APPROXIMATELY UNITARILY EQUIVALENT
MORPHISMS AND INDUCTIVE LIMIT C*-ALGEBRAS
Marius Dadarlat
Introduction
Let X be a compact metrisable space and let B be a unital C*-algebra. We
prove that the ∗-homomorphisms from C(X) to B are classified up to stable approximate unitary equivalence by K-theory invariants. This result is used to obtain
a classification theorem for certain real rank zero inductive limits of homogeneous
C*-algebras. The classification of various classes of C*-algebras of real rank zero in
terms of invariants based on K-theory presupposes a passage from algebraic objects
to geometric objects (see [Ell], [Li1], [EGLP1], [EG], [BrD], [G], [Rø], [LiPh]). An
underlying idea of this paper is that this passage can be done by using approximate
morphisms. K-theory becomes a source of approximate morphisms thanks to the
realization of K-theory in terms of asymptotic morphisms, due to A. Connes and
N. Higson [CH]. The main results of the paper are Theorem A and Theorem B,
below.
By the universal coefficient theorem for the Kasparov KK-groups [RS],
Ext(K∗ (C(X)), K∗−1 (B)) is a subgroup of KK(C(X), B). Following [Rø], we let
KL(C(X), B) denote the quotient of KK(C(X), B) by the subgroup of pure extensions in Ext(K∗ (C(X)), K∗−1 (B)). If K∗ (C(X)) is isomorphic to a direct sum
of cyclic groups, then KL(C(X), B) coincides with KK(C(X), B).
Theorem A. Let X be a compact metrisable space. Let B be a unital C*-algebra
and let ϕ, ψ : C(X) → B be two unital ∗-homomorphisms. The following assertions
are equivalent.
(a) [ϕ] = [ψ] in KL(C(X), B).
(b) For any finite subset F ⊂ C(X) and any > 0 there exist k ∈ N, a unitary
u ∈ Uk+1 (B) and points x1 , . . . , xk in X such that
ku diag(ϕ(a), a(x1 ), . . . , a(xk )) u∗ − diag(ψ(a), a(x1 ), . . . , a(xk ))k < for all a ∈ F .
Two ∗-homomorphisms satisfying the condition (b) of Theorem A are said to be
stably unitarily equivalent. If condition (b) is satisfied without the ∗-homomorphism
η(a) = diag(a(x)1 , . . . , a(xk )), then we say that ϕ and ψ are approximately unitarily
equivalent. Theorem A generalizes certain results in [Li1-3] and [EGLP]. If ϕ and
ψ are ∗-monomorphisms and B is a purely infinite, simple C*-algebra, then one can
This research was partially supported by NSF grant DMS-9303361
Typeset by AMS-TEX
1
2
drop the ∗-homomorphism η from condition (b), see Theorem 1.7. The apparition
of pure extensions in this context is related to the phenomenon of quasidiagonality,
see [Br], [Sa], [BrD]. The author believes that future generalizations of Theorem A
should be based on Voiculescu’s noncommutative Weyl-Von Neumann theorem [V].
We use a suitable version of Theorem A for asymptotic morphisms to prove the
following theorem.
Theorem B. Let A, B be two simple C*-algebras of real rank zero. Suppose that A
Lkn
M[n,i] (C(Xn,i )),
and B are inductive limits A = lim
An and B = lim
Bn , An = i=1
−
→
−
→
Lln
Bn =
i=1 M{n,i} (C(Yn,i )). Suppose that Xn,i , Yn,i are finite CW complexes,
0
K (Xn,i ) and K 0 (Yn,i ) are torsion free and supn,i {dim(Xn,i ), dim(Yn,i )} < ∞.
Then A is isomorphic to B if and only if
(K∗ (A), K∗ (A)+ , Σ∗ (A)) ∼
= (K∗ (B), K∗ (B)+ , Σ∗ (B)).
Theorem B is proved by showing that the C*-algebras A and B are isomorphic to
certain inductive limits of subhomogeneous C*-algebras with 1-dimensional spectrum that are known to be classified by ordered K-theory [Ell]. In the process we
use the semiprojectivity of the dimension-drop C*-algebras proved by T. Loring
[Lo-1]. In a remarkable recent paper [EG], Elliott and Gong classify the simple C*algebras of real rank zero that are inductive limits of homogeneous C*-algebras with
spectrum 3-dimensional finite CW complexes. Roughly speaking, this classification
result is achieved in two stages. The first stage corresponds to passing from Ktheory to homotopy and uses the connective KK-theory introduced by A. Nemethi
and the author [DN]. The second stage corresponds to showing that homotopic ∗homomorphisms are stably approximately unitarily equivalent. A similar approach
is taken in [G]. In this paper we show that these techniques can be combined with
techniques of approximate morphisms to produce classification results for inductive
limits of homogeneous C*-algebras with higher dimensional spectra. A key tool of
our approach is a suspension theorem in E-theory due to T. Loring and the author
[DL], (see also [D2]). The idea of using homotopies of asymptotic morphisms in
the study of approximate unitary equivalence of ∗-homomorphisms appears also in
[LiPh]. The study of inductive limits of matrix algebras of continuous functions
was proposed by E. G. Effros [Eff].
1. Approximately unitarily equivalent morphisms and K-theory
In this section we prove Theorem A and give an application for ∗-monomorphisms
from C(X) to purely infinite, simple C*-algebras.
We need the following result from [EGLP].
Theorem 1.1. [EGLP] Let X be the cone over a compact metrisable space and let
B be a unital C*-algebra. For any finite subset G ⊂ C(X), any > 0 and any unital
∗-homomorphism ρ : C(X) → B, there are two unital ∗-homomorphisms with finite
dimensional image, η : C(X) → Ms−1 (B) and ξ : C(X) → Ms (B), such that
kdiag(ρ(a), η(a)) − ξ(a)k < for all a ∈ G.
The following result is implicitly contained in [EG].
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Theorem 1.2. Let X be a finite, connected CW complex. For any finite subset
F ⊂ C(X) and any > 0, there exist r ∈ N, a unital ∗-homomorphism τ : C(X) →
Mr−1 (C(X)) and a unital ∗-homomorphism µ : C(X) → Mr (C(X)) with finite
dimensional image, such that
kdiag(a, τ (a)) − µ(a)k < for all a ∈ F .
Proof. By [DN] there exist R ∈ N and a unital ∗-homomorphism σ : C(X) →
MR−1 (C(X)) such that ψ : C(X) → MR (C(X)), defined by ψ(a) = diag(a, σ(a)) is
homotopic to an evaluation map a 7→ a(x0 )1R . Let Ψ : C(X) → MR (C(X × [0, 1]))
be a ∗-homomorphism implementing a homotopy from ψ to the evaluation map
a 7→ a(x0 )1R . Since Ψ(a)(x, 1) = a(x0 )1R for all a ∈ C(X), one can identify Ψ
with a ∗-homomorphism Ψ : C(X) → MR (C(X̂)) where X̂ = X × [0, 1]/X × {1}
is the cone over X. Moreover, the natural inclusion X ∼
= X × {0} ,→ X̂ induces a
restriction ∗-homomorphism ρ : MR (C(X̂)) → MR (C(X)) and ψ factors as ψ = ρΨ.
