The Range of Approximate Unitary Equivalence Classes of Homomorphisms from AH-algebras

0.22 MB·English

by  Huaxin Lin

The Range of Approximate Unitary Equivalence Classes of
Homomorphisms from AH-algebras
Huaxin Lin
8
0
0 Abstract
2
∗
Let C be a unital AH-algebra and A be a unital simple C -algebra with tracial rank

n
zero. It has been shown that two unital monomorphisms φ,ψ : C A are approximately
a →
unitarily equivalent if and only if
J

5
[φ]=[ψ] in KL(C,A) and τ φ=τ ψ for all τ T(A),
2 ◦ ◦ ∈

] where T(A) is the tracial state space of A. In this paper we prove the following: Given
A κ KL(C,A)withκ(K0(C)+ 0 ) K0(A)+ 0 andwithκ([1C])=[1A]andacontinuous
∈ \{ } ⊂ \{ }
O affine map λ : T(A) Tf(C) which is compatible with κ, where Tf(C) is the convex set of
→
. all faithful tracial states, there exists a unital monomorphism φ:C A such that
h →
t
a [φ]=κ and τ φ(c)=λ(τ)(c)
m ◦
e
[ for all c Cs.a. and τ T(A). Denote by Monau(C,A) the set of approximate unitary
∈ ∈
equivalence classes of unital monomorphisms. We provide a bijective map

1
v Λ:Mone (C,A) KLT(C,A)++,
au
8 →
5 whereKLT(C,A)++isthesetofcompatiblepairsofelementsinKL(C,A)++andcontinuous
8
3 affine maps from T(A) to Tf(C).
. Moreover,werealizedthattherearecompactmetricspacesX,unitalsimpleAF-algebras
1
0 A and κ KL(C(X),A) with κ(K0(C(X))+ 0 ) K0(A)+ 0 for which there is no
∈ \{ } ⊂ \{ }
8 homomorphism h:C(X) A so that [h]=κ.
→
0
:
v 1 Introduction
i
X
r Recall that an AH-algebra is a C∗-algebra which is an inductive limit of C∗-algebras C , where
a n
C = P M (C(X ))P for some finite CW complex X and projections P M (C(X )).
n n r(n) n n n n r(n) n
∈
Note that every unital separable commutative C∗-algebra is an AH-algebra and every AF-
algebra is an AH-algebra. It was shown in [11] (see also Theorem 3.6 of [13]) that two unital
monomorphisms φ,ψ : C A, where A is a unital simple C∗-algebra with tracial rank zero,
→
are approximately unitarily equivalent if and only if
[φ] = [ψ] and τ φ(c) = τ ψ(c)
◦ ◦
for all c C and τ T(A). This result plays a role in the study of classification of amenable
s.a.
∈ ∈
C∗-algebras, or otherwise known as the Elliott program. It also has applications in the study of
dynamical systems both classical and non-commutative ones (see [11]). It is desirable to know
the range of the approximately unitary equivalence classes of monomorphisms from a unital
AH-algebra C into a unital simple C∗-algebra with tracial rank zero. For example, one may
ask if given any κ KL(C,A) and any continuous affine map λ : T(A) T(C) there exists a
∈ →
monomorphism φ such that [φ] = κ and τ h(c) = λ(τ)(c) for all c C and τ T(A).
s.a.
◦ ∈ ∈
1
When C is a finite CW complex, it was shown (see also a previous result of L. Li [6]) in [10]
that, for any κ KK(C,A) with κ(K (C) 0 ) K (A) 0 and with κ([1 ]) = [1 ],
0 + 0 + C A
∈ \{ } ⊂ \{ }
there exists a unital monomorphism φ: C A such that [φ] = κ. It should be noted that both
→
conditions that κ([1 ]) = [1 ] and κ(K (C) 0 ) K (A) 0 are necessary for the existence
C A 0 0
\{ } ⊂ \{ }
of such φ. One of the earliest such results (concerning monomorphisms from C(T2) into a unital
simpleAF-algebra) ofthiskindappearedinapaperofElliottandLoring([3]seealso[2]). Itwas
shown in [10] that the same result holds for the case that C is a unital simple AH-algebra which
has real rank zero, stable rank one and weakly unperforated K (C). Therefore, it is natural to
0
expect that it holds for general unital AH-algebras.
