K-Theory 6: 267-284, 1992.
© 1992 Kluwer Academic Publishers, Printed in the Netherlands,
267
Approximately Central Matrix Units and the
Structure of Noncommutative Tori
BRUCE BLACKADAR, ALEXANDER KUMJIAN,
and M I K A E L R O R D A M
Department of Mathematics, University of Nevada, Reno, NV 89557, U.S.A. and Department of Mathematics
and Computer Science,Odense University, DK-5230 Odense M, Denmark
(Received: February 1992)
Abstract. The property of approximate divisibility for C*-algebras is introduced and studied. Simple
approximately divisible C*-algebras are shown to have nice nonstable K-theory properties. Nonrational noncommutative tori are shown to be approximately divisible. It follows that every simple
noncommutative torus (in particular, every irrational rotation algebra) has stable rank one and real
rank zero.
Key words. C*-algebra, central sequence, nonstable K-theory, noncommutative torus.
1. Introduction
The nonstable K-theory of simple C*-algebras has been of recent interest; there has
been much work on comparison theory, stable rank, and real rank. The main thrust
of most of this work has been to show that nonstable K-theory is completely
degenerate for many simple C*-algebras. A common technique in much of this work
is to exploit the fact that simple C*-algebras often have large approximately central
matrix subalgebras.
In this paper, we will introduce a version of this property which is present in a
large class of simple C*-algebras, and which implies that nonstable K-theory is
trivial. The results we prove generalize those of [Ro 1], [Ro 21, [BK], [Kw], [Pt],
and ICE], among others.
D E F I N I T I O N 1.t. A finite-dimensional C*-algebra is completely noncommutative if
it has no commutative central summands (i.e. no Abelian central projections).
D E F I N I T I O N 1.2. A separable unital C*-algebra is approximately divisible if, for
every xl .... , x, s A and e > 0, there is a completely noncommutative finite-dimensional C*-subalgebra B of A, containing the unit of A, such that [][xi, ylJt < ~: for
i = 1..... n and all y in the unit bail of B.
The term 'approximately divisible' refers to the (nontrivial) fact that there is a dense
subalgebra of the algebra so that, up to equivalence, for any n every positive element
in the subalgebra can be written as an orthogonal sum of positive elements, each of
268
BRUCE BLACKADARET AL.
which is an orthogonal sum of at least n equivalent positive elements (Corollary
2.10).
Our main results are:
T H E O R E M 1.3. Let A be an approximately divisible (separable unital) C*-aIgebra.
Then
(a) A can be written as [ u A , ] - , where A', c~ A,+I contains a completely noncommutative unitaI finite-dimensional C*-subaIgebra (in particular, the A, are
strictly increasing). Conversely, any (separable unital) C*-algebra which can be
written in such a way is approximately divisible.
(b) There is a decreasing sequence (D,) of unital subalgebras of A such that D, is an
AF algebra with no Abelian projections and A = [w(A c~ D~)]-.
T H E O R E M 1.4. Let A be a simple approximately divisible (separable unital) C*algebra. Then
(a) I f A is finite, it is stably finite. I f A is infinite, it is purely infinite. Any unitat
C*-algebra which is Morita equivalent to A (in particular, pAp for any projection
p ~ A) is approximately divisible.
(b) A has the (SP) property: every nonzero hereditary C*-subalgebra of A contains a
nonzero wojection.
(c) A has stable rank 1 if and only if it is finite.
(d) A satisfies all of the Fundamental Comparability Questions of EB15] (cf FRo 2]),
i.e. comparability of positive elements is completely determined by the dimension
.functions. In particular, if p and q are nonzero projections of A with ~(p) < ~(q)
for all quasitraces r on A, then p -~ q.
(e) A has real rank 0 if and only if the projections of A distinguish quasitraces (i.e. if
"cl and ~z are quasitraees on A with ~I(P) =zz(p) for all projections p ~ A, then
zl = rz.) In particular, if A has a unique quasitrace, then A has real rank O.
(f) I f A is exact, then A has real rank zero if and only if the projections of A
distinguish the tracial states of A. In particular, if A has a unique tracial state,
then A has real rank zero.
[See Section 3 for a brief discussion of comparison theory and the definition of a
quasitrace.]