Set G = Ψ(F ). Since X̂ is contractible we can use Theorem 1.1 to produce unital
∗-homomorphisms
η : MR (C(X̂)) → M(s−1)R (C(X)),
ξ : MR (C(X̂)) → MsR (C(X))
with finite dimensional image such that
kdiag(ρ(d), η(d)) − ξ(d)k < for all d ∈ G. Hence
kdiag(ρΨ(a), ηΨ(a)) − ξΨ(a)k < for all a ∈ F . Set r = sR, τ (a) = diag(σ(a), ηΨ(a)) and µ(a) = ξΨ(a). Then
kdiag(a, τ (a)) − µ(a)k < for all a ∈ F . Definition 1.3. A C*-algebra A is said to have property (P ), if for any finite subset
F ⊂ A and any > 0, there exist r ∈ N, a ∗-homomorphism τ : A → Mr−1 (A) and
a ∗-homomorphism µ : A → Mr (A) with finite dimensional image such that
kdiag(τ (a), a) − µ(a)k < for all a ∈ F .
It is clear that any finite dimensional C*-algebra has property (P ). Theorem
1.2 shows that if X is a finite CW complex, then C(X) has property (P ). Is is not
hard to see that the class of C*-algebras with property (P ) is closed under direct
sums and tensor products.
For C*-algebras C, D let M ap(C, D) denote the set of all linear, contractive,
completely positive maps from C to D. If G is a finite subset of C and δ > 0 we
say that ϕ ∈ M ap(C, D) is δ-multiplicative on G if kϕ(ab) − ϕ(a)ϕ(b)k < δ for all
a, b ∈ G.
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Lemma 1.4. Let A be a C*-algebra with property (P ). Let > 0 and let F ⊂ A be
a finite set. There are δb > 0 and a finite subset Fb ⊂ A such that if B is any unital
C*-algebra and ϕ0 , ϕ1 , . . . , ϕn is a sequence of maps in M ap(A, B) such that ϕj is
b
is δ-multiplicative
on Fb for j = 0, . . . , n, then there exist k ∈ N, a ∗-homomorphism
η : A → Mk (B) with finite dimensional image and a unitary u ∈ Uk+1 (B) such that
ku diag(ϕ0 (a), η(a)) u∗ − diag(ϕn (a), η(a))k < + max
max
a∈F 0≤j≤n−1
kϕj+1 (a) − ϕj (a)k
for all a ∈ F .
Proof. For given F ⊂ A and > 0, let r ∈ N, τ and µ be as in Definition 1.3.
Then D = µ(A) is a finite dimensional C*-subalgebra of Mr (A). By elementary
perturbation theory (see [Bra]), there is a finite subset G of D containing µ(F )
and there is δ > 0 such that whenever E is a C*-algebra and Ψ ∈ M ap(D, E)
is δ-multiplicative on G, there exists a ∗-homomorphism Ψ0 : D → E satisfying
kΨ0 (d) − Ψ(d)k < for all d ∈ G.
For s ∈ N set ϕs,j = ϕj ⊗ ids : Ms (A) → Ms (B). It easily seen that one can
b
find δb > 0 and Fb ⊂ A finite such that if ϕj ∈ M ap(A, B) is δ-multiplicative
on Fb
then ϕr,j is δ-multiplicative on G. Since ϕr,j is δ-multiplicative on G, there is a
∗-homomorphism ψj : D → Mr (B) such that kϕr,j (d) − ψj (d)k < for all d ∈ G
and j = 0, . . . , n.
Define L, L0 : A → Mnr (B) by
L = diag(ϕr−1,0 τ, ϕ0 , ϕr−1,1 τ, ϕ1 , . . . , ϕr−1,n−1 τ, ϕn−1 )
L0 = diag(ϕ0 , ϕr−1,0 τ, ϕ1 , ϕr−1,1 τ, . . . , ϕn−1 , ϕr−1,n−1 τ )
Note that L is unitarily equivalent to L0 . Thus there is a permutation unitary
u ∈ Unr+1 (B) such that
udiag((L0 , ϕn )u∗ = diag(ϕn , L).
(1)
Let
λ = max
max
a∈F 0≤j≤n−1
kϕj+1 (a) − ϕj (a)k
Since kϕj+1 (a) − ϕj (a)k ≤ λ for all a ∈ F
(2)
kdiag(ϕ0 (a), L(a)) − diag(L0 (a), ϕn (a))k ≤ max kϕj+1 (a) − ϕj (a)k = λ.
j
Using (1) and (2) we obtain
(3)
kudiag(ϕ0 (a), L(a))u∗ − diag(ϕn (a), L(a))k ≤ λ
for all a ∈ F .
On the other hand
(4)
kL(a) − diag(ϕr,0 µ(a), . . . , ϕr,n−1 µ(a))k
= kdiag(ϕr,0 (τ (a) ⊕ a − µ(a)), . . . , ϕr,n−1 (τ (a) ⊕ a − µ(a)))k
≤ kτ (a) ⊕ a − µ(a)k < 5
for all a ∈ F , since ϕr,j are norm decreasing. Note that kϕr,j µ(a) − ψj µ(a)k < for all a ∈ F since µ(F ) ⊂ G and kϕr,j (d) − ψj (d)k < for all d ∈ G. This implies
(5)
kdiag(ϕr,0 µ(a), . . . , ϕr,n−1 µ(a)) − diag(ψ0 µ(a), . . . , ψn−1 µ(a))k < for all a ∈ F . The ∗-homomorphism defined by η = diag(ψ0 µ, . . . , ψn−1 µ) has finite
dimensional image. Using (4) and (5) we obtain
kL(a) − η(a)k < 2
(6)
for all a ∈ F . Combining (3) and (6) we find
kudiag(ϕ0 (a), η(a))u∗ − diag(ϕn (a), η(a))k < 4 + λ
for all a ∈ F .
Suppose that the C*-algebra A is unital. Suppose that the ∗-homomorphism µ
from Definition 1.3 and the maps ϕj are unital. Then it easily seen that one can
arrange for the ∗-homomorphism η to be unital.
The notion of asymptotic morphism due to Connes and Higson led to a geometric
realization of E-theory [CH]. Let A, B be separable C*-algebras. Roughly speaking,
an asymptotic morphism from A to B is a continuous family of maps ϕt : A → B,
t ∈ T = [1, ∞), which satisfies asymptotically the axioms for ∗-homomorphisms.
A homotopy of asymptotic morphisms ϕt , ψt : A → B is given by an asymptotic
(0)
(1)
morphism Φt : A → B[0, 1] such that Φt = ϕt and Φt = ψt . Here B[0, 1] denotes
the C*-algebra of continous functions from the unit interval to B. The homotopy
classes of asymptotic morphisms from A to B are denoted by [[A, B]] and the class of
ϕt by [[ϕt ]]. Two asymptotic morphisms ϕt , ψt are said equivalent if ϕt (a)−ψt (a) →
0, as t → ∞ for all a ∈ A. Equivalent asymptotic morphisms are homotopic. In this
paper we deal exclusively with asymptotic morphisms from nuclear C*-algebras. It
was observed in [D] that if A is nuclear then any asymptotic morphism from A
to B is equivalent to a completely positive linear asymptotic morphism. This is
a consequence of the Choi-Effros theorem [CE], and it applies for homotopies of
asymptotic morphisms as well. Henceforth, throughout the paper, by an asymptotic
morphism we will mean a completely positive linear asymptotic morphism unless
stated otherwise. Let M∞ denote the dense ∗-subalgebra of the compact operators
K obtained as the union of the C*-algebras Mn . Using approximate units it is not
hard to see that any asymptotic morphism from A to B ⊗ K is equivalent to an
asymptotic morphism ϕt : A → B ⊗ M∞ for which there is a function α : T → N
such that ϕt (A) ⊂ B ⊗ Mα(t) . The map α is called a dominating function for ϕt .
