K-Theory 9: 1t7-137, 1995.
© 1995 KluwerAcademicPublishers. Printedin the Netherlands.
~17
Approximately Unitarily Equivalent Morphisms
and Inductive Limit C*-Algebras*
MARIUS DADARLAT
DepartmentofMathematics, PurdueUniversity,WestLafayette, IN 47907, U.S.A.
(Received: May 1994)
Abstract. It is shown that two unital ,-homomorphisms from a commutative C*-algebra C(X)
to a unital C*-algebra B are stably approximately unitarily equivalent if and only if they have
the same class in the quotient of the Kasparov group KK(C(X),B) by the closure of zero. A
suitable generalization of this result is used to prove a classificationresult for certain inductive limit
C*-algebras.
Key words: *-homomorphisms, unitary equivalence, Kasparov groups, C'*-algebras, real rank
zero.
O. Introduction
Let X be a compact metrizable 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 b y / ( - t h e o r y 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 jEll], [Lil], [EGLP1], [EG], [BrD], [G], [R0],
[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 Theorems A and
B, below.
By the universal coefficient theorem for the Kasparov K K - g r o u p s [RS],
Ext(K.(C(X)), t(._I(B)) is a subgroup of K t f ( C ( X ) , B). Following [R0], we
let K L ( C ( X ) , B) denote the quotient of K K ( C ( X ) , B) by the subgroup of pure
extensions in Ext(K.(C(X)), K . _ I ( B ) ) . If K.(C(X)) is isomorphic to a direct
sum of cyclic groups, then K L ( C ( X ) , B) coincides with K K ( C ( X ) , B).
T H E O R E M A. Let X be a compact metrizable space. Let B be a unital C*algebra and let q~, ~: C( X ) --+ B be two unital *-homomorphisms. The following
assertions are equivalent.
* This research was partially supported by NSF grant DMS-9303361.
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MARIUSDADARLAT
(a) [~] = [¢] in K L ( C ( X ) , B).
(b) For any finite subset F C C ( X ) and any e > 0 there exist k E N, a unitary
u ~ Uk+I(B) andpoints X l , . . . , X k in X such that
Nudiag(~(a),a(xl),...,a(xk))u* - diag(¢(a),a(xl),...,a(xk))tl < e
for all a C 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
rl(a ) = diag(a(xl),..., a(xk)), then we say that ~p and ¢ are approximately unitarily equivalent. Theorem A generalizes certain results in [Lil-3] and [EGLP].
If ~ and ¢ are .-monomorphisms and B is a purely infinite, simple C*-algebra,
then one can drop the ,-homomorphism 71from condition (b), see Theorem 1.7.
The apparition of pure extensions in this context is related to the phenomenon of
quasidiagonality, see [Br], [Sa], ~rD]. The author believes that future generalizations of Theorem A should be based on Voiculescu's noncommutative Weyl-Von
Neumann Theorem IV].
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 and B are inductive limits
kr,.
A = li___mAnand B = li__mBn,An = ( ~ M[n,i](C(Xn,i)),
i=l
in
i=l
Suppose that X~,i,Ya,i are finite C W complexes, K°(X~,;) and K°(Y~,i) are
torsion free and supn,i { dim( Xn,i )~dim(Yn,i ) ) < cv. Then a is isomorphic to B if
and only if
(I¢.(a), K.(A)+, e.(A)) u (K.(B), U.(B)+, r,.(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 one-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. Lofing [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 C W complexes. Roughly speaking,
this classification result is achieved in two stages. The first stage corresponds to
APPROXIMATELY UNtTARILYEQUIVALENTMORPHISMS
~_19
passing from K-theory 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 [Eft].
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 metrizable space and
let B be a unital C*-algebra. For any finite subset G C C ( X ), any ~ > 0 and any
unital ,-homomorphism p: C( X ) -~ B, there are two unital ,-homomorphisms
with finite-dimensional image, r/:C(X) --+ M s - I ( B ) and ~ : C ( X ) ---+ Ms(B),
such that
[[diag(p(a), ~/(a))- ~(a)[[ < e
for all a E G.
The following result is implicitly contained in [EG].
THEOREM 1.2, Let X be a finite, connected C W complex. For any finite subset
F C C ( X ) a n d a n y e > O, thereexistr E N, aunital*-homomorphismr:C(X) --+
MT_I(C(X)) and a unital ,-homomorphism #: C ( X ) --+ M r ( C ( X ) ) with finite
dimensional image, such that
Ildiag(a,r(a))-/*(a)[[ < e
for all a E F.
Proof. By [DN] there exist/~ C ~ and a unital *-homomorphism cr: C ( X ) -+
M R - I ( C ( X ) ) such that ~b:C ( X ) -+ MR(C(X)),definedby ~b(a) = diag(a, a(a))
is homotopic to an evaluation map a ~-+ a(zo)lR. Let ~: C ( X ) --+ M R ( C ( X x
[0, 1])) be a ,-homomorphism implementing a homotopy from ~b to the evaluation map a ~ a(x0)lR. Since ~(a)(a:, 1) = a(z0)lR for all a E C ( X ) ,
one can identify ~I~ with a ,-homomorphism ~: C ( X ) -+ MR(C(f()) where
= X × [0, 1]/X x {1} is the cone over X. Moreover, the natural inclusion
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MARIUSDADARLAT
X -~ X x {0} ~ A? induces a restriction ,-homomorphism p: M R ( C ( f ( ) ) ---+
M n ( C ( X ) ) and ¢ factors as ¢ = p~/. Set G = ~ ( F ) . Since J~ is contractible we
can use Theorem 1.1 to produce unital ,-homomorphisms
~ M(s_I)R(C(X)),
(:Mn(C(2))~
MsR(C(X))
with finite-dimensional image such that
[[diag(p(d),zl(d))- ~(d)[[ < E
for all d E G. Hence
[Idiag(p
(a),
7¢l(a))
-
(a)ll <
for all a E F. Set r = sR, r ( a ) = diag(a(a), 7/krl(a)) and #(a) = ~9(a). Then
[[diag(a, r(a)) - ~(a)l t < e
for all a E F.
