On locally AH algebras

91 Pages·2012·0.65 MB·English

On locally AH algebras
Huaxin Lin
1 Abstract
1
0 Weshowthateveryunitalamenableseparablesimple C∗-algebra withfinite tracialrank
2 which satisfies the UCT has in fact tracial rank at most one. We also show that unital
∗
c separable simple C -algebras which are “tracially” locally AH with slow dimension growth
e are -stable. As a consequence, unital separable simple C∗-algebras which are locally AH
D Z
with no dimension growth are isomorphic to a unital simple AH-algebra with no dimension

1 growth.
3

] 1 Introduction
A
O
The program of classification of amenable C -algebras, or the Elliott program, is to classify
∗
.
h amenable C -algebras up to isomorphisms by their K-theoretical data. One of the high lights
∗
at of the success of the Elliott program is the classification of unital simple AH-algebras (induc-
m tive limits of homogeneous C -algebras) with no dimension growth by their K-theoretical data
∗
[ (known as the Elliott invariant) ([16]). The proof of this first appeared near the end of the last

century. Immediately after the proof appeared, among many questions raised is the question
3
v whetherthesameresultholdsforunitalsimplelocally AH-algebras (seethedefinition3.5below)
5
with no dimension growth. It should be noted that AF-algebras is locally finite dimensional.
4
4 But(separable)AF-algebras areinductivelimitsoffinitedimensionalC∗-algebras. Theso-called
0 AT-algebras are inductive limits of circle algebras. More than often, these AT-algebras arise
.
4 as local circle algebras (approximated by circle algebras). Fortunately, due to the weak-semi-
0 projectivity of circle algebras, locally AT-algebras are AT-algebras. However, the situation is
1
completely different for locally AH algebras. In fact it was proved in [11] that there are unital
1
: C -algebras which are inductive limits of AH-algebras but themselves are not AH-algebras. So
v ∗
i in general, a locally AH algebra is not an AH algebra.
X
On the other hand, however, it was proved in [26] that a unital separable simple C -algebra
∗
r
a which is locally AH is a unital simple AH-algebra, if, in addition, it has real rank zero, stable
rank one and weakly unperforated K -group and which has countably many extremal traces.
0
In fact these C -algebras have tracial rank zero. The tracial condition was late removed in
∗
[58]. In particular, if A is a unital separable simple C -algebra which is locally AH with no (or
∗
slow) dimension growth and which has real rank zero must be a unital AH-algebra. In fact such
C -algebras have stable rank one and have weakly unperforated K (A). The condition of real
∗ 0
rank zero forces these C -algebras to have tracial rank zero. More recently, classification theory
∗
extends to those C -algebras that are rationally tracial rank at most one ([59], [32], [37] and
∗
[35]). These are unital separable simple amenable C -algebras A such that A U have tracial
∗
⊗
rank at most one for some infinite dimensional UHF algebra U. An important subclass of this
(whichincludes,forexample,theJiang-Sualgebra )istheclassofthoseunitalseparablesimple
Z
C -algebras A such thatA U have tracial rankzero. By now we have some mechanical tools to
∗
⊗
verifycertainC -algebrastohavetracialrankzero(see[26],[5],[58]and[38])andbasedonthese
∗
results, we have some tools to verify when a unital simple C -algebra is rationally tracial rank
∗
zero ([55] and [54]). However, these result could not be applied to the case that C -algebras are
∗
1
of tracial rank one, or rationally tracial rank one. Until now, there is no effective way, besides
Gong’s decomposition result ([18]), to verify when a unital separable simple C -algebra has
∗
tracial rank one (but not tracial rank zero). In fact, as mentioned above, we did not even know
when a unital simple separable locally AH algebra with no dimension growth has tracial rank
one. This makes it much hard to decide when a unital simple separable C -algebra is rationally
∗
tracial rank one.
Closely related problem is whether a unital separable simple C -algebra with finite tracial
∗
rank is in fact of tracial rank at most one. This is an open problem for a decade. If the problem
has an affirmative answer, it will make easier, for many cases, to decide certain unital simple
C -algebras to have tracial rank at most one.
∗
The purpose of this research is to solve these problems. Our main results include the follow-
ing:
Theorem 1.1. Let A be a unital separable simple C -algebra which is locally AH with no
∗
dimension growth. Then A is isomorphic to a unital simple AH-algebra with no dimension
growth.
We actually prove the following.
Theorem 1.2. Let A be a unital separable simple amenable C -algebra with finite tracial rank
∗
which satisfies the Universal Coefficient Theorem. Then A is isomorphic to a unital simple
AH-algebra with no dimension growth. In particular, A has tracial rank at most one.
To establish the above, we also prove the following
Theorem 1.3. Let A be a unital separable simple C -algebra in then A is -stable, i.e.,
∗ 1
C Z
A= A .
∼
⊗Z
(See 3.6 below for the definition of .)