This applies also to homotopies and yields a bijection
[[A, B ⊗ M∞ ]] → [[A, B ⊗ K]].
We consider here only asymptotic morphisms that are dominated by functions α as
above. Recall that if A is nuclear, then the Kasparov group KK(A, B) is isomorphic
to [[SA, SB ⊗ K]] (see [CH]).
Let X be a finite connected CW complex with base point x0 and let B be a unital
C*-algebra. Then by the suspension theorem of [DL], [[C0 (X \ x0 ), B ⊗ M∞ ]] ∼
=
KK(C0 (X \x0 ), B). Let ϕt : C0 (X \x0 ) → B⊗M∞ be an asymptotic morphism and
let α be a dominating function for ϕt . For each t ∈ T we let ϕα
t : C(X) → B ⊗Mα(t)
denote the unital extension of ϕt : C0 (X \x0 ) → B ⊗Mα(t) with ϕα
t (1) = 1B ⊗1α(t) .
β
α
Note that if α(t) ≤ β(t) then ϕt = 1B ⊗ 1α(t) ϕt 1B ⊗ 1α(t) .
6
Theorem 1.5. Let X be a finite connected CW complex with base point x0 and let
B be a unital C*-algebra. Let ϕt , ψt : C0 (X \ x0 ) → B ⊗ M∞ be two asymptotic
morphisms. Suppose that [[ϕt ]] = [[ψt ]] in KK(C0 (X \ x0 ), B). Then for any finite
set F ⊂ C(X) and any > 0 there are t0 ≥ 1 and maps α, β : [1, ∞) → N with
α dominating both ϕt and ψt such that for any t ≥ t0 there exist a unitary u ∈
U (B⊗Mα(t) ⊗Mβ(t)+1 ) and a unital ∗-homomorphism η : C(X) → B⊗Mα(t) ⊗Mβ(t)
with finite dimensional image such that
∗
α
kudiag(ϕα
t (a), η(a))u − diag(ψt (a), η(a))k < for all a ∈ F .
Proof. Using the suspension theorem in E-theory of [DL], we find an asymptotic
(0)
(1)
morphism Φt : C0 (X \ x0 ) → B[0, 1] ⊗ M∞ such that Φt = ϕt and Φt = ψt .
Let α : [1, ∞) → N be a function dominating ϕt , ψt , and Φt . We are going to use
e t : C(X) →
Lemma 1.4 and the notation from there. We find t0 such that if Φ
∼
b
e t is δ-multiplicative
(B[0, 1] ⊗ M∞ ) is the unital extension of Φt , then Φ
on Fb
e t commutes with 1B ⊗ 1α(t) , it follows that
for all t ≥ t0 . Since the image of Φ
b
b
Φα
t : C(X) → B[0, 1] ⊗ Mα(t) is δ-multiplicative on F for all t ≥ t0 . Let t ≥ t0 be
fixed. By uniform continuity we can find a sequence of points s0 = 0, . . . , sn = 1 in
the unit interval such that
max
max
a∈F 0≤j≤n−1
(sj ),α
kΦt
(sj+1 ),α
(a) − Φt
(a)k < .
(s ),α
By applying Lemma 1.4 for the sequence of unital maps Φt j
we find a unital
∗-homomorphism η with finite dimensional image and a unitary u such that
∗
α
kudiag(ϕα
t (a), η(a))u − diag(ψt (a), η(a))k < 2
for all a ∈ F .
In the proof of Theorem A we are going to apply Theorem 1.5 for ∗-homomorphisms.
Note that if ϕ : C(X) → B is a ∗-homomorphism, whose restriction to C0 (X \ x0 )
is denoted by ϕ too, then ϕα (a) = ϕ(a) + a(x0 )(1α(t) − ϕ(1)).
1.6 The Proof of Theorem A.
Recall from [F] that an extension of abelian groups
π
0−
→K−
→G−
→H−
→0
is called pure if its restriction to any finitely generated subgroup of H is trivial.
The isomorphism classes of pure extensions form a subgroup of Ext(H, K). The
universal coefficient theorem of [RS] gives an exact sequence
0−
→ Ext(K∗ (A), K(B)∗−1 ) −
→ KK(A, B) → Hom(K∗ (A), K∗ (B)) −
→0
for A in a large class of separable nuclear C*-algebras and any C*-algebra B that
has a countable approximate unit. The quotient of KK(A, B) by the subgroup
of pure extensions in Ext(K∗ (A), K∗−1 (B)) is denoted by KL(A, B) [Rø]. The
7
subgroup of pure extensions was studied and characterized in the setting of [BDF]
in [KS].
a) ⇒ b) Suppose first that X is a finite CW complex. Since K∗ (C(X)) is
finitely generated any pure extension of K∗ (C(X)) is trivial, hence KL(C(X), B) =
KK(C(X), B). Let X1 , . . . , Xm be the connected components of X. Let ei be
the unit of C(Xi ). Using the definition of the K0 group we find R ∈ N, a
∗-homomorphism ξ : C(X) → MR (B) and a unitary v ∈ UR+1 (B) such that
vdiag(ϕ(ei ), ξ(ei ))v ∗ = diag(ψ(ei ), ξ(ei )) for i = 1, . . . , m. Note that since ϕ and
ψ are unital, ξ can be chosen to be unital. After replacing ϕ,ψ by vdiag(ϕ, ξ)v ∗
and diag(ψ, ξ), we may assume that ϕ(ei ) = ψ(ei ), i = 1, . . . , m. Let ϕi , ψi denote
the restrictions of ϕ and ψ to C(Xi ). Then [ϕi ] = [ψi ] ∈ KK(C(Xi ), B). Let
δ > 0 and let Fi = F ∩ C(Xi ). Using Theorem 1.5 for each i we find k(i) ∈ N,
a ∗-homomorphism ηi : C(Xi ) → Mk(i) (B) with finite dimensional image and a
unitary ui ∈ Uk(i)+1 (B) such that
kui diag(ϕi (a), ηi (a))u∗i − diag(ψi (a), ηi (a))k < δ
L
for all a ∈ Fi . Let k = k(1) + . . . + k(m) and define η 0 : C(X) =
i C(Xi ) →
0
Mk (B) by η (a1 , . . . , am ) = diag(η1 (a1 ), . . . , ηm (am )). By conjugating (ui , 1, . . . , 1)
by suitable permutation unitaries we find unitaries vi ∈ Uk+1 (B) such that
kvi diag(ϕi (a), η 0 (a))vi∗ − diag(ψi (a), η 0 (a))k < δ
for all a ∈ Fi . Let pi = diag(ϕi (ei ), η 0 (ei )) = diag(ψi (ei ), η 0 (ei )). By chosing δ
small enough we can perturb vi to unitaries wi ∈ Uk+1 (B) such that wi pi wi∗ = pi
and
kwi diag(ϕi (a), η 0 (a))wi∗ − diag(ψi (a), η 0 (a))k < for all a ∈ Fi , i = 1, . . . , m. Let p = 1k − η 0 (1). Let u ∈ Uk+1 (B) be the unitary
w1 p1 + . . . + wm pm + diag(0, p). Fix x0 ∈ X and define η : C(X) → Mk (B) by
η(a) = η 0 (a) + a(x0 )p. Then η is a unital ∗-homomorphism with finite dimensional
image and
kudiag(ϕ(a), η(a))u∗ − diag(ψ(a), η(a))k < for all a ∈ F .