[]
DEFINITION 1.3. A C*-algebra A is said to have property ( H ) , if for any finite subset F C A and any e > 0, there exist r E N, a .-homomorphism 7-: A --+ M r - 1(A)
and a .-homomorphism #: A --+ M r ( A ) with finite-dimensional image such that
Ildiag(7-(a),a)-
(a)ll < E
for all a E F.
It is clear that any finite dimensional C*-algebra has property (H). Theorem 1.2
shows that if X is a finite C W complex, then C ( X ) has property (H). It is not
hard to see that the class of C*-algebras with property ( H ) is closed under direct
sums and tensor products.
For C*-algebras C, D let Map(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/~ > 0 we
say that ~p E Map(C, D) is 6-multiplicative on G if [l~(ab) - ~(a)~(b)l I < 6 for
all a, b E G.
LEMMA 1.4. Let A be a C*-algebra with property ( H ). Let e > 0 and let F C A
be a finite set. There are ~ > 0 and a finite subset F C A such that if B is any unital
C*-algebra and qa0, ~Pl, • • •, ¢Pn is a sequence o f maps in Map(A, B) such that ~oj is
6-multiplicative on [" for j = 0 , . . . , n, then there exist k E N, a ,-homomorphism
7: A --+ Mk( B) with finite dimensional image and a unitary u E Uk+l( B) such
that
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APPROXIMATELY UNITARILY EQUIVALENT MORPHISMS
]]u diag(q~o(a), zl(a))u* - diag(q~n(a), ~/(a))ll
< c + max
max
a ~ F O<~j<~n - 1
l[~j+~(a) - ~j(a)t i
for all a E F.
Proof. For given F C A and E > 0, let r E 1~, 7- and # be as in Definition 1.3.
Then D = #(A) is a finite dimensional C*-subalgebra of Mr(A). By elementary
perturbation theory (see [BraD, there is a finite subset G of D containing # ( F )
and there is ~ > 0 such that whenever E is a C*-algebra and • E Map(D, E )
is ~-multiplicative on G, there exists a ,-homomorphism k~: D --+ E satisfying
I[~'(d) - ~(d)i I < e for all d E G.
For s E N set qo~,j = ~j ® ida: M~(A) --+ Ms(B). It is easily seen that one can
find ~ > 0 and _P C A finite such that if Tj E Map(A, B) is ~-multiplicative on
/~ then ~ , j is ~-multiplicative on G. Since ~ , j is ~5-multiplicative on G, there is a
• -homomorphism ¢j: D ~ M~.(B) such that l l ~ , j ( d ) - ¢ j ( d ) l l < c for all d E G
andj = O,...,n.
Define L, L': A --+ Mnr(B) by
L = diag(~_l,oT-, ~0, ~ - 1 , 1 r , ~ 1 , . . . , ~-1,,~-1r, cpn-1),
L t = diag(~o, ~ - l , o r , ~1, ~-1,17", • •., ~n-1,
~r-I,n-lT)
•
Note that L is unitarily equivalent to U. Thus, there is a permutation unitary
u E Unr+I(B) such that
u d i a g ( r ' , ~,~)u* = d i a g ( ~ , L).
(1)
Let
A=max
max
a E F O<<.j<~n - 1
II~j+~(a)-~j(a)ll
Since II~j+l(a) - ~ ( a ) t l ~ ~ for an a E F
lldiag(~o(a),L(a))
- diag(L'(a), ~ ( a ) ) l l
~< rnax I I ~ j + l ( a ) - ~j(a)ll
=
A.
3
(2)
Using (1) and (2), we obtain
Iludiag(~yo(a),L(a))u* - diag(~n(a),L(a))ll <~ A
for all a E F.
(3)
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MARIUSDADARLAT
On the other hand,
IlL(a) - diag(~,,0#(a),..., qa~,~_l#(a))[I
= [[diag(~,0(r(a) • a - Iz(a)),..., qa,,~-l(r(a) • a - ~(a)))ll
~< liT(a) • a - ~ ( a ) l l < E
(4)
for all a E F, since ~ , j are norm decreasing. Note that I I ~ r , j / 4 a ) - C j ~ ( a ) f l <
for all a E F since # ( F ) C G and II~,,j(d) - CAd)ll < : for all d E G. This
implies
Ildiag(qo~,o/~(a),..., qo~,~_l/z(a)) - d i a g ( ¢ 0 # ( a ) , . . . , ¢~-1~(a))11 < ¢
(5)
for all a E F. The ,-homomorphism defined by ~ = d i a g ( ¢ 0 # , . . . , ¢,~-1#) has
finite-dimensional image. Using (4) and (5) we obtain
I l L ( a ) - ~(a)ll < 2e
(6)
for all a E F. Combining (3) and (6), we find
[[u diag(qo0(a), o(a))u* - diag(qa,~(a), ~(a))l[ < 4e + A
for all a E F.