1
C
The article is organized as follows. Section 2 serves as a preliminary which includes a
number of conventions that will be used throughout this article. Some facts about a subgroup
SU(M (C(X))/CU(M (C(X))) have been discussed. The detection of those unitaries with
n n
trivial determinant at each point which are not in the closure of commutator subgroup play
new role in the Basic Homotopy Lemma which will be presneted in section 11. In section 3,
we introduce the class of simple C -algebras which may be described as tracially locally AH
1 ∗
C
algebras of slow dimension growth. Several related definitions are given. In section 4, we discuss
some basic properties of C -algebras in class . In section 5, we prove, among other things,
∗ 1
C
that C -algebras in have stable rank one and the strict comparison for positive elements. In
∗ 1
C
section 6, we study the tracial state space of a unital simple C -algebra in . In particular, we
∗ 1
C
show that every quasi-trace of a unital separable simple C -algebra in extends to a trace.
∗ 1
C
Moreover, we show that, for a unital simple C -algebra A in , the affine map from the tracial
∗ 1
C
state space to state space of K (A) maps the extremal points onto the extremal points. In
0
section 7, we discuss the unitary groups of simple C -algebras in a subclass of . In section
∗ 1
C
8, using what have been established in previous sections, we combine an argument of Winter
([60])and an argument of Matui and Sato ([39]) to prove Theorem 1.2 above. In section 9 we
present some versions of so-called existence theorem. In section 10, we present a uniqueness
statement that will beproved in section 12 and an existence type resultregarding the Bott map.
The uniqueness theorem holds for Y being a finite CW complex of dimension zero as well as
the case that Y = [0,1]. An induction on the dimension d will be presented in the next two
sections. In section 11, we present a version of The Basic Homtopy Lemma which was first
studied intensively in [4] and later in [31]. A new obstruction for the Basic Homotopy Lemma
in this version will be dealt with which was earlier mentioned in section 2. In section 12, we
prove the uniqueness statement in section 10. In section 13 we present the proofs for Theorem
1.1 and 1.3. Section 14 serves an appendix to this article.
2
2 Preliminaries
2.1. Let A be a unital C -algebra. Denote by T(A) the convex set of tracial states of C.
∗
Let Aff(T(A)) be the space of all real affine continuous functions on T(A). Denote by M (A)
n
the algebra of all n n matrixes over A. By regarding M (A) as a subset of M (A), define
n n+1
×
M (A) = M (A). If τ T(A), then τ Tr, where Tr is standard trace on M , is a trace
∞ ∪∞n=1 n ∈ ⊗ n
on M (A). Throughout this paper, we will use τ for τ Tr without warning.
n
⊗
We also use QT(A) for the set of all quasi-traces of A.
Let C and A be two unital C -algebras with T(C) = and T(A) = . Suppose that h :
∗
6 ∅ 6 ∅
C A is a unital homomorphism. Define an affine continuous map h : T(A) T(C) by
♯
→ →
h (τ)(c) = τ h(c) for all τ T(A) and c C.
♯
◦ ∈ ∈
Definition 2.2. Let C be a unital C -algebra with T(C) = . For each p M (C) define
∗ n
6 ∅ ∈
pˇ(τ) = (τ Tr)(p) for all τ T(A), where Tr is the standard trace on M . This gives positive
n
⊗ ∈
homomorphism ρ :K (C) Aff(T(C)).
C 0
→
Apositivehomomorphism s : K (A) CisastateonK (A)ifs([1 ]) = 1.LetS(K (A))be
0 0 A 0
→
the state space of K (A). Define r :T(C) S(K (A)) by r (τ)([p]) = τ(p) for all projections
0 A 0 A
→
p M (A) (for all n 1).
n
∈ ≥
Definition 2.3. Let A and B be two C -algebras and ϕ: A B be a positive linear map. We
∗
→
will use ϕ(K) : A M (B) for the map Φ(K)(a) = ϕ(a) 1 . If a B, we may write a(K)
→ K ⊗ MK ∈
K
for the element a 1 and sometime it will be written as diag(a,a,...,a).
⊗ MK
Definition 2.4. In what follows, we may identify T with the unitzcir}c|le a{nd z C(T) with the
∈
identity map on the circle.
Definition 2.5. Let A be a unital C -algebra. Following [17], define
∗
F K (A) = ϕ (z ) K (A) :ϕ Hom(C(Sn),M (A)) ,
n i i b i
{ ∗ ∈ ∈ ∞ }
where z is a generator (the bott element) of K (C(Sn)), if i = 0 and n even, or i = 1 and n
b i
odd. F K (A) is a subgroup of K (A), i= 0,1.
n i i
Definition 2.6. Fix an integer n 2, let z be a generator of K (C(S2n 1)). Let z bea unitary
1 − b
≥
in M (C(S2n 1)) which represents z. We fix one such unitary that z SU (C(S2n 1)), i.e.,
n − b n −
∈
det(u(x)) = 1 for all x S2n 1. In case n = 2, one may write
−
∈
z w¯
z = − , (e2.1)
b w z¯
(cid:18) (cid:19)
where S3 = (z,w) C2 : z 2+ w 2 = 1 .