Let C be the unitization of , the algebra of compact operators on l2. Then it does not have
K
a faithful tracial state. Consequently, it can not be embedded into any unital UHF-algebra, or
anyunitalsimpleC∗-algebra whichhasatleastonetracialstate(Ithasbeenshownthataunital
AH-algebraC canbeembeddedintoaunitalsimpleAF-algebra ifandonlyifC admitsafaithful
tracial state –see [13]) . This example at least suggests that for general unital AH-algebras, the
problem is slightly more complicated than the first thought. Moreover, we note that to provide
the range of approximately unitary equivalence classes of unital monomorphisms from C, we
also need to consider the map λ : T(A) T(C). Let X be a compact metric space and let
→
C = C(X). Suppose that h : C A is a unital monomorphism and suppose that τ T(A).
→ ∈
Then τ h induces a Borel probability measure on X. Suppose that κ KL(C,A) is given. It
◦ ∈
is clear that not every measure µ can be induced by those h for which [h] = κ. Thus, we should
consider a compatible pair (κ,γ) which gives a more complete information on K-theory than
either κ or γ alone.
ThemainresultofthispaperistoshowthatifC isaunitalAH-algebra,Aisanyunitalsimple
C∗-algebra with tracial rank zero, κ KL(C,A)++ (see 2.3 below) with κ([1 ]) = [1 ] and
C A
∈
λ : T(A) Tf(C), where Tf(C) is the convex set of faithful tracial states, which is a continuous
→
affine map and is compatible with κ, there is indeed a unital monomorphism φ : C A such
→
that
[φ] =κ in KL(C,A) and φ = λ.
T
We also show that the existence of λ is essential to provide homomorphisms φ. In fact,
we find out that there are compact metric spaces X, unital simple AF-algebras A and κ
∈
KL(C(X),A)++ with κ([1C]) = [1A]) for which there are no λ : T(A) Tf(C(X)) which is
→
compatible with κ. Moreover, we discovered that there are no homomorphism h : C A (not
→
just monomorphisms) such that [h] = κ. This further demonstrates that tracial information is
an integral part of K-theoretical information.
2 Notation
2.1. Let A be a unital C∗-algebra. Denote by T(A) the tracial state space of A. Denote by
Aff(T(A)) the space of all real affine continuous functions on T(A). If τ T(A), we will also
∈
use τ for the tracial state τ Tr on M (A) for all integer k 1, where Tr is the standard trace
k
⊗ ≥
on M . If a A , denote by aˇ a real affine function in Aff(T(A)) defined by aˇ(τ) = τ(a) for
k s.a.
∈
all τ T(A).
∈
Let C be another unital C∗-algebra. Suppose that γ : Aff(T(C)) Aff(T(A)) is a positive
→
linear map. We say it is unital if γ(1 )(τ) = 1. We say it is strictly positive, if a Aff(T(A))
C +
∈ \
0 , then γ(a)(τ) > 0 for all τ T(A).
{ } ∈
Supposethat φ :C A is a unital homomorphism. Denote by h : T(A) T(C) the affine
T
→ →
continuous map induced by h, i.e.,
h (τ)(c) = τ h(c) for all c C.
T
◦ ∈
2
It also induces a positive linear map h :Aff(T(C)) Aff(T(A)) defined by
♯
→
h (aˇ)(τ) = τ h(a) for all a C and τ T(A),
♯ s.a
◦ ∈ ∈
where aˇ(τ) = τ(a) for τ T(A).
∈
If λ : T(A) T(C) is an affine continuous map, then it gives a unital positive linear map
→
λ :Aff(T(C)) Aff(T(A)) by
♯
→
λ (f)(τ) = f(λ(τ)) for all f Aff(T(C)) and for all τ T(A).
♯
∈ ∈
Conversely, a unital positive linear map γ : Aff(T(C)) Aff(T(A)) gives an affine continuous
→
map γ : T(A) T(C) by
T
→
f(γ (τ)) = γ(f)(τ) for all f Aff(T(C)) and τ T(C).