[]
Suppose that the C*-algebra A is unitai. Suppose that the ,-homomorphism #
from Definition 1.3 and the maps ~j are unital. Then it easily seen that one can
arrange for the ,-homomorphism r/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
~t: A ~ B , t E T = [1,oo), which satisfies asymptotically the axioms for ,homomorphisms. A homotopy of asymptotic morphisms ~t, ¢~: A --~ B is given
by an asymptotic morphism ~t: A ~ B[0, 1] such that ~ 0 ) = ~t and ~ l ) = ¢~.
Here B[0, 1] denotes the C*-algebra of continuous 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 ~t by [[~]]. Two asymptotic morphisms ~t, ~bt are said
equivalent if ~ t ( a ) -- Ct(a) ~ 0, as t ~ oo for all a E 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 contractive completely positive linear asymptotic morphism unless stated otherwise. Let Moo denote the dense
APPROXIMATELY UNITARILY EQUIVALENT MORPHISMS
123
• -subalgebra of the compact operators E obtained as the union of the C*-algebras
M,~. Using approximate units it is not hard to see that any asymptotic morphism
from A to B ® E is equivalent to an asymptotic morphism ~t: A ~ B ® Moo for
which there is a function a: T ~ N such that (pt(A) C B ® M~(t). The map ct is
called a dominating function for 7~t. This applies also to homotopies and yields a
bijection
[[A,B ® M~]] ~ [[A, B ®/C]].
We consider here only asymptotic morphisms that are dominated by functions
a as above. Recall that if A is nuclear, then the Kasparov group K K ( A , B) is
isomorphic to [[SA, S B ®/C]] (see [CH]).
Let X be a finite connected C W complex with base point x0 and let B be a unital
C*-algebra. Then by the suspension theorem of [DL], [[Co(X\xo), B ® Mc~]]
K K ( C o ( X \ x o ) , B). Let ~t: Co(X\xo) ~ B ® M ~ be an asymptotic morphism
and let a be a dominating function for ~t. For each t E T we let ~ ' : C ( X ) --,
B ® M~(t) denote the unital extension of Tt: Co(X\xo) ~ B ® Mc~(~) with
~p~(1) = 1B ® 1~(~). Note that i f a ( t ) ~</3(t) then p~ = 1B ® l~(t)qot~lB ® l~(t).
THEOREM 1.5. Let X be a finite connected C W complex with base point xo and
let B be a unital C*-algebra. Let ~t , Zbt:Co( X \ xo) ~ B ® M ~ be two asymptotic
morphisms. Suppose that [[~t]] = [[g3t]] in K K ( Co( X \ xo ) , B ). Then for any finite
set F C C ( X ) and any e > 0 there are to >>.1 and maps a,/3: [1, c~) ~ N with
dominating both qot and ~bt such thatfor any t >1to there exist a unitary u E U ( B ®
M~(t) ® Mp(t)+l) anda unital *-homomorphism rl: C( X ) ~ B ® M~(t) ® M~(t)
with finite dimensional image such that
Iludiag(~(a),~(a))u*- diag(¢~(a), ~/(a))ll < e
for all a E F.
Proof. Using the suspension theorem in E-theory of [DL], we find an asymptotic
morphism ~t:Co(X\xo) ~ B[0, 1] ® M ~ such that ¢~I°) = 7~t and ~}1) = Ct.
Let a: [1, oc) ---> N be a function dominating 7~t, Ct, and ~t. We are going to
use Lemma 1.4 and the notation from there. We find to such that if ~t: C ( X ) --+
(B[0, 1] ® M ~ ) ~ is the unital extension of ~t, then ~t is (~-multiplicative on P
for all t ) to. Since the image of ~ commutes with i s ® la(t), it follows that
• ~: C ( X ) --. B[0, 1] ® M~(t) is ~-multiplicative on P for all t/> to. Let t ) to be
fixed. By uniform continuity we can find a sequence of points so = 0 , . . . , Sn = t
in the unit interval such that
max
max
a E F O<~j<~n - 1
H~s3)'~(a)-
< ~.
By applying Lemma 1.4 for the sequence of unital maps ~I "~)'~ we find a unital
,-homomorphism ~/with finite dimensional image and a unitary u such that
[lu d i a g ( ~ ( a ) , 'q(a))u* - diag(~b~(a), da))tl < 2E
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MARIUSDADARLAT
for all a E F.
[:3
In the proof of Theorem A we are going to apply Theorem 1.5 for ,-homomorphisms. Note that if ~p: C(X) ~ B is a ,-homomorphism, whose restriction to
Co(X\xo) is denoted by ~ too, then ~ ( a ) = ~(a) + a(xo)(la(t) - ~(1)).
1.6.
THE PROOF OF THEOREM A
Recall from [F] that an extension of Abelian groups
O ~ K -+ G-L, H -+ O
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(//, K). The
universal coefficient theorem of [RS] gives an exact sequence
0--+ Ext(K.(A),K(B)._I)-+ 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 K K ( A , B) by the subgroup
of pure extensions in Ext(K.(A), K._I(B)) is denoted by KL(A, B) [Re]. The
subgroup of pure extensions was studied and characterized in the setting of [BDF]
in [KS].