In the general case we embed X into the Hilbert cube I ω and write X = ∩ Xn ,
Xn = Pn × I ωn where Pn are finite CW complexes with Pn+1 ⊂ Pn × I and
ωn = {n + 1, n + 2, . . .}. Let gi : I ω → I be the ith -coordinate map. By abuse
of notation we let gi also denote the restrictions of gi to X and Pn . Without loss
of generality we may assume that the set F in the statement of the Theorem is
equal to {g1 , g2 , . . . , gn } for some n. Let now n be fixed and choose m ≥ n such
that for any y ∈ Pm there is x ∈ X such that |gi (x) − gi (y)| < for i = 1, . . . , n.
Let ρm : C(Pm ) → C(X) be induced by the X ,→ Pm × I ωm → Pm with the
second arrow standing for the canonical projection. Consider the ∗-homomorphisms
ϕρm , ψρm : C(Pm ) → B , the finite set {g1 , . . . , gn }⊂ C(Pm ) and > 0. Since
the Kasparov product induces a product at the level of the KL-groups [Rø] and
KL(C(Pm ), B) = KK(C(Pm ), B), it follows that the ∗-homomorphisms ϕρm , ψρm
give rise to the same KK-element. By the first part of the proof we find a unital ∗homomorphism η0 : C(Pm ) → Mk (B) with finite dimensional image and a unitary
u ∈ Uk+1 (B) such that
kudiag(ϕρm (gi ), η0 (gi ))u∗ − diag(ψρm (gi ), η0 (gi ))k < 8
P`
for i = 1, . . . , n. The map η0 can be expressed as η0 (a) =
j=1 a(yj )qj with
yj ∈ Pm and mutually orthogonal projections qj . The integer m was chosen so
that we can find points xj ∈ X with |gi (yj ) − gi (xj )| < for i = 1, . . . , n. Define
P`
η : C(X) → Mk (B) by η(a) = j=1 a(xj )qj . Then
kudiag(ϕ(gi ), η(gi ))u∗ − diag(ψ(gi ), η(gi ))k < 2
for i = 1, . . . , n.
So far we have proven that if [ϕ] = [ψ] ∈ KL(C(X), B), then for any > 0 and
F ⊂ C(X) finite, there exist k ∈ N, a unital ∗-homomorphism η : C(X) → Mk (B)
with finite dimensional image and a unitary u ∈ Uk+1 (B) such that
kudiag(ϕ(a), η(a))u∗ − diag(ψ(a), η(a))k < for all a ∈ F . Next we show that η can be chosen of the form η(a) = diag(a(x1 ), . . . , a(xk ))
with x1 , . . . , xk ∈ X. This is done in two steps.
First we note that if w ∈ Uk+r (B), ξ : C(X) → Mr (B) is a ∗-homomorphism
with finite dimensional image ηb = wdiag(η, ξ)w∗ , and u
b = 1 ⊕ w(u ⊕ 1r )1 ⊕ w∗ ,
then
kb
udiag(ϕ(a), ηb(a))b
u∗ − diag(ψ(a), ηb(a))k < for all a ∈ F . This remark shows that we have the freedom to change η by taking
direct sums and conjugating by unitaries. The proof of (a) ⇒ (b) is completed by
using the following elementary Lemma:
Lemma. Let η : C(X) → Mk (B) be a unital ∗-homomorphism with finite dimensional image. Then there are a unital ∗-homomorphism with finite dimensional
image ξ : C(X) → Mr (B) and a unitary w ∈ Uk+r (B) such that wdiag(η, ξ)w∗ is
equal to to a ∗-homomorphism of the form ηb(a) = diag(a(x1 ), . . . , a(xk+r )).
Proof. There are distinct points x1 , . . . , xn in X and a partition
Pn of 1k into mutually orthogonal projections p1 , . . . , pn such that η(a) =
i=1 a(xi )pi for all
a ∈ C(X).P The proof is done by induction after
= a(x1 )p2 +
Pnn. Define ξ(a)
n
0
0
a(x2 )pP
+
a(x
)p
,
η
(a)
=
a(x
)(p
+
p
)
+
a(x
)p
,
ξ
(a)
=
a(x2 )(p1 +
1
i i
1
1
2
i i
i=3
i=3
n
p2 ) + i=3 a(xi )pi . Then it is easily seen that diag(η, ξ) is unitarily equivalent to
diag(η 0 , ξ 0 ). The formulae for η 0 and ξ 0 involves n − 1 distinct points each.
(b) ⇒ (a) This is a generalization of Proposition 5.4 in [Rø]. The proof is based
on an adaptation of the argument given in [Rø] and it remains valid if we replace
C(X) by any C*-algebra in the class N of [RS]. To save notation set A = C(X).
The first step is to show that K∗ (ϕ) = K∗ (ψ). This is standard and follows easily
from b) by using functional calculus and the definition of K-theory. Therefore by
the universal coefficient theorem of [RS] the difference element z = [ϕ] − [ψ] lies
in the image of Ext(K∗ (A), K∗−1 (B)) in KK(A, B). It is shown in [Rø] that the
element z is given by
(7)
π
∗
0−
→ K∗ (SB) −
→ K∗ (E) −→
K∗ (A) −
→0
where E is the C*-algebra of all pairs (f, a), where a ∈ A, f ∈ B[0, 1], f (0) = ϕ(a),
f (1) = ψ(a). The map π : E → A is given by π(f, a) = a. Our aim is to show
that z corresponds to a pure extension. By assumption there exist a sequence of
9
∗-homomorphisms ηi : C(X) → Mn(i) with finite dimensional image and a sequence
of unitaries ui ∈ Um(i) (B) where m(i) = 1 + n(1) + . . . + n(i), such that if
ϕi = diag(ϕ, η1 , . . . , ηi ),
ψi = diag(ψ, η1 , . . . , ηi )
then
lim kui ϕi (a)u∗i − ψi (a)k = 0
(8)
i→∞
for all a ∈ C(X). Note that K∗ (ϕi ) = K∗ (ψi ). As above we have a ”difference”
extension
π(i)
0−
→ Mm(i) (SB) −
→ Ei −−→ A −
→0
corresponding to the pair ϕi , ψi . There is a natural embedding γi : Ei → Ei+1 ,
γi (f, a) = (f ⊕ ηi+1 (a), a). This yields a commutative diagram
0 −−−−→
Mm(i) (SB)


y
−−−−→
π(i)
Ei −−−−→

γi
y
A −−−−→ 0

id
y A
π(i+1)
0 −−−−→ Mm(i+1) (SB) −−−−→ Ei+1 −−−−→ A −−−−→ 0
Taking the inductive limit of these extensions we obtain an extension
π
0−
→ K ⊗ SB −
→ E∞ −−∞
→A−
→0
whose KK-theory class is equal to z (by the continuity of K-theory and the five
lemma). With this construction in hands, by using (8) and reasoning as in the
proof of Proposition 5.4 in [Rø] one shows that if g ∈ K∗ (A) and ng = 0 for some
n then there is i ≥ 1 and h ∈ K∗ (Ei ) with nh = 0 and π(i)∗ (h) = g. This shows
that the group extension
0−
→ K∗ (B) −
→ K∗ (E∞ ) −
→ K∗ (A) −
→0
is pure. We conclude that the extension (7) is pure. Another proof of (b) ⇒ (a) can be be obtained by constructing a sequence
χi ∈ M ap(A, E∞ ) such that π∞ χi = idA and kχi (ab) − χi (a)χi (b)k → 0, as i → ∞,
for all a, b ∈ A. This goes as follows. We may assume that there is a path of unitaries
ui (s), s ∈ [0, 1], in Um(i) (B) with ui (0) = 1 and ui (1) = ui . Let σi ∈ M ap(A, Ei )
be defined by
σi (a)(s) =
(ui (2s)ϕi (a)ui (2s)∗ , a)
((2 −
2s)ui ϕi (a)u∗i
for 0 ≤ s ≤ 1/2
+ (2s − 1)ψi (a), a) for 1/2 ≤ s ≤ 1
It is not hard to see that the sequence of maps χi = γi σi has the desired properties.