T
∈ ∈
Suppose that A is a unital simple C∗-algebra. Then γ is strictly positive if and only if γ
T
maps T(A) into Tf(C).
Denote by ρ : K (A) Aff(T(A)) the positive homomorphism induced by ρ ([p])(τ) =
A 0 A
→
τ(p) for all projections p M (A) and τ T(A).
∞
∈ ∈
Let A and C be two unital C∗-algebras and let κ : K (C) K (A) be a unital positive
0 0 0
→
homomorphism (κ ([1 ]) = [1 ]). Suppose that λ : T(A) T(C) is a continuous affine map.
0 C A
→
We say that λ is compatible with κ , if τ(κ([p])) = λ(τ)(p) for all projections p in M (A).
0 ∞
Similarly, a unital positive linear map γ :Aff(T(C)) Aff(T(A)) is said to be compatible with
→
κ , if γ(pˇ)(τ) = τ(κ([p]) for all projections p in M (C). γ is compatible with κ if and only if
0 ∞ 0
γ is so.
T
Two projections in A are equivalent if there exists a partial isometry w A such that
∈
w∗w = p and ww∗ = q.
2.2. Let A be a unital C∗-algebra and let C be a separable C∗-algebra which satisfies the
universal coefficient theorem. By a result of Dadarlat and Loring ([1]),
KL(C,A) = Hom (K(C),K(A)), (e2.1)
Λ
where, for any C∗-algebra B,
∞
K(B) = i=0,1Ki(B)M i=0,1Ki(B,Z/nZ).
⊕ ⊕
n=2
We will identify two objects in (e2.1). Denote by
KF,k(C) = M Ki(B)M⊕i=0,1Ki(B,Z/nZ).
i=0,1 n|k
If K (C) is finitely generated (i = 0,1), then there is k 1 such that
i 0
≥
Hom (K(C),K(A)) = Hom (F K(C),F K(A))
Λ ∼ Λ k0 k0
(see [1]).
Definition 2.3. Denote by KL(C,A)++ the set of those κ Hom (K(C),K(A)) such that
Λ
∈
κ(K (C) 0 ) K (A) 0 .
0 + 0
\{ } ⊂ \{ }
Denote by KL (C,A)++ the set of those κ KL(C,A)++ such that κ([1 ]) = [1 ].
e C A
∈
3
Definition 2.4. Letκ KL (C,A)++ andletλ : T(A) T(C)beacontinuousaffinemap. We
e
∈ →
say that λ is compatible with κ if λ is compatible with κ . Let γ : Aff(T(C)) Aff(T(A))
|K0(C) →
be a positive linear map. We say γ is compatible with κ if γ is compatible with κ , i.e.,
|K0(C)
τ κ([p]) = γ(pˇ)(τ) for all projections p M (C).
∞
◦ ∈
2.5. Let C = C(X) for some compact metric space X. One has the following short exact
sequence:
0 kerρ K (C) C(X,Z) 0.
C 0
→ → → →
It is then easy to see that, for every projection p M (C), there is a projection q C and an
∞
∈ ∈
integer n such that ρ ([p]) = nρ ([q]). It follows that if C is a unital AH-algebra, then for every
A A
projection p M (C), there is a projection q C and an integer n 1 such that
∞
∈ ∈ ≥
ρ ([p]) = nρ ([q]).
A A
Note also that in this case Aff(T(C)) = C . Therefore, in this note, instead of considering
s.a
a unital positive linear maps γ : Aff(T(C)) Aff(T(A)), we may consider unital positive
→
linear maps γ : C Aff(T(A)). Moreover, γ is compatible with some κ KL(C,A)++, if
s.a
→ ∈
γ(p)(τ) = τ(κ([p])) for all projections p C and τ T(A).
∈ ∈
2.6. Let φ,ψ : C A be two maps between C∗-algebras. Let ǫ > 0 and C be a subset.
→ F ⊂
We write
φ ψ on ,
ǫ
≈ F
if
φ(c) ψ(c) < ǫ for all c .
k − k ∈ F
2.7. Let L : C A be a linear map. Let δ > 0 and C be a (finite) subset. We say L is
→ G ⊂
δ- -multiplicative if
G
L(ab) L(a)L(b) < δ for all a,b .
k − k ∈ G
Definition 2.8. Let A be a unital C∗-algebra. Denote by U(A) the unitary group of A. Let
B A be another C∗-algebra and φ :B A be a map. We write φ = adu for some u U(A)
⊂ → ∈
if φ(b) = u∗bu for all b B.