(a) =~ (b) Suppose first that X is a finite C W complex. Since K . (C (X)) is finitely generated any pure extension of K . ( C ( X ) ) is trivial, hence K L ( C ( X ) , B) =
K K ( 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 E N, a ,homomorphism ( : C ( X ) ~ MR(C1B) and a unitary v E UR+I(B.) such that
vdiag(~(ei),~(ei))v* = diag(~b(ei),~(ei)) for i = 1 , . . . , r a . Note that since
and ¢ are unital, ~ can be chosen to be unital. After replacing ~o, ¢ by v diag(~y, ~)v*
and diag(¢, ~), we may assume that ~(ei) = ~b(ei), i = 1,..., m. Let ~oi,¢i denote
the restrictions of !P and ¢ to C(Xi). Then [~i] = [~bi] E KK(C(Xi), B). Let
> 0 and let Fi = F N C(Xi). Using Theorem 1.5 for each i we find k(i) E N,
a ,-homomorphism r/i: C(Xi) ~ Mk(o(B) with finite dimensional image and a
unitary ui E Uk(i)+l(B) such that
Ilu~ diag(Ti(a), 71i(a))u~ - diag(~bi(a), ~(a))ll <
for all a E Fi. Let k = k(1) + ... + k(ra) and define r/': C(X) = @iC(Xi) -+
Mk(B) by r/'(al,...,a,~) = diag@l(al),...,r/m(am)). By conjugating (ul,
1 , . . . , 1) by suitable permutation unitaries we find unitaries vl E Uk+l(B) such
that
Ilvi diag(~vi(a), ~'(a))v~ - diag(¢i(a), ~/(a))ll < 6
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APPROXIMATELYUNITARILYEQUIVALENTMORPHISMS
for all a E Fi. Let Pi = diag(~i(ei), 7/'(ei)) = diag(¢i(ei), rl'(ei)). By choosing
small enough we can perturb v~ to unitaries wi E Uk+I(B) such that wipiw* = Pi
and
liwi d i a g ( ~ ( a ) , ~f(a))w~ - diag(~bi(a), ~/'(a))ll < e
for all a E F / , i = 1 , . . . , m . Letp = lk --~'(1). Let u E Uk+I(B)be the unitary
w~pl + . . . + Wmpm + diag(O, p). Fix x0 E X and define r/: C ( X ) --~ Mk(B) by
~?(a) = ~'(a)+a(xo)p. Then ~7is a unital ,-homomorphism with finite-dimensional
image and
llu diag(
(a),
rl(a))u*
-
diag(¢(a),
(a))ll < e
for all a E F.
In the general case, we embed X into the Hilbert cube I ~° and write X =
MX~, X~ = P~ × I "~n where R, are finite C W complexes with P~+I C P~ × I
and w~ = {n + 1, n + 2,...}. Let gi: I ~ ~ 1 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 {gl,g2,...,g~} for some n. Let now n be fixed and choose m /> n
such that for any y E P ~ there is x E X such that lgi(x) - gi(Y)] < e for
i = 1 , . . . , n. Let Pro: C(Pm) ---, C ( X ) be induced by the X ~ P m × I ~m -+
Pm with the second arrow standing for the canonical projection. Consider the ,homomorphisms ~PPm,ePm: C(Pm) --+ B, the finite set {91,... ,gn} C C(Pm)
and ¢ > 0. Since the Kasparov product induces a product at the level of the
KL-groups [R0] and KL(C(Pm), B) = K K ( C ( P m ) , B), it follows that the , homomorphisms ~Pm, zbpra give rise to the same KK-element. By the first part
of the proof we find a unital ,-homomorphism r/0: C(Pm) --+ Ma(B) with finitedimensional image and a unitary u E Uk+I(B) such that
lludiag(~pm(g~), 'qo(g~))u* - diag(¢p,~(gi), ~o(g~))ll <
c
for i = 1 , . . . , n . The map V0 can be expressed as ~]0(a) = E~=la(yj)qj with
yj E Pm and mutually orthogonal projections qj. The integer m was chosen so
that we can find points xj E X with lgi(Yj) - gi(xj)l < c for i = 1 , . . . , n. Define
r/: C ( X ) ---+ Mk(B) by ~(a) = E~=la(xj)qj. Then
diag( (g ),
- diag(¢(gl),
?(gi))ll
< 2v
fori= 1,...~n.
So far we have proven that if [~2] = [¢] E K L ( C ( X ) , B ) , t h e n f o r a n y e > 0and
F C C ( X ) finite, there exist k E H, aunital ,-homomorphism r/: C ( X ) --, Mk(B)
with finite dimensional image and a unitary u E Uk+l (B) such that
Nu diag(~p(a), rl(a))u* - diag(¢(a), ~/(a))ll <
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MARIUSDADARLAT
for all a E F. Next we show that r/ can be chosen of the form r/(a) =
d i a g ( a ( x l ) , . . . , a(xk)) with x l , . . . , xk E X. This is done in two steps.
First we note that if w E Uk+~(B), (: C ( X ) --+ M~(B) is a .-homomorphism
with finite-dimensional image, ¢/= w diag(r/, ~)w*, and ¢t = 1 @ w(u ® 1~)1 @ w*,
then
/l~ diag(~(a), ¢/(a))z2* - diag(¢(a), ~a))ll < c
for all a E F. This remark shows that we have the freedom to change ~ by taking
direct sums and conjugating by unitaries. The proof of (a) =~ (b) is completed by
using the following elementary Lemma:
LEMMA. Let ~/: C ( X ) --+ Mk( B ) be a unital ,-homomorphism with finite dimen-
sional image. Then there are a unital ,-homomorphism with finite-dimensional
image ~: C( X ) -+ M r ( B ) and a unitary w E Uk+r( B ) such that w diag(~, ~)w*
is equal to a .-homomorphism of the form ~)(a) = diag(a(x 1), • • •, a(x k+r)).