Many standard simple C*-algebras are approximately divisible. For example, it is
easy to see that every simple unital AF algebra is approximately divisible (cf.
Proposition 4.1); more generally, many (if not all) of the approximately homogeneous
simple unital C*-algebras [-BBEK] with real rank 0 are approximately divisible.
Also, if A and B are separable unital C*-algebras and A is approximately divisible,
then so is A ® B. We also have the following:
T H E O R E M 1.5, Every nonrational noncommutative torus [Rf2] is approximately
divisible. So if A is a simple noncommutative torus, then A has stable rank 1 and real rank O.
269
APPROXIMATELY CENTRAL MATRIX UNITS
The first draft of this paper (written in the fall of 1990) did not include part (f) of
Theorem 1.4, and the last conclusion about simple noncommutative tori in Theorem
1.5 was only proved for a generic bicharacter. In the spring of 1991, U. Haagerup
[ H a ] proved that every quasitrace on an exact C*-algebra is a trace, yielding the
improved results stated here.
The organization of the paper is as follows. In Section 2, we prove the technical
approximation results necessary for the theorems, and obtain Theorem 1.3. Section 3
contains the proof of Theorem 1.4. Then in Section 4, we establish the approximate
divisibility of the noncommutative tori (Theorem 1.5).
To streamline terminology, throughout this paper we will assume, unless otherwise
specified, that all C*-algebras are separable and unital, and that all C*-subalgebras
contain the unit and that all homomorphisms are unitaL
2. Approximation Results
First recall that if A is a C*-algebra and B is a finite-dimensional C*-subalgebra,
then there is a natural conditional expectation En, from A onto B' ~ A defined by
EB,(x) = f
uxu*du,
Jr:(B)
where U(B) is the unitary group of B, with normalized Haar measure du. The proof
of the next simple proposition is left to the reader.
P R O P O S I T I O N 2.1. (a) I f B is a finite-dimensional C*-subaIgebra of A and x ~ A,
then EB,(x) ~ B' c~ A and IIEB,(x)!t <. Itx ll; /f II[x, u] !l < 8 for all u e U(B), then
flx - E~,(x) l! < ~.
(b) I f B1 and B2 are commuting finite-dimensional C*-subaIgebras of a C*-algebra
A, then EBI and E ~ commute.
If B is a finite-dimensional C*-algebra, then a set {eTs} of matrix units for B
satisfies the usual set of identities for all i, j, k, l, r, s:
r , =eik,"
e,sejk
e~sekl~
s = 0
if j ¢ k or r ¢ s,
(e[s)* = es~,' E e'it = 1.
i,r
D E F I N I T I O N 2.2. A set {~efs} of elements of a C*-algebra iE, indexed as for a set
of matrix units for a finite-dimensional C*-algebra B, will be called a set of
approximate matrix units of type B within ~ if the ~er.,japproximately satisfy within
the same identities as the matrix units e[s of B, i.e. Itae,)ce)~-- ~e[klt < e for at1 i, j, k, r,
etc.
L E M M A 2.3. For any finite-dimensional C*-aIgebra B and e > 0, there is a ~ > 0
such that, whenever ~ is a C*-algebra and {ce[s} is a set of approximate matrix units of
type B i n / E within 3, then there is a set {e[s} of exact matrix units for a copy of B in ~E
r
r
satisfying IIeis
- ( e ~s
11< e for all i, j, r.
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BRUCE BLACKADARET AL.
Proof. While an elementary brute-force proof can be obtained (cf. [G1, 1.10],
[Bt, 2.1]), we give a more elegant argument using shape theory [B12]. Fix a set of
matrix units {e~j} of B. For each n, let Cn be the universal C*-algebra generated by
r n )}, indexed as the matrix units of B, such that the f~j(n) are
elements {f~j(
approximate matrix units within t/n. There are natural quotient maps ~z, from
C = C1 onto Cn sending f~j(1) to f~j(n), and a quotient map ~z: C ~ B sending f~(1)
to e~'j. Let J, [resp. J] be the kernel of re, [resp. re]; then J = [ u J , ]
Since B is
semiprojective [B12, 2.30], the identity map from B to C/J can be lifted to a (unital)
embedding ~: B ~ C/Jn for some sufficiently large n. Since f'ij(n) and ~(e~j) have the
same image in C/J, for any e we can find a k such that the image of ~(e~j) in C,+k
differs from f ~ ( n + k) by less than e for all i, ], r. Set 6 = 1/(n + k). If sE is a
C*-algebra and {~e~j}is a set of approximate matrix units of type B in ~ within 6,
then there is a homomorphism of Cn+k into ~E sending f~j(n) to ce~i. The images of
the ~(e~j) under this homomorphism give the required matrix units in ./E.