{ ∈ | | | | }
2.7. Let C be a unital C -algebra. Denote by U(C) the unitary group of C and denote by
∗
U (C) the subgroup of U(C) consisting of unitaries which connected to 1 by a continuous path
0 C
of unitaries. Denote by CU(C) the closure of the normal subgroup generated by commutators
of U(C). Let u U(C). Then u¯ is the image of u in U(C)/CU(C). Let U(A) be a subset.
∈ W ⊂
Denote by the set of those elements u¯ such that u . Denote by CU (C) the intersection
0
W ∈ W
CU(C) U (C). Note that U(A)/CU(A) is an abelian group.
0
∩
We use the following metric on U(A)/CU(A) :
dist(u¯,v¯)= inf uv w :w CU(A) .
∗
k − k ∈ }
3
Using de la Harp-Scandalis determinant, by K. Thomsen (see [52]), there is a short splitting
exact sequence
0 Aff(T(C))/ρ (K (C)) U(M (C))/CU(M (C)) K (C) 0. (e2.2)
→ C 0 → ∪∞n=1 n n → 1 →
Suppose that r 1 is an integer and U(M (A))/U(M (A)) = K (A), one has the following
r r 0 1
≥
short splitting exact sequence:
0 Aff(T(C))/ρ (K (C)) U(M (C))/CU(M (C)) K (C) 0. (e2.3)
C 0 r r 1
→ → → →
For u U (C), we will use ∆(u) for the de la Harp and Skandalis determinant of u, i.e.,
0
∈
the image of u in Aff(T(C))/ρ (K (C)). For each C -algebra C with U(C)/U (C) = K (C),
C 0 ∗ 0 1
we will fix one splitting map J : K (C) U(C)/CU(C). For each u¯ J (K (C)), select
c 1 c 1
→ ∈
and fix one element u U(C) such that u = u¯. Denote this set by U (K (C)). Denote by
c c c 1
∈
Π : U(C)/CU(C) K (C) the quotient map. Note that Π J = id .
c → 1 c◦ c K1(C)
If A is a unital C -algebra and ϕ : C A is a unital homomorphism, then ϕ induces a
∗
→
continuous homomorphism
ϕ : U(C)/CU(C) U(A)/CU(A).
‡
→
If g Aff(T(A)), denote by g the image of g in Aff(T(A))/ρ (K (A)).
A 0
∈
Definition 2.8. Let A and B be two unital C -algebras. Let G U(M (A))/CU(M (A))
∗ 1 m m
⊂
be a subgroup. Let γ : G U(M (B))/CU(M (B)) be a homomorphism and let Γ :
1 m m
→
Aff(T(A)) Aff(T(B)) be an affine homomorphism. We say that Γ and λ are compati-
→
ble if γ(g¯) = Γ(g) for all g Aff(T(A)) such that g G U (M (B))/CU(M (B))
1 0 m m
∈ ∈ ∩ ⊂
Aff(T(A))/ρ (K (A)). Let λ : T(B) T(A) be continuous affine map. We say γ and λ are
A 0
→
compatible if γ and the map from Aff(T(A)) Aff(T(B)) induced by λ are compatible. Let
→
κ Hom (K(A),K(B)). We say that κ and γ are compatible if κ (z) = Π γ(z) for all
∈ Λ |K1(A) c ◦
z G . We say that κ and λ are compatible if ρ (κ ([p]) = λ(τ)([p]) for all projections
∈ 1 B |K0(A)
p M (A).
∈ ∞
Definition 2.9. Let X be a compact metric space and let P M (C(X)) be a projection in
m
∈
M (C(X))suchthatP(x) = 0forallx X,wherem 1isaninteger. LetC = PM (C(X))P
m m
6 ∈ ≥
and let r 1 be an integer. Denote by SU (C) the set of those unitaries u M (C) such that
r r
≥ ∈
det(u(x)) = 1 for all x X. Note that SU (C) is a normal subgroup of U (C).
r r
∈
The following is an easy fact.
Proposition 2.10. Let C be as in 2.9, let Y be a compact metric space and let P M (C(Y))
1 n
∈
be a projection such that P (y) = 0 for all y Y. Let B = P M (C(Y))P . Suppose that
1 1 n 1
6 ∈
ϕ :C B is a unital homomorphism. Then ϕ maps SU (C) into SU (B) for all integer r 1.
r r
→ ≥
It is also easy to see that CU(M (C(C))) SU (C) U (M (C)). Moreover, one has
r ⊂ ∪∞k=1 k ∩ 0 k
the following:
Proposition 2.11. Let X be a compact metric space and let C = PM (C(X))P be as in 2.9.
m
Then
SU (C) U (M (C)) CU(M (C)) for all integer r 1.
r 0 r r
∩ ⊂ ≥
Proof. Let u SU (C) U (M (C)). Write u = k exp(√ 1h ), where h M (C) . Put
∈ r ∩ 0 r j=1 − j j ∈ r s.a.