Proof There are distinct points Xl,...,Xn in X and a partition of lk into
mutually orthogonat projections P l , - . . , P~ such that r/(a) = E~=la(xi)p~ for all
a E C(X). The proof is done by induction after n. Define ~(a) = a(xl)p2 +
a(x2)Pl + ~=3a(xi)Pi, ~t(a) = a(xl)(Pl +/92)+ ~n=3a(xi)Pi,~t(a) = a(x2)(Pl+
P2) + E~=3a(xi)Pi. Then it is easily seen that diagQ/, () is unitarily equivalent to
diagQf, ~'). The formulae for 7/' and ~t involves n - 1 distinct points each.
(b) =~ (a) This is a generalization of Proposition 5.4 in [R¢]. The proof is based
on an adaptation of the argument given in [Re] and it remains valid if we replace
C ( X ) by any C*-algebra in the class A/" of [RS]. To save notation set A = C(X).
The first step is to show that K,(¢p) = K,(~b). 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 = [qD]- [¢] lies
in the image of E x t ( K , ( A ) , K , _ I ( B ) ) in K K ( A , B). It is shown in [Re] that the
element z is given by
0 ~ K . ( S B ) -+ K . ( E ) ~r, K . ( A ) --+ 0
(7)
where E is the C*-algebra of all pairs (.f, a), where a E A, f E B[0, 1], f ( 0 ) =
~(a), f ( 1 ) = ~b(a). The map 7r:E -+ A is given by ~r(f,a) = a. Our aim is to
show that z corresponds to a pure extension. By assumption there exist a sequence
of .-homomorphisms ~/i: C ( X ) --+ Mn(i) with finite dimensional image and a
sequence of unitaries ui E Ura(i)(B) where m(i) = 1 + n(1) + - . . + n(i), such
that if
qoi = diag(~, ~/h..., r/i),
~bi = diag(¢, 7h,..., ~/i)
then
lim Ilui~i(a)u7 - ~b~(a)N = 0
i--+oo
(8)
127
APPROXIMATELYUNITARILYEQUIVALENTMORPHISMS
for all a C C ( X ) . Note that K,(cpi) = K.(~bi), As above we have a 'difference'
extension
~(~)
o ~ M.~(i)(SB) ~ Ei ---+ A ~ 0
corresponding to the pair ~0i, ~'i. There is a natural embedding 7i: Ei -+ Ei+I,
7 i ( f , a) = ( f ® ~li+l( a), a). This yields a commutative diagram
0-
, M~(i)(SB )
-
•
Ei-,,
,(i)
,A
, 0
ida
0
'~ Mm(i+I)(SB)
' ~i+1
~(i+1) . A
o, 0
Taking the inductive limit of these extensions we obtain an extension
O -+ IC ® S B ~ Eoo ~% A --+ O
whose K K - t h e o r y 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 IRe] one shows that if g E K , ( A ) and n9 = 0 for some n
then there is i >/ 1 and h E K , ( E i ) with nh = 0 and 7r(i),(h) = g. This shows
that the group extension
0 ~ K.(B)-+ K.(Eoo)~
K.(A)--+ 0
is pure. We conclude that the extension (7) is pure.
O
Another proof of (b) ~ (a) can be obtained by constructing a sequence Xi E
~ 0, as i ~ ~ ,
Map(A, E ~ ) such that Ir~xi = idA and IIx~(ab) - x d a ) ~ d b ) l l
for all a, b C A. This goes as follows. We may assume that there is a path ofunitaries
ui(s), s E [0, 1], in U~(1)(B) with u d 0 ) = I and u d l ) = ul. Let al E Map(A, E~)
be defined by
ai(a)(s) = ~ (ui(28)~i(a)ui(2s)*,a)
for0 ~< 8 ~< 1/2
[ ( ( 2 - 2~)ui~oi(a)u7 + (2~ - 1)¢;(a), a) for 1/2 ~< ~ ~< 1
Let 0i denote the embedding of Ei into E ~ . It is not hard to see that the sequence of
maps Xi = 0iai has the desired properties. The sequence X~ 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.
128
MARIUSDADARLAT
THEOREM 1.7. Let X be a compact metrizable space. Let B be a purely infinite,
simple, unital C*-algebra and let qo, ~b: C ( X ) --+ B be two unital ,-monomorphisms. Then [[~]] = [[4]] in K L ( C ( X ) , B ) if and only if ~ is approximately
unitarily equivalent to 4. That is, if and only if there is a sequence o f unitaries
un E U( B ) such that flu~(a)u: - ¢(a)ll -~ O f or all a E C ( X ) .
Proof Fix e > 0 and F C C ( X ) finite. Using Theorem A all we have to do is
to find a unitary wo E U ( B ) such that ilwo(a)wG - ¢(a)ll < 5c for all a E F.
Let k E N,~7:C(X) --, M k ( B ) and u E Uk+I(B) be provided by Theorem A.
Since r/is unital and has finite dimensional image there exist L E N, distinct points
x l , . . . , XL in X and a partition of lk into mutually orthogonal nonzero projections
P l , . . . ,PL E M~:(B) such that
L
rl(a) = ~
a(xl)Pi
i=l
for all a E C ( X ) . D e f i n e 3`: C ( X ) --+ M k + I ( B ) by 7 = u~u*. Set q = 3'(1) and
qi = upiu*. Then
L
url(a)u* = E
L
a(xi)qi
q + E qi = lk+l.
and
i=1
(9)
i=l
We are going to use repeatedly a result of Zhang [Zhl] asserting that a simple,
purely infinite C*-algebra has real rank zero. Since 3' is a monomorphism and B
has real rank zero we can apply Lemma 4.1 in [EGLP] to get nonzero, mutually
orthogonal projections d, d l , . . . , d L in q M~ + l ( B )q such that d + d l + . . . + d L = q
and
7(a) - d T ( a ) d - EL a(xi)di
< ~
(lO)
i=1
for all a E F. Similarly, there are nonzero, mutually orthogonal projections
e, el, . . . , eL in B such that c + el + . . . + eL = l a n d
¢(a)
L
,
Off)
i=I
for all a E 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 E 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 el by
nonzero subprojections ei <~ ci and c by e = 1 - el . . . . .