∈
Let φ,ψ : C A be two maps. We say that φ and ψ are approximately unitarily equivalent
→
if there exists a sequence of unitaries u A such that
n
{ } ⊂
lim adu φ(c) = ψ(c) for all c C.
n
n→∞ ◦ ∈
3 Approximate unitary equivalence
We begin with the following theorem
Theorem 3.1. (Theorem 3.6 of [13] and see also Theorem 3.4 of [11]) Let C be a unital AH-
algebra and let A be a unital simple C∗-algebra with tracial rank zero. Suppose that φ,ψ :C A
→
are two unital monomorphisms. Then there exists a sequence of unitaries u A such that
n
{ } ⊂
lim adu ψ(c) = φ(c) for all c C,
n
n→∞ ◦ ∈
if and only if
[φ] = [ψ] in KL(C,A) and τ φ = τ ψ for all τ T(A).
◦ ◦ ∈
We need the following variation of results in [11].
4
Theorem 3.2. Let C be a unital AH-algebra, let A be a unital simple C∗-algebra with TR(A) =
0 and let γ :C Aff(T(A)) be a unital strictly positive linear map.
s.a.
→
For any ǫ > 0 and any finite subset C, there exist η > 0, δ > 0, a finite subset C,
F ⊂ G ⊂
a finite subset C and a finite subset K(C) satisfying the following:
s.a.
H ⊂ P ⊂
Suppose that L ,L : C A are two unital completely positive linear maps which are δ- -
1 2
→ G
multiplicative such that
[L ] = [L ] , (e3.2)
1 P 2 P
| |
τ L (g) γ(g)(τ) < η for all g , i= 1,2. (e3.3)
i
| ◦ − | ∈ H
Then there is a unitary u A such that
∈
adu L L on . (e3.4)
2 ǫ 2
◦ ≈ F
Proof. Write C = ∞ C , where C = P M (C(X ))P , where X is a compact subset of a
∪n=1 n n n r(n) n n
finite CW complex and where P M (C(X )) is a projection. Let ǫ > 0 and a finite subset
n r(n) n
∈
C be fixed. Without loss of generality, we may assume that C . Let η > 0 such that
1 0
F ⊂ F ⊂
f(x) f(x′) < ǫ/8 for all f ,
| − | ∈ F
if dist(x,x′) < η . Let x ,x ,...,x X be η /2-dense in X. Suppose that O O = if
0 1 2 m 0 i j
{ } ⊂ ∩ ∅
i = j, where
6
O = x X :dist(x,x ) < η /2s , j = 1,2,...,m
j j 0
{ ∈ }
for some integer s 1.
≥
Choose non-zero element g (C ) such that 0 g 1 whose support lies in O ,
j 1 s.a j j
∈ ≤ ≤
j = 1,2,...,m. Note such g exists (by taking those in the center for example). Choose
j
σ = min inf γ(g )(τ) :τ T(A) :1 j m .
0 j
{ { ∈ } ≤ ≤ }
Since γ is strictly positive, σ > 0. Set σ = min σ /2,1/2s . Then, by Corollary 4.8 of [11], such
0 0
{ }
δ > 0, η > 0, , and exists.
G H P
Lemma 3.3. Let X be a compact metric space, let A be a unital simple C∗-algebra with
TR(A) = 0 and let γ : C(X) Aff(T(A)) be a unital strictly positive linear map.
s.a.
→
Then, for any ǫ > 0 and any C(X), there exists δ > 0, a finite subset C(X) , a
s.a.