[]
The next proposition is routine.
P R O P O S I T I O N 2.4. Let B be a finite-dimensional C*-algebra and e > O. Then there
is a ~ > 0 such that, whenever {e~j} are matrix units for a copy of B inside any
C*-algebra ~ , and {ae~j} are any elements of JE with Ilcei~- e~jIt < 6for all i, j, r, then
{ae~'j}forms a set of approximate matrix units of type B within ~ in JE.
P R O P O S I T I O N 2.5. Let B be a finite-dimensional C*-subatgebra of a C*-aloebra A,
with d the vector space dimension of B. I f x ~ A satisfies IIIx, e~-j] II < ~ for all e~j in a set
of matrix units for B, then II[x, y-Ill < de for all y in the unit ball of B.
Proof. There are d matrix units e~i, and every element in the unit ball of B can be
written as Zcq~ei~ with I~j[ ~< 1.
[]
For the rest of the approximation results, it is convenient to work with completely
noncommutative finite-dimensional (cncfd) C*-algebras of limited size:
D E F I N I T I O N 2.6. A standard cncfd C*-algebra is a C*-algebra isomorphic to M2,
M3, or M2 @ M3.
It is easy to see that every cncfd C*-algebra contains a (unital) standard cncfd C*subalgebra. In fact, if n >I 7, then M , contains a unital copy of M2 @ M3. So we have:
P R O P O S I T I O N 2.7. A C*-algebra is approximately divisible if and only if for every
xl ..... x n s A and e > O, there is a standard cncfd C*-subalgebra B of A with
II[Xk, e~j] II < ~ .for all k, i, j, r, for a set of matrix units {e~j} 02["B. Alternately, A is
approximately divisible if and only if there is a sequence (B,) of standard cncfd
C*-subalgebras of A, such that any sequence consisting of a matrix unit chosen from
each B~ is a central sequence in A.
We can now combine the previous results to yield the main technical fact used in
the proof of Theorem 1.3.
APPROXIMATELY CENTRAL MATRIX UNITS
271
L E M M A 2.8. Let A be an approximately divisible C*-atgebra, xl, ..., x~ e A, C a
finite-dimensional C*-subalgebra of A, and e > O. Then there is a standard cncfd
C*-subalgebra B of A, such that II[Xk, y] II < e for all k and for all y in the unit ball of
B, and such that B exactly commutes with C.
Pro@ Set el = e/(78maxkIlx~ll). By Lemma 2.3 choose e2 such that whenever
{£gr~j} is a set of approximate matrix units within e2, of the type of a standard cncfd
C*-algebra, in a C*-algebra Ai;, there are actual matrix units for a standard cncfd
C*-subalgebra of A~ approximating the a~j within e.~.Then by 2.4 choose e3 such that
if {f~j} is a set of matrix units for a standard cncfd C*-subalgebra of a C*-algebra A~;
r
and ce~i
are elements of A~ approximating the f ~r within e,3, then {~ rij} is a set of
approximate matrix units within e2. Finally, let d be the vector space dimension of C
and {c~q} be a set of matrix units for C. Set
6 = rain (e/39, el/d, e3/d)
and choose a standard cncfd C*-subalgebra D of A such that If[Xk, fit] [I < 6 and
r 11< 6 for all k, i, j, r, p, q, s, where {f ~ .j} is a set of matrix units for D.
11[c~, f~j]
Then by 2.5 N[y,f~j][I < rain (el, e3) for all y in the unit ball of C, so if cer.