R(x) = rankP(x) for all x X. Note R(x) = 0 for all x X. It follows that
∈ 6 Q ∈
k
1
( )T ( h (x)) =N(x) Z, (e2.4)
x j
2π√ 1 ∈
− j=1
X
4
whereT isthestandardtraceonM .NotethatN(x) C(X).Thereforethereisaprojection
x rR(x)
∈
Q M (C) such that
L
∈
rankQ(x) = N(x) for all x X. (e2.5)
∈
Let τ T(C). Then
∈
τ(f)= t (f)dµ for all f PM (C(X))P, (e2.6)
x τ m
∈
ZX
where t is the normalized trace on M and µ is a Borel probability measure on X. Let Tr
x R(x) τ
be the standard trace on M . Then
L
ρ (Q)(τ) = (t Tr)(Q(x))dµ (e2.7)
C x τ
⊗
ZX
N(x)
= dµ (e2.8)
τ
R(x)
ZX
for all τ T(C), Define a smooth path of unitaries u(t) = k exp(√ 1h (1 t)) for t [0,1].
∈ j=1 − j − ∈
So u(0) = u and u(1) = id . Then, with T being the standard trace on M ,
Mr(C) Q r
1 1 du(t) 1 1 k
( ) (τ T) u(t) dt = ( ) (τ T)( h )dt (e2.9)
∗ j
2π√ 1 ⊗ dt 2π√ 1 ⊗
− Z0 − Z0 j=1
X
1 k 1 Tx( kj=1hj(x))
= ( ) (t T)( h (x))dµ = ( ) dµ (e2.10)
x j τ τ
2π√ 1 ⊗ 2π√ 1 R(x)
− ZX j=1 − ZX P
X
N(x)
= dµ = ρ (Q)(τ) for all τ T(C). (e2.11)
τ C
R(x) ∈
ZX
By a result of Thomsen ([52]), this implies that
u= 1 U(M (C)))/CU(M (C).
r r
∈
In other words,
SU (C) U (C) CU(M (C)).
r 0 r
∩ ⊂
Definition 2.12. Let X be a finite CW complex. Let X(n) be the n-skeleton of X and let
s : C(X) C(X(n)) be the surjective map induced by restriction, i.e., s (f)(y) = f(y)
n n
→
for all y X(n). Let P M (C(X)) be a projection for some integer l 1 and let C =
l
∈ ∈ ≥
PM (C(Y))P. Denote still by s : C P(n)M (C(X(n)))P(n), where P(n) = P . Put C =
l n → l |X(n) n
P(n)M (C(X(n)))P(n). Note that C is a quotient of C, and C is a quotient of C . It was
l n n 1 n
−
proved by Exel and Loring ([17]) that F K (C) = ker(s ) .
n i n 1 i
− ∗
Suppose that X has dimension N. Let
I = kerr = f C :f = 0 .
N N−1 { ∈ |X(N−1) }
Then I is an ideal of C. There is an embedding j : I C which maps λ 1+f to λ 1 +f
N N N C
→ · ·
for all f I . Define, for 1< n < N,
N
∈
e
I = f C : f = 0 .
n { ∈ n |X(n−1) }
Again, there is an embedding j :I C .
n n n
Note that I = P(n)′M (C(Y ))P(→n)′, where Y = Sn Sn Sn (there are only finitely
n ∼ l n n ···
many of Sn). e
W W W
e
5
Lemma 2.13. Let X be a compact metric space. Then
Tor(K (C(X))) F K (C(X)). (e2.12)
1 3 1
⊂
Proof. We first consider the case that X is a finite CW complex. Let Y be the 2-skeleton of X
and let s : C(X) C(Y) be the surjective homomorphism defined by f f for f C(X).
Y
→ 7→ | ∈
Then, by Theorem 4.1 of [17],
F K (C(X)) = kers . (e2.13)
3 1 1
∗
Since K (C(Y)) is torsion free, Tor(K (C(X))) kers . Therefore
1 1 1
⊂ ∗
Tor(K (C(X))) F K (C(X)). (e2.14)
1 3 1
⊂
Forthegeneralcase,letg Tor(K (C(X))beanon-zeroelement. WriteC(X) = lim (C(X ),ϕ ),
1 n n n
∈ →∞
whereeachX isafiniteCWcomplex. Thereisn andg K (C(X )suchthat(ϕ ) (g ) =
n 0 ′ ∈ 1 n0 n0,∞ ∗1 ′
g. Let G be the subgroup generated by g . There is n n such that (ϕ ) is injective on
1 ′ 1 ≥ 0 n1,∞ ∗1
(ϕ ) (G ). Let g = (ϕ ) (g ). Put G = (ϕ ) (G ). Then G Tor(K (C(X ))).
n0,n1 ∗1 1 1 n0,n1 ∗1 ′ 2 n0,n1 ∗1 1 2 ⊂ 1 n1
From what has been proved, G F K (C(X )). It follows from part (c) of Proposition 5.1 of
2 ∈ 3 1 n1
[17] that ϕ (G ) F K (C(X)). It follows that g F K (C(X)).
n1,∞ 2 ⊂ 3 1 ∈ 3 1
Lemma 2.14. Let X be a compact metric space and let G K (C(X)) be a finitely generated
1
⊂
subgroup. Then G = G G F K (C(X)), where G is a finitely generated free group.
