0 j,1 j,1 11
j =1,...,t. The maps ν and ν have orthogonal images and therefore add up to
0 1
a unital ∗-homomorphism
ν :M ⊕M →(1 −p)A(1 −p).
t t+1 A A
Next, one observes that
α+2β α+β α
> > +β;
1−α−2β 1−α−β 1−α
we may therefore fix s,t∈N such that
α+2β s s α
> > > +β.
1−α−2β t t+1 1−α
By hypothesis, we have
τ(1 −p)<1−α−2β ∀τ ∈T(A),
A
so
s
·τ(1 −p)<α+2β <τ(p)∀τ ∈T(A).
A
t
Since
1 1
r(ν (e )+ν (e ))< ·f = ·r(1 −p),
0 11 1 11 A
t t
from comparisonwe obtain s pairwise orthogonalsubprojections of p which are all
Murray–von Neumann equivalent to the projection ν (e )+ν (e ). This clearly
0 11 1 11
yields an embedding
κ:M ⊕M →A
t+s t+1+s
withthedesiredproperties,whereι isthesumoftheupperleftcornerembeddings
0
and ι is the sum of the lower right corner embeddings.
1
Proof: (of Lemma 3.2) If A is tracially AF, the assertion obviously holds with
any n > 1. Conversely, suppose there is n ∈ N such that the assertion in the
lemma holds. Let 0 6= a ∈ A , a finite subset F ⊂ A and ε > 0 be given. By our
+
assumption on A, there is a nonzero projection q ∈aAa. Set
η :=min{τ(q)|τ ∈T(A)}.
Note that η >0, since T(A) is compact and the positive function τ 7→τ(q) is con-
tinuous and everywhere nonzero (A is simple, hence every tracial state is faithful).
For i∈N, choose strictly positive numbers ε such that ε <ε and define
i N i
i k P
1 1
α := 1− ;
i
n (cid:18) n(cid:19)
kX=0
note that (1−α )(1− 1)=1−α , whence
i n i+1
α 1
i
(2) (1−α )+(1−α )=1− ∀i
i+1 i+1
1−α n
i
and that
i→∞
α −→ 1.
i
CLASSIFICATION, Z-STABILITY AND DECOMPOSITION RANK 9
We shall inductively construct finite-dimensional C∗-algebras B ⊂ A, i ∈ N, with
i
the following properties:
a) kb1 −1 bk< i ε ∀b∈F
Bi Bi k=0 k
b) dist(1 b1 ,B )P< i ε ∀b∈F
Bi Bi i k=0 k
c) τ(1Bi)>αi ∀τ ∈T(PA).
Having done so, (i) and (ii) of Definition 3.1 will hold by a) and b), respectively.
Since α →1, by c) there is K ∈N such that τ(1 −1 )<η. From comparison
i A BK
we see that 1 −1 is Murray–vonNeumann equivalent to q ∈aAa, whence (iii)
A BK
of 3.1 holds and A is tracially AF.
We proceed to construct the B . B obviously exists by assumption. Suppose
i 0
that B (satisfying properties a), b) and c)) has been constructed for some i ∈ N.
i
We show how to obtain B .
i+1
First, set
1
β := min{τ(1 )−α |τ ∈T(A)},
2 Bi i
thenβ >0,againsince T(A)iscompactandthe positive functionτ 7→τ(1 )−α
Bi i
is continuous and nonzero.
By Lemma 3.4, there are s,t ∈ N, finite-dimensional C∗-algebras D := M ⊕
t+s
M ,C :=M ⊕M andC :=M ⊕M and∗-homomorphismsι :C →D,
t+1+s 0 t t+1 1 s s 0 0
ι :C →D and κ:D →A satisfying
1 1
s α
i
(3) > +β,
t+1 1−α
i
(4) κι (1 )=1 −1 ,
0 C0 A Bi
ι (1 )+ι (1 )=1
0 C0 1 C1 D
and
(5) t·θι (1 )<s·θι (1 )<(t+1)·θι (1 )
1 C1 0 C0 1 C1
for any θ ∈T(D). In particular,
(6) κι (1 )≤1 .
1 C1 Bi
Set
F :=F ∪B (B )∪B (κ(D))
1 i 1
(where B (.) denotes the unit ball of a C∗-algebra); choose γ >0 such that
1 e
α
i
(7) 2γ +β+1 <β(1−α )
i+1
(cid:18)1−α (cid:19)
i
and
(8) 7γ <ε .
i+1
Choose 0 < ϑ < γ such that the assertion of Proposition 3.3 holds for both the
finite-dimensional C∗-algebras B and D.
i
By hypothesis, there is a finite-dimensional C∗-algebra F ⊂A such that
d) kb1 −1 bk<ϑ∀b∈F
F F
e) dist(1 b1 ,F)<ϑ∀b∈F
F F e
f) τ(1 )> 1 ∀τ ∈T(A).
F n e
10 WILHELMWINTER
By d) in connection with Proposition 3.3, there is a ∗-homomorphism
̺:B →(1 −1 )A(1 −1 )
i A F A F
such that
(9) k̺(b)−(1 −1 )b(1 −1 )k<γ·kbk∀06=b∈B .
A F A F i
Similarly, we obtain a ∗-homomorphism
κ˜ :D →(1 −1 )A(1 −1 )
A F A F
such that
(10) kκ˜(d)−(1 −1 )κ(d)(1 −1 )k<γ·kdk∀06=d∈D.
A F A F
Set
B :=̺(B )⊕F ;
i+1 i
we proceed to check that B statisfies properties a), b) and c) above (with i+1
i+1
in place of i).
First, we have
kb1 −1 bk
Bi+1 Bi+1
= kb(1 +̺(1 ))−(1 +̺(1 ))bk
F Bi F Bi
≤ kb̺(1 )−̺(1 )bk+kb1 −1 bk
Bi Bi F F
d),(9)
< kb(1 −1 )1 (1 −1 )−(1 −1 )1 (1 −1 )bk
A F Bi A F A F Bi A F
+2γ+ϑ
d)
< k(1 −1 )(b1 −1 b)(1 −1 )k
A F Bi Bi A F
+2γ+3ϑ
(a),(8) i+1
< ε ,
k
kX=0
so a) above holds (with i+1 in place of i). Next, we check b):
dist(1 b1 ,B )
Bi+1 Bi+1 i+1
= dist((̺(1 )+1 )b(̺(1 )+1 ),B )
Bi F Bi F i+1
d)
≤ dist(̺(1 )b̺(1 )+1 b1 ,B )+2ϑ
Bi Bi F F i+1
≤ dist(̺(1 )b̺(1 ),̺(B ))+dist(1 b1 ,F)+2ϑ
Bi Bi i F F
= dist(̺(1 )(1 −1 )b(1 −1 )̺(1 ),̺(B ))+dist(1 b1 ,F)+2ϑ
Bi A F A F Bi i F F
(9)
< dist(̺(1 )̺(b)̺(1 ),̺(B ))+dist(1 b1 ,F)+γ+2ϑ
Bi Bi i F F
i
b),e)
< ε +γ+3ϑ
k
Xk=0
(8) i+1
< ε .
k
Xk=0
We will not prove c) directly; instead, we assume that
(11) τ(1 )≤α
Bi+1 i+1