The sequence χi gives an approximate splitting of the extension E∞ . Then one
shows as in [BrD] that the corresponding extension in K-theory is pure.
Suppose that the C*-algebra B is purely infinite and simple. Then Theorem A
can be modified as follows.
10
Theorem 1.7. Let X be a compact metrisable space. Let B be a purely infinite,
simple, unital C*-algebra and let ϕ, ψ : C(X) → B be two unital ∗-monomorphisms.
Then [[ϕ]] = [[ψ]] in KL(C(X), B) if and only if ϕ is approximately unitarily
equivalent to ψ. That is, if and only if there is a sequence of unitaries un ∈ U (B)
such that kun ϕ(a)u∗n − ψ(a)k → 0 for all a ∈ C(X).
Proof. Fix > 0 and F ⊂ C(X) finite. Using Theorem A all we have to do is
to find a unitary w0 ∈ U (B) such that kw0 ϕ(a)w0∗ − ψ(a)k < 5 for all a ∈ F .
Let k ∈ N, η : C(X) → Mk (B) and u ∈ Uk+1 (B) be provided by Theorem A.
Since η is unital and has finite dimensional image there exist L ∈ N, distinct points
x1 , . . . , xL in X and a partition of 1k into mutually orthogonal nonzero projections
p1 , . . . , pL ∈ Mk (B) such that
η(a) =
L
X
a(xi )pi
i=1
for all a ∈ C(X). Define γ : C(X) → Mk+1 (B) by γ = uϕu∗ . Set q = γ(1) and
qi = upi u∗ . Then
uη(a)u∗ =
(9)
L
X
a(xi )qi
and q +
L
X
i=1
qi = 1k+1 .
i=1
We are going to use repeatedly a result of Zhang [Zh1] asserting that a simple,
purely infinite C*-algebra has real rank zero. Since γ is a monomorphism and B
has real rank zero we can apply Lemma 4.1 in [EGLP] to get nonzero, mutually
orthogonal projections d, d1 , . . . , dL in qMk+1 (B)q such that d + d1 + . . . + dL = q
and
(10)
kγ(a) − dγ(a)d −
L
X
a(xi )di k < i=1
for all a ∈ F . Similarly, there are nonzero, mutually orthogonal projections
c, c1 , . . . , cL in B such that c + c1 + . . . + cL = 1 and
0
(10 )
kψ(a) − cψ(a)c −
L
X
a(xi )ci k < i=1
for all a ∈ F . Since B is simple and purely infinite, di is equivalent to a subprojection ei of ci . Recall that two projections e and f in a C*-algebra A are called
equivalent if there is a partial isometry v ∈ A such that v ∗ v = e and vv ∗ = f . The
proof of Lemma 4.1 in [EGLP] shows that (10’) remains true if we replace ci by
nonzero subprojections ei ≤ ci and c by e = 1k − e1 − . . . − eL . Therefore we may
assume that di is equivalent to ci in Mk+1 (B). Since Mk+1 (B) is simple and purely
infinite, di +qi is equivalent to a subprojection of di . Similarly ci ⊕pi is equivalent to
a subprojection of ci . Actually, we can find partial isometries v, w ∈ Mk+1 (B) such
that v(di + qi )v ∗ + d0i = di and w(ci ⊕ pi )w∗ + c0i = ci for some nonzero projections
d0i , c0i , i = 1, . . . , L. Define Tγ : C(X) → M2k+2 (B) by
∗
Tγ (a) = diag(dγ(a)d +
L
X
i=1
a(xi )di +
L
X
i=1
a(xi )qi ,
L
X
i=1
a(xi )d0i )
11
Using v one obtains a partial isometry V ∈ M2k+2 (B) with V ∗ V = 1k+1 +
V V ∗ = q and such that
L
X
∗
(11)
V Tγ (a)V = dγ(a)d +
PL
i=1
d0i ,
a(xi )di
i=1
Similarly, if Tψ : C(X) → M2k+2 (B) is defined by
∗
Tψ (a) = diag(cψ(a)c +
L
X
a(xi )ci ⊕
i=1
L
X
a(xi )pi ,
i=1
L
X
a(xi )c0i )
i=1
then there is a partial isometry W ∈ M2k+2 (B), W ∗ W = 1k+1 +
1B such that
∗
0
(11 )
W Tψ (a)W = cψ(a)c +
L
X
PL
0
i=1 ci ,
WW∗ =
a(xi )ci .
i=1
Fix a ∈ A. From (10), (11) and (10’), (11’) we obtain
(12)
kV ∗ γ(a)V − Tγ (a)k < (120 )
kW ∗ ψ(a)W − Tψ (a)k < On the other hand, using (10) and (10’) we obtain
(13)
kTγ (a) − diag(u(ϕ(a) ⊕ η(a))u∗ ,
L
X
a(xi )d0i )k < i=1
0
(13 .)
kTψ (a) − diag(ψ(a) ⊕ η(a),
L
X
a(xi )c0i )k < i=1
By construction [di ] = [ci ] and [qi ] = [pi ] in K0 (B). This implies [d0i ] = [c0i ]
in K0 (B). Since these are proper projections in Mk+1 (B) we can find a unitary
u0 ∈ Uk+1 (B) such that u0 d0i u∗0 = c0i for i = 1, . . . , L (see [Cu], [Zh2]). Let U =
1k+1 ⊕ u0 ∈ U2k+2 (B). Then
(14) kU diag(u(ϕ(a)⊕η(a))u∗ ,
L
X
a(xi )d0i )U ∗ −diag(ψ(a)⊕η(a),
i=1
L
X
a(xi )c0i )k < i=1
∗
since ku(ϕ(a) ⊕ η(a))u − ψ(a) ⊕ η(a)k < . Using (13), (13’) and (14) we obtain
(15.)
kU Tγ (a)U ∗ − Tψ (a)k < 3
Then from (12), (12’) and (15)
kU V ∗ uϕ(a)u∗ V U ∗ − W ψ(a)W ∗ k < 5.
Finally if we set w0 = W ∗ U V ∗ u, then
kw0 ϕ(a)w0∗ − ψ(a)k < 5
for all a ∈ F . Certain special cases of Theorem 1.7 have been previously proven in [Li1-3] and
[EGLP] and the above proof uses ideas from those papers. For other applications
of the asymptotic morphisms in the study of ∗-homomorphisms we refer the reader
to [DL], [D1] and [LiPh].