1 3 1 1
⊕ ∩
Proof. As in the proof of 2.13 we may assume that X is a finite CW complex. Let Y be
the 2-skeleton of X. Let s : C(X) C(Y) be the surjective map defined by the restriction
→
s(f)(y)= f(y)forallf C(X)andy Y.Then,byTheorem4.1of[17],kers = F K (C(X)).
1 3 1
∈ ∈ ∗
Therefore G/G F K (C(X)) is isomorphic to a subgroup of K (C(Y)). Since dimY = 2,
3 1 1
∩
Tor(K (C(Y))) = 0 . Therefore G/G F K (C(X)) is free. It follows that
1 3 1
{ } ∩
G = G G F K (C(X))
1 3 1
⊕ ∩
for some finitely generated subgroup G .
1
Definition 2.15. Let C be a unital C -algebra and let G K (C) be a finitely generated
∗ 1
⊂
subgroup. Denote by J : K (C) U(M (C))/CU(M (C)) an injective homomorphism
′ 1 → ∪∞n=1 n n
such that Π J = id , where Π is the surjective map from U(M (C))/CU(M (C))
◦ ′ K1(C) ∪∞n=1 n n
onto K (C). There is an integer N = N(G) such that J (G) U(M (C))/CU(M (C)).
1 ′ N N
∈
Let X be a compact metric space and let C = PM (C(X))P, where P M (C(X)) is a
m m
∈
projection such that P(x) = 0 for all x X. By 2.14, one may write G = G G Tor(G),
1 b
6 ∈ ⊕ ⊕
whereG is the freepart of G F K (C). Note, by 2.13, Tor(G) F K (C). Let g Tor(G) be
b 3 1 3 1
∩ ⊂ ∈
a non-zero element and let u = J (g) for some unitary u U(M (C)). Suppose that kg = 0
g ′ g N
∈
for some integer k > 1. Therefore uk CU(M (C)). It follows from 2.13 as well as 2.6, there
g ∈ N
are h ,h ,...,h M (C) such that
1 2 s N+r s.a.
∈
s
u exp(√ 1h ) SU (C),
1 j r+N
− ∈
j=1
Y
where u = 1 u and r 0 is an integer. For each x X, [detu (x)]k = 1. It follows that
1 Mr ⊕ ≥ ∈ 1
s kTr(h )(x)
j=1 j = I(x) Z for all x X.
2π√ 1 ∈ ∈
P
−
6
It follows that I(x) C(X). Therefore
∈
s
( exp(√ 1h ))k SU (C) U (M (C)) CU (C).
j r+N 0 r+N r+N
− ∈ ∩ ⊂
j=1
Y
Consequently,
s
(u exp(√ 1h ))k = 1 .
1 − j Mr+N(C)
j=1
Y
Thus there is an integer R(G) 1 and an injective homomorphism
≥
J :Tor(G) SU (C)/CU(M (C))
c(G) R(G) R(G)
→
such that Π J = id . By choosing a larger R(G), if necessarily, one obtains an injective
c(G) Tor(G)
◦
homomorphism J :G U(M (C))/CU(M (C)) such that
c(G) R(G) R(G)
→
J (G Tor(G)) SU (C)/CU(M (C)) (e2.15)
c(G) b R(G) R(G)
⊕ ⊂
and Π J = id .
c(G) G
◦
It is important to note that, if x SU (C) and [x] G 0 in K (C). Then J ([x]) = x¯.
R(G) 1 c
∈ ∈ \{ }
In fact, since [x] G Tor(G), if J ([x]) = y¯, then y SU (C). It follows that x y
b c R(G) ∗
∈ ⊕ ∈ ∈
SU (C) U (M (C)) CU(C). So y¯ = x¯. This fact will be also used without further
R(G) 0 R(G)
∩ ⊂
notice. Note also that if dimX< , then we can let R(K (C(X))) = dimX.
1
∞
Therefore one obtains the following:
Proposition 2.16. Let X, G, G and Π be as described in 2.15. Then there is an injective
b
homomorphism J : G U (C)/CU(M (C)) for some integer R(G) 1 such that
c(G) R(G) R(G)
→ ≥
Π J = id and J (G Tor(G)) SU (C)/CU(M (C)). In what follows, we may
c(G) G c(G) b R(G) R(G)
◦ ⊕ ⊂
write J instead of J , if G is understood.
c c(G)
Corollary 2.17. Let X, G and G be as in 2.15 and let Y be a compact metric space. Suppose
b
that B = P M (C(Y))P , where P M (C(Y)) is a projection and ϕ : C B is a unital
1 r 1 1 r
∈ →
homomorphism. Suppose also that z G Tor(G) and ϕ (z ) = 0. Then ϕ (J (z )) = ¯1
b b 1 b ‡ c(G) b
∈ ⊕ ∗
in U(M (C))/CU(M (C)) for some integer N 1, when dimY = d< , N can be chosen to
N N
≥ ∞
be max R(G),d .