~J =
Ec,(f~), then lla~j-f~j] t < m i n (el, ea) for all i, j, r. Then {ee;i} is a set of
approximate matrix units of type D within e2 in C' c~ A, so there is a set {e~-j} of
exact matrix units for a C*-subalgebra B of C'c~ A isomorphic to D, with
Ile~j - ~r[~ll < el for all i, j, r. Then
r
r
II[xk, e~j]l; <~ III-X k,f~)]tl
+ It[Xk, eefj- f rij]ll + ]![Xk, e~j--0ef~]ll
< 6 + 2tlXkll Ilcefi- f~jll + 211xkll lle[~ - cer~ll
e
2~
e
~< ~ + 211x~ll(~l + et) <~ ~ + ~ =
for all k, i, j, r. Thus II[Xk, y] I1 < e for all k and for all y in the unit ball of B, since
dim(B) ~< 13.
[]
COROLLARY 2.9. Let A be an approximately divisible C*-algebra, and p a nonzero
projection in A. Then pap is approximately divisible.
Proof. Let x~ .... , x, ~ pap and e > 0. Choose a standard cncfd C*-subalgebra B of
A which approximately commutes with x , , . . . , x , within ~ and which exactly
commutes with p (i.e. with C*(p, 1).) Then pB is a standard cncfd C*-subalgebra of
pAp which approximately commutes with x~ .... , x~ within ~.
[]
Proof of Theorem 1.3. (a) Let A be an approximately divisible C*-algebra. Fix a
countable dense set {x~, x2, ... } in A. Inductively choose a sequence (B,) of standard
cncfd C*-subalgebras of A such that II[x~, y] [I < 2 -~ for 1 ~< k ~< n and all y in the
unit ball of B,, and such that B, commutes with B~,...,B,_~ (C*(B1 .... ,B,,_a)
is finite-dimensional since the B~ commute). Set A, = [ ~ > , B ~ ] ' c~A. It is clear
that A , c _ A , + ~ and that B , + ~ _ A ' , c ~ A , + I (in fact, A,=B'~+~c~A,+~). It
remains to show that [wA,]- = A; it suffices to show that each Xk is in [ ~ A , ] - .
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BRUCE BLACKADAR ET AL.
Fix Xk and e > 0, and choose n ~> k such that 2 - " ~< e. Define a sequence ( y ' ) by
Ym = EB'~+t(EB'.+2(... (EB;,+,,(Xk)).-.)).
Then
IIY" - Ym+l It ~ IIXk -- EBa+m+~(Xk)[[ < 2 -I"+m+l)
since the EB~ are norm-decreasing, and so (ym) converges to some y e A, and
! [ X k - - Y } I < ~ I I x k - - Y l I I + ~ t I Y , . - - Y m + l l l < ~ 2 - ( " + " + 1 ) = 2 - " ~<~ra
m=O
Also, y ' E [Bn+ 1 u "'" U B n + m ] ' N A since the EB~ commute, so y e A,.
The converse statement is trivial.
(b) Let the sequence (B.) be as in (a), and set
D, = [~)kC*(B,, B,+ i,.-., B,+k)]-.
[]
An important consequence of this expression as an inductive limit is the following
observation, which is motivation for the name 'approximately divisible'.
C O R O L L A R Y 2.10. Let A and {A,} be as in 1.3. Then for every n and k, there is a
cncfd C*-subalgebra B of A'. n A such that every central summand of B is at least
k × k. So C*(A,, B) is isomorphic to a direct sum of large matrix algebras over An. In
other words, the inclusion t of A , into A factors as follows, for some kl,..., k~ >1 k:
An
,A
?
1
i~=l id
/
//
i=l
Proof. Fix n and k. For each m, let C" = C*(Bn+3"-2, Bn+3m-l,Bn+3~). Then the
cncfd C*-algebra Cm ~ A~ c~ A n + 3." has all its central summands at least 8 x 8, and
so each central summand contains a (unital) copy of M2 (9 M 3. Putting these
together, we obtain a unital copy of M2 (~ M 3 in C" in which each nonzero
projection has central support 1 in C'. Let Dm be this copy of M2 @ Ma. The D"
commute for different m and if p is a nonzero projection in D ' , then the natural
homomorphism from C*(A,, 1)1,..., 17,,_ 1) to pC*(A,, D1 ..... Din- 1) is injective.
Choose m so that 2m>~ k, and let B = C*(D1 .... ,D'). Then B is finite-dimensional, and each central summand of B has size 2J3" - j ~> 2 " ) k
for some j,
0 ~<j ~< m. The cutdown of A, by any central projection of B is naturally isomorphic
to A,.