eL. Therefore we may
assume that di is equivalent to c~ in Mk+l (B). Since Mk+l (B) is simple and purely
infinite, d i + qi is equivalent to a subprojection of di. Similarly ci ® Pi is equivalent
APPROXIMATELY UNITARILY EQUIVALENT MORPHtSMS
129
to a subprojection of ci. Actually, we can find partial isometries v, w C Mk+I(B)
!
such that v(di + qi)v* + d~ = di and w(ci ® pi)w* + ci = ci for some nonzero
projections d~, e~, i = 1 , . . . , L. Define T.y: C(X) --+ Mzk+z(B) by
(
L
L
L
~)
Tv(a ) = diag d7(a)d* + ~ a(z~)di + ~ a(xi)q,:, ~ a(x;)d
i=1
i=1
i=1
Using v one obtains a partial isometry V C m2k+z(B) with
EL=ld~, VV* = q and such that
.
/
V*V = lk+l +
L
VT, v(a)V* = dT(a)d + E a(xi)di.
(11)
i=l
Similarly, if T0:
C(X) ~ M2k+2(B) is defined by
Te(a) = diag c¢(a)c* + ~ a(z~)c~ ® ~ a(xJpi, ~., a(xg)c
i=1
i=!
then there is a partial isometry W E M2k+2(B),
1B such that
,
i=1
W*W
L
; WW
= lk+l + ~i=lCi,
*
=
L
WT~(a)W* = ce(a)c + ~ a(x~)c~.
(11')
i=1
Nx a C A. From (10), (11) and Off), (11') we obtain
llv*7(a)V - ~:~(a)]l < E,
tlW*¢(a)W
-
(12)
T¢(a)ll < e.
(12')
On the other hand, using (10) and (t0 t) we obtain
IIT.v(a)-diag(u(cp(a)®'rl(a))u*, L~=la(xi)d~) l < e,
(13)
11
(13')
(
Ti~(a) - diag ~(a) ® r~(a),~_a(xi)c
i=1
:)
< e.
By construction [di] = [ci] and [qi] = [Pi] in K0(B). This implies [d~] = [c}]
in Ko(B). Since these are proper projections in Mk+I(B) we can find a unitary
130
MARIUSDADARLAT
uo E Uk+I(B) such that uodiu o* = c~i for i = 1, ... , L (see [Cu], [Zh2]). Let
U = lk+l ® u0 E U2k+2(B). Then
L
-diag(¢(a)®rl(a),~=la(xi)c~) l < e,
since Ilu(qo(a) ® rl(a))u* - ~(a) ®
obtain
llUT(a)U*- T
(14)
(a)ll < ~. Using (13), (13') and (14) we
(15)
,(a)ll < 3e.
Then from (12), (12') and (15)
llfW*u (a)u*Wf*
- W*
(a)Wll <
Finally if we set w0 = WUV*u, then
II 0 (a)w;- ¢(a)ll <
for all a E F.
[3
Certain special cases of Theorem 1.7 have been previously proven in [Lil-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].
Remarks. 1.8. (a) By a result of T. Loring [Lo2], if X is a finite connected
CW complex with base point x0 and O,~ is a Cuntz algebra with n >/ 4 even,
then all the elements of KK(Co(X\xo), 0,~) are induced by ,-homomorphisms
Co(X\xo) --+ On. By combining this result with Theorem 1.7 we obtain a bijection
between KK(Co(X\xo), 0~) 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\Yo) × IR. Let Q(tI) be the Calkin algebra. There are infinitely
many unitary equivalence classes of unital ,-monomorphisms from C(X) to
Q(11). These classes are parametrized by the Brown-Douglass-Fillmore group
Ext(C(X)) ~ KK(Co(X\xo), Q(H)) ~ Ext,(Z[1/2], Z). However, since all the
extensions in Ext~(Z[1/2], Z) are pure, it follows that KL(Co(X\xo), Q(H)) = 0
and any two unital ,-monomorphisms from C(X) to the Calkin algebra are approximately unitarily equivalent (see [Br]).
APPROXIMATELYUNITARILYEQUIVALENTMORPHISMS
131
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*algebras in D will be denoted D. A C*-algebra will be called an AD-algebra if it
is isomorphic to an inductive limit of C*-algebras in D.
By the nonunital dimension-drop interval we mean
~,~ = { f E C([0,1],M,~) { f(O) = O, f(1) is scalar}.
The unital dimension-drop interval ~,~ is the unitization of ~,~. The K-theory of lI,~
is easily computed, K0(]i,~) = 0, Kl(lIm) ~ Z/ra.
LEMMA 2.1. Let X, Y be finite, connected C W complexes and let k, m, r E N.
Let 7: M k ( C ( X ) ) --+ P M m ( C ( Y ) ) P be a unital *-homomorphism where P
is a projection in Mm(C(Y)). Let u: Mk(C(X)) --, Mkr(C(X)) be a unital ,-homomorphism of the form u(a) = diag(a,a(xl),...,a(xr_l)),xi E X.
Suppose that rank(P) > lOkrdim(Y). Then there are *-homomorphisms 70,
71: )dk(C(X)) --+ P M m ( C ( Y ) ) P with orthogonal images such that 70 has finite
dimensional image, 71 factors through u and 7 is homotopic to 7o + 71.