12
Remarks 1.8.
a) By a result of T. Loring [Lo2], if X is a finite connected CW complex with
base point x0 and On is a Cuntz algebra with n ≥ 4 even, then all the elements
of KK(C0 (X \ x0 ), On ) are induced by ∗-homomorphisms C0 (X \ x0 ) → On . By
combining this result with Theorem 1.7 we obtain a bijection between KK(C0 (X \
x0 ), On ) and the set of all approximate unitary equivalence classes of unital ∗monomorphisms from C(X) to On .
b) Let Y be the dyadic solenoid and let X be the one point compactification
of (Y \ y0 ) × R. Let Q(H) be the Calkin algebra. There are infinitely many unitary equivalence classes of unital ∗-monomorphisms from C(X) to Q(H). These
classes are parametrized by the Brown-Douglass-Fillmore group Ext(C(X)) ∼
=
KK(C0 (X \ x0 ), Q(H)) ∼
= Ext1Z (Z[1/2], Z). However, since all the extensions in
Ext1Z (Z[1/2], Z) are pure, it follows that KL(C0 (X \ x0 ), Q(H)) = 0 and any two
unital ∗-homomorphism from C(X) to the Calkin algebra are approximately unitarily equivalent (see [Br]).
2. Inductive limit C*-algebras
Consider the following list of C*-algebras: the scalars, C; the circle algebra, C(T);
the unital dimension-drop interval, defined below; and all C ∗ -algebras arising from
these by forming matrix algebras and taking finite direct sums. The collection of
all these C*-algebras will be denoted D. The collection of all quotients of C*b A C ∗ −algebra will be called an AD-algebra if it
algebras in D will be denoted D.
is isomorphic to an inductive limit of C*-algebras in D.
By the non-unital dimension-drop interval we mean
Im = {f ∈ C([0, 1], Mm ) | f (0) = 0, f (1) is scalar}.
The unital dimension-drop interval eIm is the unitization of Im . The K-theory of Im
is easily computed, K0 (Im ) = 0, K1 (Im ) ∼
= Z/m.
Lemma 2.1. Let X, Y be finite, connected CW complexes and let k, m, r ∈ N.
Let γ : Mk (C(X)) → P Mm (C(Y ))P be a unital ∗-homomorphism where P is
a projection in Mm (C(Y )). Let ν : Mk (C(X)) → Mkr (C(X)) be a unital ∗homomorphism of the form ν(a) = diag(a, a(x1 ), . . . , a(xr−1 )), xi ∈ X. Suppose that rank(P ) > 10kr dim(Y ). Then there are ∗-homomorphisms γ0 , γ1 :
Mk (C(X)) → P Mm (C(Y ))P with orthogonal images such that γ0 has finite dimensional image, γ1 factors through ν and γ is homotopic to γ0 + γ1 . If P is a
trivial projections and rank(P ) is divisible by kr, then one can take γ0 = 0.
Proof. The result is a consequence of Theorems 4.2.8 and 4.2.11 in [DN]. Lemma 2.2. Let H be a finite subset of C(S 2 ), let > 0 and let X be a finite,
connected CW complex. There is m0 such that if m ≥ m0 , then for any unital
∗-homomorphism σ 0 : C(S 2 ) → Mm (C(X)) there exist a unital ∗-homomorphism
σ : C(S 2 ) → Mm (C(X)) and a C*-subalgebra D of Mm (C(X)) isomorphic to a
quotient of a direct sum of matrix algebras over C(S 1 ), such that σ 0 is homotopic
to σ and dist(σ(a), D) < for all a ∈ H.
Proof. If X is a product of spheres, the result appears in [EGLP1]. In particular
for X = S 2 we find n and such that the unital ∗-homomorphism ν 0 : C(S 2 ) →
13
Mn (C(S 2 )), ν 0 (a) = (a, a(x0 ), . . . , a(x0 )) is homotopic to a ∗-homomorphism ν with
ν(H) approximately contained to within in a circle algebra. The more general
situation we consider here is reduced to the case X = S 2 by using Lemma 2.1.
Indeed, by Lemma 2.1 there is m0 such that any ∗-homomorphism σ 0 : C(S 2 ) →
Mm (C(X)), m ≥ m0 , is homotopic to a direct sum between a ∗-homomorphism
with finite dimensional image and a ∗-homomorphism that factors through ν. Proposition 2.3. Let X be a finite, connected CW complex. Let F be a finite
subset of C(X) and let > 0. Suppose that the K-theory group K0 (C(X)) is
torsion free. Then there exist r ∈ N, a ∗-homomorphism η : C(X) → Mr−1 (C(X))
b
with finite dimensional image and a C*-subalgebra D ⊂ Mr (C(X)) with D ∈ D
such that
dist(a ⊕ η(a), D) < for all a ∈ F . The ∗-homomorphism η can be chosen of the form
η(a) = diag(a(x1 ), . . . , a(xr−1 ).
Proof. There are c, d ≥ 0, m(i) ≥ 2 with K0 (C(X)) ∼
= Zc+1 , K1 (C(X)) ∼
= Zd ⊕
Z/m(1) ⊕ . . . ⊕ Z/m(k). Let N = c + d + k and set
Bj = C0 (S 2 \ pt) if 1 ≤ j ≤ c,
Bj = C0 (S 1 \ pt) if c < j ≤ c + d,
Bj = Im(j−c−d) if c + d < j ≤ N ,
B = B1 ⊕ . . . ⊕ BN .
If K1 (C(X)) is torsion free, then we consider only Bj for 1 ≤ j ≤ c + d. By
construction, K∗ (C0 (X \ x0 )) is isomorphic to K∗ (B). The universal coefficient
theorem of [RS] shows that C0 (X \ x0 ) is KK-equivalent to B. Using the E-theory
description of KK-theory of [CH] and the suspension theorem of [DL], we find
an asymptotic morphism ϕt : C0 (X \ x0 ) → B ⊗ M∞ yielding a KK-equivalence.
Let χ ∈ KK(B, C0 (X \ x0 )) be such that χ[[ϕt ]] = [[idC0 (X\x0 ) ]]. The element χ
can be written as χ = χ1 + . . . χN with χj ∈ KK(Bj , C0 (X \ x0 )). It is known
that every element in KK(Bj , C0 (X \ x0 )) can be realized as a ∗-homomorphism
Bj → C0 (X \ x0 ) ⊗ ML for a suitable integer L.
[C0 (S 1 \ pt), C0 (X \ x0 ) ⊗ M∞ ] ∼
= KK(C0 (S 1 \ pt), C0 (X \ x0 )) [Ros]
2
[C0 (S \ pt), C0 (X \ x0 ) ⊗ M∞ ] ∼
= KK(C0 (S 2 \ pt), C0 (X \ x0 )) [Se], [DN]
[Im , C0 (X \ x0 ) ⊗ M∞ ] ∼
= KK(Im , C0 (X \ x0 )) [DL].
It follows that there are ∗-homomorphisms ψj0 : Bj → C0 (X \ x0 ) ⊗ ML with
ej → C(X) ⊗ ML be the unital extension of ψ 0 and let ψj
[ψj0 ] = χj . Let ψej : B
j
ej ⊗ M∞ → C(X) ⊗ ML ⊗ M∞ .
denote the ∗-homomorphism ψj = ψej ⊗ id∞ : B
Define
e
ψ : ⊕N
j=1 (Bj ⊗ M∞ ) → C(X) ⊗ ML ⊗ MN ⊗ M∞
,
ψ(b1 , . . . , bN ) = diag(ψ1 (b1 ), . . . , ψN (bN )).