F ⊂ G ⊂
set S ,S ,...,S of mutually disjoint clopen subsets with n S = X, satisfying the following:
1 2 n ∪i=1 i
For any two unital homomorphisms φ ,φ : C(X) pAp with finite dimensional range for
1 2
→
some projection p A with τ(1 p)< δ such that
∈ −
[φ (χ )] = [φ (χ )] in K (A), i = 1,2,...,n, (e3.5)
1 Si 2 Si 0
τ φ (g) γ(g)(τ) < δ and (e3.6)
1
| ◦ − |
τ φ (g) γ(g)(τ) < δ (e3.7)
2
| ◦ − |
for all g and for all τ T(A), there exist a unitary u U(pAp) such that
∈ G ∈ ∈
adu φ φ on . (e3.8)
1 ǫ 2
◦ ≈ F
Proof. This follows from 3.2 immediately. There is a sequence of finite CW complex X such
n
that C(X) = lim (C(X ),h ), where each h is a unital homomorphism. Fix ǫ > 0 and a
n→∞ n n n
finite subset C(X). Without loss of generality, we may assume that h ( ) for some
K K
F ⊂ F ⊂ F
integer K 1 and a finite subset .
K
≥ F
5
Given any finitesubset K(C(X)), one obtains a finitesubset K(C(X )) such that
k k
P ⊂ Q ⊂
[h ]( )= for somek 1. Letp ,p ,...,p bemutually orthogonal projections corresponding
k k 1 2 n
Q P ≥
to the connected components of X . To simplify notation, without loss of generality, we may
k
assume that k = K.
TherearemutuallydisjointclopensetsS ,S ,...,S ofX with n S = X suchthath (p ) =
1 2 n ∪i=1 i k i
χ , i= 1,2,...,n. Since φ and ψ are homomorphisms with finite dimensional range, if
Si
[φ(χ )] =[ψ(χ )] in K (A),
Si Si 0
then
[φ h ] = [ψ h ] in KL(C(X ),A).
k k k
◦ ◦
This, in particular, implies that
[φ] = [ψ] .
P P
| |
This above argument shows that the lemma follows from 3.2.
Definition 3.4. Let X bea compact metric space which is a compact subsetof some finite CW
complex Y. Then there exists a decreasing sequence of finite CW complexes X Y such that
n
⊂
X X and lim dist(X ,X) = 0.
n n
⊂ n→∞
Denote by s : C(X ) C(X ) (for n > m) and s : C(X ) C(X) be the surjective
m,n m n n n
→ →
homomorphisms induced by the inclusion X X and X X , respectively.
n+1 n n
⊂ ⊂
Lemma 3.5. Let Y be a finite CW complex and X Y be a compact subset. For any ǫ > 0,
⊂
any finite subset C(X), there exists a finite subset K(C(X)), an integer k 1 and an
F ⊂ P ⊂ ≥
integer N 1 satisfying the following:
≥
For any unital homomorphisms φ,ψ :C(X ) A (m k) for any unital simple C∗-algebra
m
→ ≥
with TR(A) = 0 for which
[φ] = [ψ] in KL(C(X ),A),
Q Q m
| |
where Q K(C(X )) is a finite subset such that [s ]( ) = , then there exists a unitary
m m
⊂ Q P
U M (A) such that
N+1
∈
adU (φ Φ s ) (ψ Φ s ) on s−1( ),
◦ ⊕ ◦ m ≈ǫ ⊕ ◦ m m F
where Φ :C(X) M (A) is defined by
N
→
Φ(f)= diag(f(x ),f(x ),...,f(x )) for all f C(X ), (e3.9)
1 2 N 1
∈
where x ,x ,...,x is a finite subset of X.