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( S2), let ~ > 0 and let X be a finite,
connected C W complex. There is mo such that if m ) too, then for any unital
*-homomorphism a': C ( S 2) ~ Mm( C( X ) ) there exist a unital *-homomorphism
a: C ( S 2) --+ M m ( C ( X ) ) a n d a C*-subalgebra D of Mm(C(X)) isomorphictoa
quotient of a direct sum of matrix algebras over C( S 1), such that (r~ is homotopic
to cr and dist(cr(a),D) < Efor all a E 11.
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 # : C ( S 2) --+
M,~(C(S2)), u'(a) = (a, a(x0),..., a(xo)) is homotopic to a *-homomorphism
u with u(11) approximately contained to within z 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 unital *-homomorphism
a': C ( S 2) -+ Mm(C(X)), m ) rao, is homotopic to a direct sum between a ,homomorphism with finite-dimensional image and a ,-homomorphism that factors
through u.
[]
PROPOSITION 2.3. Let X be a finite, connected C W complex. Let F be a finite
subset of C ( X ) and let ~ > O. Suppose that the K-theory group Ko( C ( X ) ) is torsion free. Then there exist r C N, a *-homomorphism r/: C ( X ) ~ M~_I(C(X))
with finite dimensional image and a C* -subalg ebra D C M~ ( C ( X ) ) with D E
such that
dist(a ® r/(a), D) <
132
MARIUS DADARLAT
for all a E F. The ,-homomorphism 77can be chosen of the form
rf(a ) = diag( a( x l ) , . . . , a( Xr_ l ).
Proof There are c, d/> 0, re(i) /> 2 with Ko(C(X)) TM Z c+l, K I ( C ( X ) ) -~
Z d @ Z / m ( 1 ) ® - - - ® Z/m(k). Let N = c + d + k and set
Bj = Co(S2\pt) if 1 ~< j ~< c,
Bj = Co(S1\pt) i f c < j ~< c + d,
Bj = ]Im(j_c_d) if c + d < j ~ N,
B = B1 @ ' " ® B:v.
If K 1 ( C ( X ) ) is torsion free, then we consider only Bj for 1 ~< j ~< c + d. By
construction, K.(Co(X\xo)) is isomorphic to K.(B). The universal coefficient
theorem of [RS] shows that Co(X \:co) 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 qPt: Co(X\:co) ~ B @ Moo yielding a KK-equivalence.
Let X C K K ( B , Co(X\xo)) be such that X[[~¢]] = [[idco(x\x0)]]. The element X
can be written as X = X~ + "" "XN with Xj E KK(Bj,Co(X\:co)). It is known
that every element in K K ( B j , Co(X\:co)) can be realized as a ,-homomorphism
Bj --. Co(X\:co) ® ML for a suitable integer L.
[Co( SI\pt), Co(Xkxo) ® Moo] ~- KK(Co(SI\Pt), Co(X\xo)) [Ros],
[Co( S2\pt), Co(Xkxo)® Moo] -~ KK(Co( S2kPt), Co(X\xo)) [Se], [DN],
[~m, Co( X \ xo) ® Moo] ~- g g (]Im, Co( X \ :co)) [DL].
It follows that there are ,-homomorphisms ¢j.0. Bj --+ Co(Xkxo) ® ML with
[~b°] = Xj. Let ~)j: f3j -+ C ( X ) ® ML be the unital extension of ¢o and let ¢j
denote the ,-homomorphism ~)j = ~j ® idoo " B j ® Moo -+ C ( X ) ® ML ® Moo.
Define
N
¢: (~(J[3j @ Moo) -+ C ( X ) ® ML ® MN® M~o
j=l
¢(bl,...
, bN) = d i a g ( ~ b l ( b l ) , . . . , ~)N(bN)).
Form the diagram
Co(X\xo)
®j(By ® Moo)
"~ ,, C ( X ) ® ML ® MN ® Moo
APPROXIMATELYUNITARILYEQUIVALENTMORPHISMS
~33
where 7(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 hornotopy
of asymptotic morphisms ff~: Co(X\xo) -+ C(X) ® ML[O, 1] ® MN ® m ~ with
=
and
=
Let F C C(X) and ~ be as in the statement of Proposition 2.3. Let F and
be given by Lemma 1.4. As in the proof of Theorem 1.5, let ~ be a dominating
function for ~ and find to such that ~ is $-multiplicative on _P for all t ~> to.
Now let t >/to be fixed. By increasing ~(t), we may assume that the image of~t is
contained in ®~Y=I/)J ® M~(~). Moreover we may assume that ~2~ is ~-multiplicative
on _~. We obtain the following diagram
C ( X ) --
~
• C ( X ) @ ML (~ MN ~ Ms(t)
where the superscript ~ indicates that the corresponding maps have been unitalized as described before Theorem 1.5. The restriction of ¢ to @j (/~j ® Ms(t) )
is denoted by ¢ too. Note that ~0),~ = 7~ and ~1),~ = ¢~p~,. The next
step is to modify the above diagram so that ¢ will become homotopic to a ,homomorphism which factors approximately through an algebra in 79. Set C =
~j=IN (/~j Q m~(~)), E = C(X) ® ML ® MN ® Ms(t) and H = ~p~(F) (recall that
t is fixed). By using Lemma 2.2 we find m, such that ifT0: E --+ Mm(E) is defined
by 7o(c) = diag(e, c(xo),..., c(xo)), with x0 6 X, then the ,-homomorphism 70¢
is homotopic to a ,-homomorphism a: C -+ Mm(E) such that a(H) is approximately contained, to within e, in a subalgebra D of M,d.(E) with D E/5. Actually,
one applies Lemma 2.2 for the partial ,-homomorphism 70¢j, 1 ~< j ~< c. Let
el: C --+ ~fm(E)[O, 1] be a homotopy of ,-homomorphisms with k9(°) = 7o¢ and
~(1) = a. Next we consider the path of maps F(s) in Map(C(X), Mm(E)) obtained
by the juxtaposition o f T 0 ~ ~)'~ with ~ ( s ) ~ . It is clear that 7o7 `~is the initial point
of F (~) and a ~ is the terminal point. Since ~ and ¢p~ are $-multiplicative on
/~, it follows that [(~) is ~-multiplicative on/~ for each s. This enables us to use
Lemma 1.4 to show that 7o7 ~ is stably approximately unitarily equivalent to t r ~ .