Form the diagram
C0 (X \ x0 )


ϕt y
ej ⊗ M∞ )
⊕j ( B
γ
−−−−→ C(X) ⊗ ML ⊗ MN ⊗ M∞
14
where γ(a) = a ⊗ e for a fixed one-dimensional projection e. By construction, the
above diagram commutes at the level of KK-theory. Therefore there is a homotopy
of asymptotic morphisms Φt : C0 (X \ x0 ) → C(X) ⊗ ML [0, 1] ⊗ MN ⊗ M∞ with
(0)
(1)
Φt = γ and Φt = ψϕt .
Let F ⊂ C(X) and be as in the statement of Proposition 2.3. Let Fb and δb
be given by Lemma 1.4. As in the proof of Theorem 1.5, let α be a dominating
b
b
function for Φt and find t0 such that Φα
t is δ-multiplicative on F for all t ≥ t0 .
Now let t ≥ t0 be fixed. By increasing α(t), we may assume that the image of ϕt is
α
b
e
contained in ⊕N
j=1 Bj ⊗ Mα(t) . Moreover we may assume that ϕt is δ-multiplicative
on Fb. We obtain the following diagram
C(X)


ϕα
t y
γα
−−−−→ C(X) ⊗ ML ⊗ MN ⊗ Mα(t)
ej ⊗ Mα(t) )
⊕j ( B
where the superscript α indicates that the corresponding maps have been unitalej ⊗ Mα(t) ) is
ized as described before Theorem 1.5. The restriction of ψ to ⊕j (B
(0),α
(1),α
denoted by ψ too. Note that Φt
= γ α and Φt
= ψϕα
t . The next step is to
modify the above diagram so that ψ will become homotopic to a ∗-homomorphism
b Set C = ⊕N (B
ej ⊗ Mα(t) ),
which factors approximately through an algebra in D.
j=1
α
E = C(X) ⊗ ML ⊗ MN ⊗ Mα(t) and H = ϕt (F ) (recall that t is fixed). By using Lemma 2.2 we find m, such that if γ0 : E → Mm (E) is defined by γ0 (c) =
diag(c, c(x0 ), . . . , c(x0 )), with x0 ∈ X, then the ∗-homomorphism γ0 ψ is homotopic
to a ∗-homomorphism σ : C → Mm (E) such that σ(H) is approximately contained,
b Actually, one applies Lemma
to within , in a subalgebra D of Mm (E) with D ∈ D.
2.2 for the partial ∗-homomorphisms γ0 ψj , 1 ≤ j ≤ c. Let Ψ : C → Mm (E)[0, 1]
be a homotopy of ∗-homomorphisms with Ψ(0) = γ0 ψ and Ψ(1) = σ. Next we consider the path of maps Γ(s) in M ap(C(X), Mm (E)) obtained by the juxtaposition
(s),α
α
(s)
of γ0 Φt
with Ψ(s) ϕα
and σϕα
t
t . It is clear that γ0 γ is the initial point of Γ
α
b
b
are
δ-multiplicative
on
F
,
it
follows
that
and
ϕ
is the terminal point. Since Φα
t
t
b
Γ(s) is δ-multiplicative
on Fb for each s. This enables us to use Lemma 1.4 to show
that γ0 γ α is stably approximately unitarily equivalent to σϕα
t . Namely, there exist
S ∈ N, a ∗-homomorphism η0 : C(X) → MS−1 (Mm (E)) with finite dimensional
image and a unitary u ∈ US (Mm (E)) such that
∗
α
kudiag(σϕα
t (a), η0 (a))u − diag(γ0 γ (a), η0 (a))k < 2
for all a ∈ F . Note that γ0 γ α is a unital ∗-homomorphism of the form
a 7→ diag(a, a(x0 ), . . . , a(x0 )) = diag(a, η1 (a)).
Since ϕα
t (F ) ⊂ H and σ(H) is approximately contained to within in D, it follows that, for all a ∈ F , diag(a, η1 (a), η0 (a)) is approximately contained to within 3
b By using the Lemma that appears in the proof of Theoin u(D⊕image(η0 ))u∗ ∈ D.
rem A, we see that η0 can be taken of the form η0 (a) = diag(a(x1 ), . . . , a(xr−1 )). Theorem 2.4. Let A be a C*-algebra of real rank zero. Suppose that A is an
Lkn
inductive limit A = lim
i=1 M[n,i] (C(Xn,i )). Suppose that
−→ (An , γm,n ), An =
15
each Xn,i is a finite connected CW complex, K0 (C(Xn,i )) is torsion free and d =
supn,i {dim(Xn,i )} < ∞. Then A is isomorphic to an AD algebra.
j,i
Proof. Let en,i denote the unit of An,i = M[n,i] (C(Xn,i )) and let γm,n
: An,i → Am,j
be the partial ∗-homomorphisms of γm,n . The C*-algebras in D are semiprojective
[Lo]. By Proposition 1.2 in [DL1], (see also Theorem 3.8 in [Lo]) it suffices to
show that for any finite subset F ⊂ An,i and > 0, there is m ≥ n, such that
for any 1 ≤ j ≤ km there exist a C*-algebra D ∈ D, and a ∗-homomorphism
j,i
j,i
σ : D → fj Am,j fj , fj = γm,n
(en,i ), such that dist(γm,n
(a), σ(D)) < for all
a ∈ F . By Theorem 1.4.14 in [EG] we can assume that F is weakly approximately
constant to within . To simplify notation say An,i = Mk (C(X)). By Proposition
2.3 there exist r ∈ N and a unital ∗-homomorphism ν : Mk (C(X)) → Mkr (C(X)) of
the form ν(a) = diag(a, a(x1 ), . . . .a(xr−1 )), and there is a unital ∗-homomorphism
ξ : D → Mkr (C(X)) with D ∈ D such that dist(ν(a), ξ(D)) < for all a ∈ F . By
Theorem 2.5 in [Su] and Lemma 2.3 in [EG] there is m ≥ n so that for each j,
1 ≤ j ≤ km , one of the following conditions are satisfied.
(i) There is a ∗-homomorphism µ : An,i → fj Am,j fj with finite dimensional
j,i
(a) − µ(a)k < for all a ∈ F .
image such that kγm,n
j,i
(ii) rank( γm,n (en,i )) = rank(fj ) = krj sj where rj > r and sj > 10d.
If we are in the first case, then we are done. Thus we may assume that (ii)
holds. Let η : Mk (C(X)) → Mk be an evaluation map. Define νj : Mk (C(X)) →
j,i
and
Mkrj (C(X)), νj (a) = ν(a) ⊕ a(x0 ) ⊗ 1k(rj −r) . We apply Lemma 2.1 for γm,n
νj . Hence we find a ∗-homomorphism ψj : Mkrj (C(X)) ⊕ Mk → fj Am,j fj such
j,i
as ∗-homomorphisms from Mk (C(X)) to fj Am,j fj ,
that ψj ν̂j is homotopic to γm,n
where ν̂j = ν ⊕ η. Since F is approximately weakly constant to within and A
has real rank zero, by Theorem 2.29 in [EG], there exist s ≥ m and a unitary
u ∈ γsm (fj )As γs,m (fj ) such that
j,i
ku γs,m ψj νj (a) u∗ − γs,m γm,n
(a)k < 70
for all a ∈ F . Define ξj : D⊕C → Mkrj (C(X))⊕Mk , ξj (d, λ) = ξ(d)⊕λ⊗1k(rj −r) ⊕
λ. The various maps that are involved here can be visualized on the diagram
F
−−−−→
An,i


νj y
j,i
γm,n
γs,m
−−−−→ fj Am,j fj −−−−→ As
ξj
D ⊕ Mk −−−−→ Mrj (An,i ) ⊕ Mk
Since dist(νj (a), ξj (D ⊕ C)) < for all a ∈ F , if we set σ = u(γs,m ψj ξj )u∗ then
j,i
dist(γs,m γm,n
(a), σ(D)) < 100.