1 2 N
{ }
Proof. Assume that the lemma were false. Then there would be a positive number ǫ > 0,
0
a finite subset C(X), an increasing sequence of finite subsets K(C(X)) with
0 n
F ⊂ {P } ⊂
= K(C(X)), a sequence of unital C∗-algebras, two subsequences R(n) , k(n) of N and
n n
∪ P { } { }
two sequences monomorphisms φ ,ψ : C(X ) A such that
n n k(n) n
→
[φ ] = [ψ ] in KK(C(X ),A ) and (e3.10)
n |Qn n |Qn k(n) n
limsup inf max u∗(φ Φ s )(f))u (φ Φ s )(f) : f s−1( ) ǫ , (e3.11)
{ { {k n n ⊕ n◦ n n − ⊕ n◦ n k ∈ m F }}} ≥ 0
n
where infimum is taken among all possible Φ : C(X) M (A ) with the form described
n R(n) n
→
above and among all possible unitaries u U(M (A)), and where K(C(X ))
n R(n)+1 n k(n)
{ } ⊂ Q ⊂
6
is a finite subset such that [s ]( ) = . Since K (C(X ) is finitely generated, by passing to
k(n) n n i n
Q P
a subsequence, if necessary, without loss of generality, we may assume (see also the end of 2.2)
that
[φ s ] = [ψ s ] in KL(C(X ),A), n = 1,2,.... (e3.12)
n+1 k(n),k(n+1) n+1 k(n),k(n+1) k(n)
◦ ◦
(m) (m) (m) (m)
Letφ = φ ,ifn m,φ = φ s ,ψ = ψ ,ifn mandψ = ψ s ,n = 1,2,....
n m n n m,n n m n n m,n
≤ ◦ ≤ ◦
(m) (m) (m) (m) (m) (m)
Denote by H ,H : C(X ) A by H (f) = φ and H (f) = ψ . Let
1 2 k(m) → Qn n 1 { n } 2 { n }
(m) (m)
π : A A / A be the quotient map. Then π H and π H both have
Qn n → Qn n Ln n ◦ 1 ◦ 2
(m)
spectrum X. Moreover, for each i, all π H gives the same homomorphism F : C(X)
◦ i i →
A / A , i= 1,2.
Qn n Ln n
Since TR(A ) = 0, A has real rank zero, stable rank one, weakly unperforated K (A ), by
n n 0 n
Corollary 2.1 of [5] and (e3.12)
(m+1) (m+1)
[H1 ◦sk(m),k(m+1)] = [H2 ◦sk(m),k(m+1)] in KL(C(Xk(m)),YAn)
n
It follows from Corollary 2.1 of [5] again that
[F1] =[F2] in KL(C,YAn/MAn).
n n
It then follows from Theorem 1.1 and the Remark 1.1 of [5] that there is an integer N 1 and
≥
a unitary W U(M ( A / A )) such that
∈ N+1 Qn n Ln n
adW (F H ) (F H ) on , (e3.13)
◦ 2⊕ 0 ≈ǫ0/2 1⊕ 0 F0
where H : C(X) M ( A / A ) is defined by H (f) = N f(x )E for all f C(X),
0 → N Qn n Ln n 0 Pi=1 i i ∈
i−1
xi X and Ei = diag(z0,}..|.,0{,1,0,...,0), i= 1,2,...,N.
∈
There is a unitary W U( A ) such that π( W ) = W. Then, for some sufficiently
{ n} ∈ Qn n { n}
large n,
W∗diag(φ (f),f(x ),f(x ),...,f(x ))W (ψ (f),f(x ),f(x ),...,f(x )) (e3.14)
n n 1 2 N n ≈ǫ0 n 1 2 N
on . This contradicts (e3.11).
0
F
Remark 3.6. There exists a positive number η > 0 and integer N > 0 which depend only on ǫ
1
and such that x ,x ,...,x and an integer N can be replaced by any η-dense finite subset
1 2 N
F { }
ξ ,ξ ,...,ξ and integer N .
{ 1 2 N1} 1
From the proof, we also know that the assumption that A has tracial rank zero can be
replaced by much weaker conditions (see Corollary 2.1 of [5]). The main difference of 3.5 and
results in [5] is that homomorphisms φ and ψ are not assumed to be from C(X) to A.
4 Monomorphisms from C(X)
Lemma 4.1. Let X be a finite CW complex and let A be a unital simple C∗-algebra with real
rank zero, stable rank one and weakly unperforated K (A). Let e ,e ,...,e C(X) be mutually
0 1 2 m
∈
orthogonal projections corresponding to connected components of X.