{ }
Proof. Suppose that u U(M (C)) such that u¯ = J (z ). Without loss of generality, one
b R(G) c(G) b
∈
may assume that ϕ(u ) U (M (B)), since ϕ (z ) = 0. By 2.16, u SU (C). It follows
b 0 R(G) 1 b b N
∈ ∗ ∈
from 2.10 that ϕ(u ) SU (B). Thus, by 2.11,
b R(G)
∈
ϕ(u ) SU (B) U (M (B)) CU(M (B)).
b R(G) 0 R(G) R(G)
∈ ∩ ⊂
It follows that ϕ (J (z )) = ¯1.
‡ c(G) b
Definition 2.18. LetAbeaunitalC -algebra andletu U (A).Denotebycel(u)theinfimum
∗ 0
∈
of the length of the paths of unitaries of U (A) which connects u with 1 .
0 A
Definition 2.19. We say (δ, , ) is a KL-triple, if, for any δ- -multiplicative contractive
G P G
completely positive linear map L : A B (for any unital C -algebra B) [L] is well defined.
∗
→ |P
Moreover, if L and L are two δ- -multiplicative contractive completely positive linear maps
1 2
G
L ,L : A B such that
1 2
→
L (g) L (g) < δ for all g , (e2.16)
1 2
k − k ∈ G
7
[L ] = [L ] .
1 2
|P |P
If K (C) is finitely generated (i = 0,1) and is large enough, then [L] defines an element
i
P |P
in KK(C,A) (see 2.4 of [31]). In such cases, we will write [L] instead of [L] , and (δ, , ) is
|P G P
called a KK-triple and (δ, ) a KK-pair.
G
Now we also assume that A is amenable (or B is amenable). Let u U(B) be such that
∈
[L(g), u] < δ for all g for some finite subset A and for some δ > 0. Then, we may
0 0 0 0
k k ∈ G G ⊂
assume that there exists contractive completely positive linear map Ψ : A C(T) B such
⊗ →
that
L(g) Ψ(g 1) < δ for all g and Ψ(1 z) u < δ
k − ⊗ k ∈ G k ⊗ − k
(see 2.8 of [31]). Thus, we may assume that Bott(L, u) is well defined (see 2.10 in [31]). In
|P
what follows, when we say (δ, , ) is a KL-triple, we further assume that Bott(L,u) is well
G P |P
defined, provided that L is δ- -multiplicative and [L(g), u] < δ for all g . In case that
G k k ∈ G
K (A) is finitely generated (i = 0,1), we may even assume that Bott(L, u) is well defined. We
i
also refer to 2.10 and 2.11 of [31] for bott (L,u) and bott (L,u). If u and v are unitary and
0 1
[u, v] < δ, we use bott (u,v) as in 2.10 and 2.11 of [31]. Let p is a projection and [p, v] <δ,
1
k k k k
we may also write bott (p,v) for the element in K (B) represented by a unitary which is close
0 1
to (1 p)+pvp.
−
Definition 2.20. If u is a unitary, we write L(u) = L(u)(L(u) L(u)) 1/2 when L(u )L(u)
∗ − ∗
h i k −
1 < 1and L(u)L(u ) 1 < 1.Inwhatfollowswewillalways assumethat L(u )L(u) 1 < 1
∗ ∗
k k − k k − k
and L(u)L(u ) 1 < 1, when we write L(u) .
∗
k − k h i
Let B be another unital C -algebra and let ϕ : A B be a unital homomorphism. Then
∗
→
ϕ L(u) = ϕ( L(u) ). Let u CU(A). Then, for any ǫ > 0, if δ is sufficiently small and is
h ◦ i h i ∈ G
sufficiently large (depending on u) and L is δ- -multiplicative, then
G
dist( L(u) ,CU(B)) < ǫ.
h i
Letδ > 0, Abeafinitesubset, U(A)beafinitesubsetandǫ > 0.We say (δ, , ,ǫ) is
G ⊂ W ⊂ G W
a -quadruple, provided the following hold: if for any δ- -multiplicative contractive completely
U G
positive linear map L : A B, L(y) is well defined,
→ h i
L(u) L(u) < ǫ/2 and L(u) L(v) < ǫ/2,
kh i− k kh i−h ik
if u, v and u v < δ. We also require that, if u CU(A) ,
∈U k − k ∈ ∩U
L(u) c < ǫ/2
kh i− k
for some c CU(B). We make one additional requirement. Let G be the subgroup of
∈ U
U(A)/CU(A) generated by u¯ : u . There exists a homomorphism λ : U U(B) such
{ ∈ U} U →
that
dist( L(u) ,λ(u)) < ǫ for all u
h i ∈ U
(see Appendix14.5 for aproofthatsuch λexists). We may denote L fora fixedhomomorphism
‡
λ. Note that, when ǫ < 1, [ L(u) ] = Π (L (u¯)) in K (B), where Π : U(B)/CU(B) K (B)
c ‡ 1 c 1
h i →
is the induced homomorphism.