This concludes the proof. In Theorem 2.4 one can allow d = ∞ under the additional assumption that A
has slow dimension growth. A similar remark applies to Theorem B.
2.5 The proof of Theorem B.
Let A and B be as in the statement of Theorem A. Theorem 2.4 shows that A
and B are AD algebras. By Theorem 7.1 in [Ell], ordered K-theory is a complete
invariant for the simple AD algebras of real rank zero. 16
References
[Bl]
B. Blackadar, K-theory for Operator Algebras, M. S. R. I. Monographs No. 5, SpringerVerlag, Berlin and New York, 1986.
[BDR]
B. Blackadar, M. Dadarlat and M. Rørdam, The real rank of inductive limit C*-algebras,
Math. Scand. 69, 211–216.
[Bra]
O. Bratteli, Inductive limits of finite-dimensional C*-algebras, Trans. Amer. Math. Soc.
171 (1972), 195-234.
[Br]
L. G. Brown, The universal coefficient theorem for Ext and quasidiagonality, Operator
Algebras and Group Representations, Pitman Press, Boston, London and Melbourne,
1983, pp. 60–64.
[BDF]
L. Brown, R. Douglas and P. Fillmore, Extensions of C*-algebras K-homology, Ann. of
Math. 105 (1977), 265-324.
[BP]
L. G. Brown and G. K. Pedersen, C*-algebras of real rank zero, J. Funct. Anal. 99
(1991), 131–149.
[Br D]
L. G. Brown and M. Dadarlat, Extensions of C*-algebras and quasidiagonality, Preprint
(1993).
[CE]
M. -D. Choi and E. G. Effros, The completely positive lifting problem for C*-algebras,
Ann. of Math. 104 (1976), 585-609.
[CH]
A. Connes and N. Higson, Deformations, morphismes asymptotiques et K-theorie bivariante, C.R. Acad. Sci. Paris 313 (1990), 101–106.
[Cu]
J. Cuntz, K-theory for certain C*-algebras, Ann. of Math. 113, 181-197.
[D]
M. Dadarlat, C*-algebras and applications, Contemporary Mathematics, vol. 145, Amer.
Math. Soc., Providence RI, 1993, pp. 393–421.
[D1]
M Dadarlat, Shape theory and asymptotic morphisms for C*-algebras, Duke Math. J.
(1994).
[D2]
M. Dadarlat, On the asymptotic homotopy type of inductive limit C*-algebras, Math.
Ann. 297 (1993), 671–676.
[DN]
M. Dadarlat and A. Nemethi, Shape theory and connective K-theory, J. of Operator
theory 23 (1990), 207–291.
[DL]
M. Dadarlat and T. A. Loring, K-homology, asymptotic representations and unsuspended E-theory, to appear in J. Funct. Analysis.
[DL1]
M. Dadarlat and T. A. Loring, Extensions of certain real rank zero C*-algebras, to
appear in Annales de L’Institut Fourier.
[Eff]
E. G. Effros, Dimensions and C*-algebras, CBMS Regional Conf. Series in Math.,
vol. 46, Amer. Math. Soc., Providence R. I., 1981.
[Ell]
G. A. Elliott, On the classification of C*-algebras of real rank zero I, J. reine angew.
Math. 443 (1993), 179-219.
[EG]
G. A. Elliott and G. Gong, On the classification of C*-algebras of real rank zero II,
Preprint 1994.
[EG1]
G. A. Elliott and G. Gong, On inductive limits of matrix algebras over the two-torus,
Preprint 1993.
[EGLP] G. A. Elliott, G. Gong, H. Lin and C. Pasnicu, Abelian C*-subalgebras of C*-algebras
of real rank zero and inductive limit C*-algebras, Preprint 1993.
[EGLP1] G. A. Elliott, G. Gong, H. Lin and C. Pasnicu, Homomorphisms, homotopy and approximations by circle algebras, in preparation.
[G]
G. Gong, On inductive limits of matrix algebras over higher dimensional spaces,I,
Preprint.
[KS]
J. Kaminker and C. Schochet, K-theory and Steenrod homology: applications to the
Brown-Douglas-Fillmore theory of operator algebras, Trans. Amer. Math. Soc. 227
(1977), 63–107.
[Li1]
H. Lin, Approximation by normal elements with finite spectrum in C*-algebras of real
rank zero, Preprint (1992).
[Li2]
H. Lin, Embedding C(S 1 × S 1 ) into purely infinite simple C*-algebras and almost commuting normal elements, Preprint.
[Li3]
H. Lin, Homomorphisms from C(X) into C*-algebras, Preprint.
[LiPh]
H. Lin and N. C. Phillips, Classification of directs limits of even Cuntz-circle algebras,
Preprint.
17
[Lo]
[Lo1]
[Lo2]
[Rø]
[Pa]
[Ros]
[RS]
[Se]
[Sa]
[Su]
[V]
[Zh1]
[Zh2]
T. A. Loring, C*-algebras generated by stable relations, J. Funct. Anal. 112 (1993),
159-201.
T. A. Loring, Stable relations II: corona semiprojectivity and dimension-drop C*-algebras,
Preprint (1993).
T. A. Loring, Normal elements of C*-algebras of real rank zero without finite spectrum
approximants, Preprint.
M. Rørdam, Classification of certain infinite simple C*-algebras, Preprint 1993.
C. Pasnicu, Homomorphisms compatible with covering maps of the two-torus, C. R.
Math. Rep. Sci. Canada 14 (1992), 143–147.
J. Rosenberg, The role of K-theory in non-commutative algebraic topology, Contemporary Mathematics, vol. 10, Amer. Math. Soc., 1983, pp. 155-182.
J. Rosenberg and C. Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized functor, Duke Math. J. 55 (1987), 431–474.
G. Segal, K-homology and algebraic K-theory, Lecture Notes in Math no. 575, Springer
Verlag, 1977, pp. 113-127.
N. Salinas, Relative quasidiagonality and KK-theory, Houston J. of Math. 18 (1992),
97–116.
H. Su, On the classification of real rank zero C*-algebras: inductive limits of matrix
algebras over non-Hausdorff graphs, thesis, University of Toronto.
D. Voiculescu, A non-commutative Weyl-Von Neumann theorem, Revue Roumaine
Math. Pures Appl. 21 (1976), 97–113.
S. Zhang, Certain C*-algebras with real rank zero and their corona and multiplier algebras, Part I, Pacific J. Math. 155 (1992), 169–197.
S. Zhang, Certain C*-algebras with real rank zero and their corona and multiplier algebras, Part II, to appear in K-Theory.
Department of Mathematics, Purdue University, West Lafayette, IN 47907 USA