Suppose that κ KK(C(X),A)++ with κ([1 ]) = [1 ]. Then, for any projection p A
C(X) A
∈ ∈
and any unital homomorphism φ :C(X) (1 p)A(1 p) with finite dimensional range such
0
→ − −
7
thatφ ([e ]) < κ([e ]),i = 1,2,...,m. Thenthereexistsaunitalmonomorphism φ : C(X) pAp
0 i i 1
→
such that
[φ +φ ]= κ in KK(C(X),A). (e4.15)
1 0
Proof. Since m κ([e ]) = [1 ] and A has stable rank one, there are mutually orthogonal
Pi=1 i A
projections p ,p ,...,p A such that
1 2 m
∈
m
Xpi = 1A and [pi] = κ([ei]), i= 1,2,...,m (e4.16)
i=1
From this it is clear that we may reduce the general case to the case that X is connected.
So now we assume that X is connected. Then it is easy to see that
κ [φ ] KK(C(X),A)++
0
− ∈
and (κ [φ ])([1 ]) = p. It follows from Theorem 4.7 of [10] that there is a monomorphism
0 C(X)
−
φ :C(X) pAp such that
1
→
[φ] = κ [φ ].
0
−
Lemma 4.2. Let X a compact metric space and let A be a unital simple C∗-algebra with tracial
rank zero. Suppose that γ : C(X) Aff(T(A)) is a unital strictly positive linear map. Let
s,a
→
S ,S ,...,S be a set of mutually disjoint clopen subsets of X with n S = X. Then for any
1 2 n ∪i=1 i
δ > 0 and any finite subset C(X) , there exists a projection p A with p = 1 and a
s.a A
G ⊂ ∈ 6
unital homomorphism h : C(X) pAp with finite dimensional range such that
→
τ h(g) γ(g)(τ) < δ for all g and τ T(A), and (e4.17)
| ◦ − | ∈ G ∈
τ h(χ ) < γ(ξ )(τ) for all τ T(A), (e4.18)
◦ Si Si ∈
i = 1,2,...,n.
Proof. Put
d= min δ,min inf γ(χ )(τ) : τ T(A) : 1 i n .
{ { { Si ∈ } ≤ ≤ }}
Since γ is strictly positive, d > 0.
Let = χ ,χ ,...,χ .Itfollowsfrom4.3of[12]thatthereisaunitalhomomorphism
G0 G∪{ S1 S2 Sn}
h : C(X) A with finite dimensional range such that
0
→
τ h(g) γ(g)(τ) < d/8n for all g (e4.19)
0
| ◦ − | ∈ G
and for all τ T(A). In particular,
∈
τ h(χ ) γ(χ )(τ) < d/8n for all τ T(A) (e4.20)
| ◦ Si − Si | ∈
i = 1,2,...,n.
Since ρ (K (A)) is dense in Aff(T(A)), there exists a projection p A such that
A 0 0
∈
d/2n < τ(p ) < d/n for all τ T(A). (e4.21)
0
∈
Note that τ(p ) < γ(χ )(τ) for all τ T(A), i= 1,2,...,n. Moreover, by (e4.20),
0 Si ∈
τ h(χ )> γ(ξ )(τ) d/8n d d/8n > τ(p ). (e4.22)
◦ Si Si − ≥ − 0
8
for all τ T(A).
Write∈h (f) = m f(x )e for all f C(X), where x X and e ,e ,...,e is a set of
mutually or0thogonaPl kp=ro1jectikonskwith m∈e = 1 . k ∈ { 1 2 k}
Pk=1 k A
Note that
h0(ξSj)= X ek.
xk∈Sj
Therefore (by (e4.22))
[p0] [ X ek]. (e4.23)
≤
xk∈Sj
By Zhang’s Riesz interpolation property (see [14] ), there are projections e′ e such that
k ≤ k
[p0] = [X e′k].
k∈Sj
By Zhang’s half projection theorem (see Theorem 1.1 of [15]), for each k, there is a projection
e′′ e′ such that
k ≤ k
[e′′]+[e′′] [e′]. (e4.24)
k k ≥ k
Thus
2[ X e′k′] ≥ [p0], i = 1,2,...,n. (e4.25)
χk∈Si
Therefore (by (e4.21) and (e4.20))
τ( X (ek −e′k′)) < τ ◦h0(χSi)−(1/2)τ(p0) (e4.26)
xk∈Si
< τ h(χ ) d/4n (e4.27)
◦ Si −
< γ(χ )(τ) d/8n for all τ T(A). (e4.28)
Si − ∈
Let p = m (e e′′). Then clearly that p = 1. Moreover,
Pk=1 k − k 6
τ(1 p) < d/4 for all τ T(A).