2.21. Let A and B be two unital C -algebra.. Suppose that A is a separable amenable C -
∗ ∗
algebra. Let K (A) be a finite subset. Then β(Q) K (A C(T)). Let be a finite
0 1
Q ⊂ ⊂ ⊗ W
subset of U(M (A C(T))) such that its image in K (A C(T)) containing β(Q). Denote
n 1
⊗ ⊗
by G( ) the subgroup generated by Q. Fix ǫ > 0. Let (δ, , ,ǫ) be a -quadruple. Let
Q G W U
J : β(G( )) U(M (A C(T)))/CU(M (A C(T))) be defined in 2.15. Let L : A B
′ N N
Q → ⊗ ⊗ →
8
be a δ- -multiplicative contractive completely positive linear map and let u U(B) such that
G ∈
[L(g), u] < δ for all g . With sufficiently small δ and large , let Ψ be given in 2.19, we
k k ∈ G G
may assume that Ψ is defined on J (β(G( ))). We denote this map by
‡ ′
Q
Bu(ϕ,u)(x) = Ψ (J (x))) for all x . (e2.17)
‡ ′
∈ Q
We may assume that [p ],[p ],...,[p ] generates , where p ,p ,...,p are assumed to be pro-
1 2 k 1 2 k
Q
jections in M (A). Let z = (1 p ) + p (1 z)(N) (see 2.3), j = 1,2,...,k. Then z is a
N j j j j
− ⊗
unitary in M (A C(T)). Suppose that A = C(X) for some compact metric space. In the
N
⊗
above, we let J = J = J and N = R(β(G( ))). Note z SU (C(X) C(T)). If
′ c(β(G( )) c j N
Q Q 6∈ ⊗
[p ] [p ] kerρ , then for each x X, there is a unitary w M such that w p (x)w = p .
i j C(X) N ∗ i j
− ∈ ∈ ∈
Then det(z z (x)) = 1. In other words, z z SU (C(X) C(T)). Note, by the end of 2.15,
i j∗ i j∗ ∈ N ⊗
J ([p ]) = z¯ , j = 1,2,...,k. If B has stable rank d, we may assume that, R(β(G( ))) d+1.
c j j
Q ≥
In what follows, when we write Bu(ϕ,u)(x), or Bu(ϕ,u) , we mean that δ is sufficiently small
|Q
and is sufficiently large so that L is well defined on J (β(x), or on J (β( )). Moreover, we
‡ c c
G Q
note that [L] = Π L . Furthermore, by choosing even smaller δ, we may also assume
′ ‡ β( )
|Q ◦ | Q
that when
[ϕ(g), u] < δ and [ϕ(g), v] < δ for all g
k k k k ∈ G
dist(Bu(ϕ,uv)(x),Bu(ϕ, u)(g)+Bu(ϕ, v)(x)) < ǫ for all x .
∈ Q
Lemma2.22. LetX beaconnectedfiniteCWcomplexofdimensiond> 0.ThenK (C(X(d 1))) =
−
∗
G K (C(X))/F K (C(X)), where G is a finitely generated free group. Consequently,
0, d 0,
∗⊕ ∗ ∗ ∗
K (C(X(1))) = S K (C(X))/F K (C(X)),
1 1 3 1
⊕
where S is a finitely generated free group.
Proof. Let I(d) = f C(X) :f = 0 . Then I(d) = C (Y), where Y = Sd Sd Sd.
{ ∈ |X(d−1) } 0 ∨ ∨···∨
In particular, K (I(d)) is free. Therefore, by applying 4.1 of [17],
∗
Ki(C(X(d−1))) ∼= G0,i Ki(C(X))/FdKi(C(X)),
⊕
where G is isomorphic to a subgroup of K (I(d)), i= 0,1.
0,i i 1
−
For the last part of the lemma, we use the induction on d. It obvious holds for d = 1.
Assume the last part holds for dimX = d 1. Suppose that dimX = d+1. Let s : C(X)
m
≥ →
C(X(m)),s : C(X) C(X(1)), s : C(X(m)) C(X(1)) be defined by the restrictions
1 m,1
→ →
(0 < m d). We have that s = s s . Note that if Y is a finite CW complex with
1 m,1 m
≤ ◦
dimension 2, then K (kersY) = 0 , where sY : C(Y) C(Y(1)) is the surjective map induced
1 1 { } 1 →
by the restriction, where Y(1) is the 1-skeleton of Y. It follows that the induced map (sY) from
1 1
K (C(Y)) into K (Y(1)) is injective. This fact will be used in the following computation∗.