[]
Remark 2.11. An almost identical proof to 1,3, using a slight variation of 2.8, shows
that if A is a C*-algebra with a central sequence of nontrivial projections, then A
APPROXIMATELYCENTRAL MATRIX UNITS
273
contains a central sequence of mutually commuting nontriviat projections, and A can
be written as [b,A.]-, where A'. c~ A.+ ~ contains a nontriviat projection. So if A has
no central projections, then A can be written as a proper (strictly increasing)
inductive limit.
If A is approximately divisible, and the approximately central finite-dimensional
C*-subalgebras can be chosen to be full matrix algebras, then A can be decomposed
as a tensor product with a U H F algebra:
P R O P O S I T I O N 2.12. Let A be an approximately divisible (separable) C*-algebra. I f
the approximately central cncfd C*-subalgebras of A can always be chosen to be full
matrix algebras, then A is isomorphic to C ® D, where C is some C*-atgebra and D is a
UHF algebra.
Proof. Given x t , . . . , x , eA, ilxkll ~< 1, given B _ A, B ~ Mm, and given e > 0, we
first show there is an M ~ B' ca A, M - Mr, such that ]l[xk, y] Jl < e for all y in the
unit ball of M. Let {e~j} be a set of matrix units for B. There is an t / > 0 such that
whenever {.)cij} is a set of matrix units of type B in A with IIeij - fii I[ < q for all i, L
there is a unitary u ~ A with Jlu - 1 !1 < e/8 and f~j = u*eiju for all i, j (take u to be the
unitary in the polar decomposition of 2iejlf~j). Then by 2.3 and 2.4 there is a ~ > 0
such that, if N is a full matrix subalgebra of A with I[[xk, y] II < 8/2 and [I[e~j, y] 11< 6
for all y in the unit ball of N, then there are matrix units {f~j} of type B in N' ca A
with I[e~j- J~[I < t/. Let u be the corresponding unitary, and set M = uNu*. Then
M ~_ B'caA and, for y e M ~ ,
II[x,, y] It <. il[x~, u*yu] II + II[x~, y - u*yu] ti
< ~ + 21tXkll IlY]I ~
4
Thus we may find a sequence of mutually commuting full matrix subalgebras (D,) so
Put C = A 1 and note that
that A = [ w A , ] where A , = [ k . @ , D j ] ' n A .
A , = C * ( A , _ I , D , ) ( ~ A , _ I ® D , ) since D, is a full matrix algebra. Then D =
C*(DI, D2 .... ) ~ ®jDj is U H F and A = C*(C,D) ~ C ® D.
[]
The situation of 2.12 is exceptional. Usually the cncfd algebras cannot be chosen to
be full matrix algebras, and in general it is not true that A,, is generated by A,_ i and B,
(this will fail if the center of B, is not contained in the center of A,). However, in some
respects an approximately divisible C*-algebra does behave like a generalized 'tensor
product' with a highly noncommutative AF algebra (one with no Abelian projections).
3. Properties of Approximately Divisible C*-Algebras
In this section, we prove the results about nonstable K-theory for approximately
divisible C*-algebras described in Theorem 1.4. We begin with the following simple
result, which proves part of 1.4(a):
274
BRUCE BLACKADAR ET AL.
P R O P O S I T I O N 3.1. Let A be an approximately divisible C*-algebra. I f A is finite,
then A is stably finite.
Proof. This follows almost immediately from 1.3 and 2.10. Write A = [ u A , ] - as
in 1.3. A is finite [resp. stably finite] if and only if each A, is finite [resp. stably finite.]
Suppose A, is not stably finite for some n, i.e. Mk(A,) contains a proper isometry for
some k. A contains a (possibly nonunital) C*-subalgebra isomorphic to Mk(A,) by
2.10, so A contains an infinite projection.
[]
We now turn to 1.4(b). This part actually follows fairly easily from 1.4(d), but we
prefer to give an elementary argument. This argument is a generalization of
[BK, 1.9] (the core of the argument goes back to [Cu 1]), and does not use the full
strength of approximate divisibility. Recall that an element a e A is full if the
two-sided ideal of A generated by a is all of A, i.e. if there are elements x~, y~
(i = 1.... ,r) with t = E~=lx~ayl. If a ~> 0 is full, then there exist x l , . . . , x , ~ A with
1 = ~ i'= l x i a x i * [ C u 1, 1.10].