− ∈
Define h(f)= m f(x )(e e′′) for all f C(X). Then
Pk=1 k k − k ∈
m
|τ ◦h(f)−τ ◦h0(f)| < τ(Xe′k′)= τ(1−p)< d/4 < δ (e4.29)
k=1
for all τ T(A).
∈
Then, by (e4.28),
τ h(χ ) < γ(χ )(τ) for all τ T(A). (e4.30)
◦ Si Si ∈
Lemma 4.3. Let X a compact metric space and let A be a unital simple C∗-algebra with tracial
rank zero. Suppose that γ : C(X) Aff(T(A)) is a unital strictly positive linear map. Let
s,a
→
S ,S ,...,S be a set of mutually disjoint clopen subsets of X with n S = X. Then for any
1 2 n ∪i=1 i
δ > 0, η > 0, for any integer N and any η-dense subset x ,x ,...,x of X and any finite
1 2 N
{ }
9
subset C(X) , there exists a projection p A with p = 1 and a unital homomorphism
s.a A
G ⊂ ∈ 6
h :C(X) pAp with finite dimensional range such that
→
τ h(g) γ(g)(τ) < δ for all g and τ T(A), and (e4.31)
| ◦ − | ∈ G ∈
τ h(χ ) < γ(χ )(τ) for all τ T(A), (e4.32)
◦ Si Si ∈
i = 1,2,...,n,
N
h(f) = Xf(xi)ei h1(f) for all f C(X), (e4.33)
⊕ ∈
i=1
where h : C(X) (1 N e )A(1 N e ) is a unital homomorphism with finite dimen-
1 → −Pi=1 i −Pi=1 i
sional range and e ,e ,...,e is a set of mutually orthogonal projections such that [e ]= [e ]
1 2 N i 1
{ } ≥
[1 p], i= 1,2,...,N.
−
Proof. Let N 1 and let η-dense subset x ,x ,...,x of X be given. Let η > 0 such that
1 2 N 0
≥ { }
f(x) f(x′) < δ/4 for all f , (e4.34)
| − | ∈ G
provided that dist(x,x′)< η .
0
Choose η > η > 0 such that B(x ,η ) intersects with one and only one S among
0 1 i 1 i
S ,S ,...,S .
1 2 n
{ }
Choose, for each i, a non-zero function f C(X) with 0 f 1 whose support is in
i
∈ ≤ ≤
B(x ,η /2). Put
i 1
d = min inf γ(f )(τ) :τ T(A) :1 i N .
0 i
{ { ∈ } ≤ ≤ }
So d > 0. Put δ = min δ/8,δ /4 and put = 1 f :i = 1,2,...,N .
0 1 0 1 C(X) i
{ } G G ∪{ }∪{ }
Now applying 4.2. We obtain a projection p A and a unital homomorphism h : C(X)
0
∈ →
pAp such that
τ h (g) γ(g)(τ) < δ for all g and (e4.35)
0 1 1
| ◦ − | ∈G
τ h (χ ) < γ(χ )(τ) (e4.36)
◦ 0 Si Si
for all τ T(A), i= 1,2,...,n. Since 1 , by (e4.35),
C(X) 1
∈ ∈G
τ(1 p)< δ < δ /4 for all τ T(A). (e4.37)
1 0
− ∈
Write h (f) = L f(ξ )q for all f C(X), where ξ X and q ,q ,...,q is a set of
0 Pj=1 j j ∈ j ∈ { 1 2 L}
mutually orthogonal projections with L q = p.
Pj=1 j
Define
e′i = X qj, i = 1,2,...,N.
ξj∈B(xi,η1/2)
It follows from (e4.35) that, for each i,
τ(e′) τ h (f ) (e4.38)
i ≥ ◦ 0 i
> γ(f )(τ) δ > 3δ /4 τ(p) (e4.39)
i 1 0
− ≥
for all τ T(A). It follows that
∈
[e′] [p], i = 1,2,...,N.
i ≥
10