1 1
We have
K (C(X(d))) = S K (C(X))/F K (C(X)) = S (s ) (K (C(X))) (e2.18)
1 0 1 d 1 0 d 1 1
⊕ ⊕ ∗
and, by 4.1 of [17],
K (C(X(1)) = S K (C(X(d))/F K (C(X(d))) (e2.19)
1 1 1 3 1
⊕
= S (s ) (K (C(X(d))) (e2.20)
1 d,1 1 1
⊕ ∗
= S (s ) (S (s ) (K (C(X)))) (e2.21)
1 d,1 1 0 d 1 1
⊕ ∗ ⊕ ∗
= S (s ) (S ) (s ) ((s ) (K (C(X)))) (e2.22)
1 d,1 1 0 d,1 1 d 1 1
⊕ ∗ ⊕ ∗ ∗
= S (s ) (S ) (s ) (K (C(X))) (e2.23)
1 d,1 1 0 1 1 1
⊕ ∗ ⊕ ∗
= S (s ) (S ) (s ) (s ) (K (C(X))) (e2.24)
1 d,1 1 0 2,1 1 2 1 1
⊕ ∗ ⊕ ∗ ◦ ∗
= S (s ) (S ) (s ) (K (C(X))/F K (C(X))) (e2.25)
1 d,1 1 0 2 1 1 3 1
⊕ ∗ ⊕ ∗
= S (s ) (S ) K (C(X))/F K (C(X))). (e2.26)
1 d,1 1 0 1 3 1
⊕ ∗ ⊕
9
PutS = S (s ) (S ).NoteK (C(X(1))isfree. SoS mustbeafinitely generated freegroup.
1 d,1 1 0 1
⊕ ∗
This ends the induction.
3 Definition of
1
C
Definition 3.1. Let A be a C -algebra and let a,b A . Recall that we write a . b if there
∗ +
∈
exists a sequence x A such that
n +
∈
lim x bx a = 0.
n k ∗n n− k
→∞
If a = p is a projection and a . b, there is a projection q Her(b) and a partial isometry v A
∈ ∈
such that vv = p and v v = q.
∗ ∗
Definition 3.2. Let 0 < d < 1. Define f C ((0, ]) by f (t) = 0 if t [0,d/2], f (t) = 1 if
d 0 d d
∈ ∞ ∈
t [d, ), and f is linear in (d/2,d).
d
∈ ∞
Definition 3.3. Denote by (0) the class of finite dimensional C -algebras and denote by (1,0)
∗
I I
theclass of C -algebras with theform C([0,1]) F, whereF (0). For an integer k 1, denote
∗
⊗ ∈ I ≥
by (k) the class of C -subalgebra with the form PM (C(X))P, where r 1 is an integer, X is
∗ r
I ≥
a finite CW complex of covering dimension at most k and P M (C(X)) is a projection.
r
∈
Definition 3.4. Denote by the class of those C -algebras which are quotients of C -algebras
k ∗ ∗
I
in (k). Let C . Then C = PM (C(X))P, where X is a compact subset of a finite CW
k r
I ∈ I
complex, r 1 and P M (C(X)) is a projection. Furthermore, there exists a finite CW
r
≥ ∈
complex Y of dimension k such that X is a compact subset of Y and there is a projection
Q M (C(Y)) such that π(Q) = P, where π : M (C(Y)) M (C(X)) is the quotient map
r r r
∈ →
defined by π(f) = f .
X
|
Definition 3.5. Let A be a unital C -algebra. We say that A is a locally AH-algebra, if for
∗
any finite subset A and any ǫ > 0, there exists a C -subalgebra C (for some k 0)
∗ k
F ⊂ ∈ I ≥
such that
dist(a,C) < ǫ for all a .
∈ F
A is said to belocally AH-algebra with nodimension growth, if there exists an integer d 0,
≥
for any finite subset A, any ǫ > 0 and any η > 0, there exists a C -subalgebra C A with
∗
F ⊂ ⊂
the form C = PM (C(X))P such that
r d
∈ I
dist(a,C) < ǫ for all a . (e3.27)
∈ F
Definition 3.6. Let g : N N be a nondecreasing map. Let A be a unital simple C -algebra.
∗
→
We say that A is in if the following holds: For any finite subset A, any ǫ > 0 and any
g
C F ⊂
a A 0 , there is a projection p A and a C -subalgebra C = PM (C(X))P with
+ ∗ r d
∈ \{ } ∈ ∈ I
1 = p such that
C
pa ap < ǫ for all a , (e3.28)
k − k ∈ F
dist(pap,C) < ǫ for all a , (e3.29)
∈ F
d+1 η
< for all x X and (e3.30)
rank(P(x)) g(d)+1 ∈
1 p . a. (e3.31)
−
If g(d) = d for all d N, we say A .
1
∈ ∈ C
10