P R O P O S I T I O N 3.2. Let A be a C*-atgebra, and a ~ A.
(a) a is futI if and only if a*a is fulL
(b) I f (a.) is a sequence converging to a, and a is full, then a. is full for sufficiently
large n.
(c) I f {A.} is an increasing sequence of C*-subalgebras with A = [ u A . ] - , and
a ~ A. is full in A, then there is an m >1 n such that a is Jull in Am.
Proof. (a) a and a*a generate the same closed two-sided ideal of A by [Pd, 1.4.5].
(b) If 1 =E~=lxiay~, set b, =Z~=~xia, yi; b, is invertible for large n, so
r
1 = El=
l(b, lxi)a,yi.
(c) If 1 = El= lxiayl with xi, Yi ~ A, choose xi, Y~e Am for some m >/n approximating x~, Yi closely enough that b = E~=lYqayi is invertible. Then 1 = Z~=I (b-lYci)aY~
in Am.
[]
For the convenience of the reader, we also recall the basic elements of the
comparison theory of positive elements in a unital C*-atgebra. See [Cu 2], [BI 5], or
[Ro 2] for details.
If a, b e A+, write a ~ b if rjbr* ~ a for some sequence (rj) in A. The relation is
symmetric and transitive. Write a ~ b if a ~ b and b ~ a. If p and q are projections,
then this definition of p ~ q agrees with the usual definition of Murray and von
Neumann, and if A is finite, then so does the definition of p .-~ q. We also have a~~ b~
( i = 1,2) and bl_t_b2 imply at + a z ~ b l +b2 [Cu2], and 11a-b11 < e implies
f~(a) ~ b, where f~ is the continuous function with f~(t) = 0 for t ~ e, f~(t) = 1 for
t/> 2e, and linear for e ~< t ~< 2e.
Pass from A+ to M~(A)+, denote by (a> the equivalence class of a with respect to
~ , and let W(A) be the set of all equivalence classes. Write (a> + (b) = (a + b) if
a _k b, and ( a ) ~< (b) if a ~ b. Then W(A) is a preordered semigroup. The equival-
APPROXIMATELY CENTRAL MATRIX UNITS
275
ence classes of projections form a subsemigroup V(A) on which the induced
preordering is the algebraic ordering.
Recall that a (normalized) quasitrace on A is a monotone, homogeneous function
• : M~o(A)+ --* R+ with ~(x*x) = T(xx*) for all x e M®(A), and ~(a + b) = r(a) + ~(b) if
a, b e M~(A)+ commute (and Z(1A) = 1). Every tracial state defines a quasitrace, and
it is a celebrated open question whether every quasitrace comes from a trace (i.e. is
linear). If A is exact (e.g. if A is nuclear), then every quasitrace comes from a trace by
the remarkable recent result of Haagerup [Ha].
The state space of W(A) is the space of dimension functions DF(A), and the lower
semicontinuous dimension functions can be identified with the space of quasitraces
QT(A) [ B H ] . There is a continuous affine map Z from QT(A) to the state space of
I/(A) defined by ;g(z)([p]) = "c(p); this map is surjective by [BR].
If a sM~(A)+, then there are affine functions 8 and d on QT(A) defined by
~(r) = ~(a) and ~i(r)= d~(a)= sup~z(J;(a)). The function ~ is continuous, and a is
lower semicontinuous but not continuous in general (a = sup~(f,(a))~). If p is a
projection, then/~ = j0 is continuous.
P R O P O S I T I O N 3.3. Let A be a C*-algebra, {A,} an increasing sequence of C*subalgebras with A = [ ~ A , ] - . Suppose, for each n and r, A', c~ A contains a (possibly
nonunital) C*-subalgebra isomorphic to M,. I f a is a full element of A, then [a*Aa]contains a nonzero projection.
Proof. Replacing a by a'a, we may assume a ~> 0, and it suffices to show that p ~ a
for some nonzero projection p in A (if Ilrar* - p Ii < 1/3, then fl/3(al/2r*ral/z) is the
desired projection). If 0 ~< a, e A, with a, ~ a, then a, is full for sufficiently large n.