Namely, there exist S 6 I~, a ,-homomorphism ~/0: C(X) ~ MS-I(Mm(E)) with
finite-dimensional image and a unitary u E Us(Mm(E)) such that
ltu d i a g ( a ~ ( a ) , 7/o(a))u* - diag(707~(a), ~/0(a))tl < 2e
for all a E F. Note that 707 ~ is a unital *-homomorphism of the form
a ~ diag(a, a(xo),..., a(xo)) = diag(a, ~1(a)).
134
MARIUSDADARLAT
Since ~ ( F )
it follows that,
to within 3E in
in the proof of
C H and a(11) is approximately contained to within c in D,
for all a C /;,diag(a, r/l(a), r/0(a)) is approximately contained
u(D @ image(r/0))u* E /). By using the Lemma that appears
Theorem A, we see that ~0 can be taken of the form r/0(a) =
diag(a(xl),...,a(xr_l)).
[~
THEOREM 2.4. Let A be a C*-algebra of real rank zero. Suppose that A is an
inductive limit
k~
A : li_m(A~, 7m,~), A,, : ( ~ M[n,i](C(X,~,i)).
4=1
Suppose that each Xn,i is a finite connected C W complex, Ko( C( Xn,i) ) is torsion
free and d = supn,i { dim( Xn,i ) ) < 0o. Then A is isomorphic to an AT) algebra.
Proof. Let e,~,i denote the unit of A~,~ = M[~,i](C(X,~,i)) and let 7,~i,n: A~,~ --+
Am,j be the partial ,-homomorphisms of 7,~,~. The C*-algebras in 79 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 C An,i and e > 0, there is m/> n, such that for
any 1 <~j <~km there exist a C*-algebra D E D, and a .-homomorphism a: D -+
fjA,~,jfj, fj = .~j,i (e ~ such that dist(7~i,~(a), a(D)) < e for all a E F. By
Theorem 1.4.14 in [EG] we can assume that F is weakly approximately constant
to within ~. To simplify notation say A~,i = Mk(C(X)). By Proposition 2.3 there
exist r E N and a unital .-homomorphism u: M k ( C ( X ) ) --+ Mkr(C(X)) of the
form u(a) = diag(a, a(xl),.., a(x~-l)), and there is a unital .-homomorphism
~: D --+ Mk~(C(X)) with D E 79 such that dist(u(a), ( ( D ) ) < e for all a E F.
By Theorem 2.5 in [Su] and Lemma 2.3 in [EG] there is ra >i n so that for each
j, 1 <~j <~km, one of the following conditions are satisfied.
(i) There is a ,-homomorphism #: A~,i --* fjAm,jfj with finite dimensional
(a)ll < E for all a E F.
image such that
(ii) rank(7~i,~(e~,i)) = rank(fj) = kq where q > lOrd.
If we are in the first case, then we are done. Thus we may assume that (ii) holds.
Define uj: M k ( C ( X ) ) ~ Mk~(C(X)) @ Mk, uj(a) = u(a) ® a(xo). We apply
Lemma 2.1 for 7~,r~ and uj. Hence we find a .-homomorphism ~bj: Mkr(C(X)) ®
Mk --+ fjA,~,jfj such that Cjuj is homotopic to .m,,~'v~"as .-homomorphisms from
M~(C(X)) to fjAm,jfj. Since F is approximately weakly constant to within e
and A has real rank zero, by Theorem 2.29 in lEG], there exist s ) m and a unitary
u E 7~m(fj)A~Ts,m(fj) such that
IluTs,m~pjuj(a)u* - %,,~7~i,~(a)I1 < 70e
APPROXIMATELYUNITARILYEQUIVALENTMORPHISMS
135
for all a • F . Define ~/: D O Mk --* M k r ( C ( X ) ) O M k , { j ( d , A ) = { ( d ) ® A. The
various maps that are involved here can be visualized on the diagram
F (. . . . . . . . . .
An,i
"[rn,n
""/s,m
"" f j A m , j f j ~ "
As
D®M~
Since dist(uj(a), { j ( D ® M k ) ) < s for all a E F , if we set cr = u(%,mCj~j)u*,
then
dist(%,mT~*n(a),c~(D)) < 100&
This concludes the proof.
[]
In T h e o r e m 2.4 one can allow d = ec under the additional assumption that A
has slow dimension growth. A similar remark applies to T h e o r e m B.
2.5 THE PROOF OF THEOREM B
Let A and B be as in the statement o f Theorem B. Theorem 2.4 shows that A and B
are A D algebras. By Theorem 7.1 in [Ell], ordered, scaled K - t h e o r y is a complete
invariant for the simple A D algebras o f real rank zero.
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[BDR]
[Bra]
[Br]
[BDF]
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