Put e = ltalt/2, and let n be large enough so that ]tan]] > ~ and ]la - a,!l < 5. Then
f~(a,) ~ a and f~(a,,) ¢ O. We may therefore assume a ~ A,. Then for some m ~> n,
r
• Let {eij: 1 <~ i,j <~r} be a set of matrix
there are x~ ..... xr~A,, with 1 = Ei=lx~axi.
units for a (possibly nonunital) copy of Mr in A',,nA, and set u = E~=ixie1~a 1/2.
Then uu* = e ~ , so u is a nonzero partial isometry; and u*u ~ a~/2Aa ~/2 is the desired
projection.
[]
Proof of Theorem 1.4(a) and (b): Theorem 1.4(b) is an immediate corollary of 3.3, and
the first statement of 1.4(a) is a special case of 3.1. Suppose A is infinite. If p is a
nonzero projection of A, then pAp is not stably finite, and hence is infinite by 3.! and
2.9. Then A is purely infinite since every nonzero hereditary C*-subalgebra of A
contains an infinite projection [Cu 1]. To prove the last statement of 1.4(a), we need
only show that a matrix algebra over an approximately divisible C*-algebra is
approximately divisible. But this is trivial since Mk(A ) ~ A ® Mk.
[]
The arguments used in the proof of 1.4(c) are slight variations of ones in [ R e 1],
and the arguments for 1.4(d) and (e) are variants of ones appearing in [Ro 2]. We
give the relevant statements in the next few lemmas and refer to the original papers
for the proofs.
276
BRUCE BLACKADAR ET AL.
L E M M A 3.4 I-Ro 1, 3.2]. Let A be a C*-algebra, and a ~ A an element which is not
one-sided invertible in A. Then a is a limit of zero divisors in A.
L E M M A 3.5. Let A be a prime C*-algebra, {A,} an increasing sequence of C*subalgebras with A = [ ~ A , ] - , and a a zero divisor in A,. Then for some m >~ n, there
is a unitary u ~ Am such that ua is orthogonal to a nonzero positive element b ~ A,,.
Proof, There are positive elements c and d in A, with ca = 0 = ad. We have
cAd ~ 0 since A is prime, and so cA,,d ¢ 0 for some m ~> n. So if C and D are the
hereditary C*-subalgebras of A,, generated by c and d respectively, by the argument
of [Ro 1, 3.4] there is a unitary u ~ A,, with uCu* n D ~ O. If 0 <~ b ~ uCu* n D, then
b is orthogonal to ua as in [Ro 1, 3.5].
[]
L E M M A 3.6. Let A be a C*-algebra. I f a ~ A is orthogonal to a fidl positive element of
A, then for all sufficiently large k there is a unitary u ~ A ® Mk such that u(a ® lk) is
nilpotent.
Proof. The proof is identical to the proof of [Ro 1, 6.4]. The only place where
simplicity is used in that proof is to obtain that the element c is full.
[]
Proof of Theorem 1.4(c). If sr(A) = 1, then A is finite [Rf 1]. Conversely, suppose A is
finite, and let A = [ u A , ] - as in 1.3. It suffices to show that if a ~ A, and 5 > 0, then
there is an invertibte b s A with Ita - b II < a We may assume a is not invertible, and
hence not one-sided invertible since A is finite. Then by 3.4 there is a zero divisor
d e A, with [[a - d ][ < 5/2. By increasing n if necessary, we may assume that there is a
unitary u e A, such that ud is orthogonal to a nonzero positive element c of A, (3.5).
Since c is full in A because A is simple, we may assume c is full in A, by again
increasing n if necessary (3.2(c)). Then by 3.6 there is a ko such that for all k ~> ko,
there is a unitary Vk e A, @ Mk such that vk(ud ® 1,) is nilpotent. Hence, by 2.10 there
is an m and a unitary v e C*(A,, B,+I, ..., 13,+,,)~_ A such that vud is nilpotent; so
there is an x e A [e.g. x = vud + (5/2)1] which is invertible and satisfies IIx - vud It <~
5/2. If b = u*v*x, then b is invertible and ]la - b I[ < e.
[]
We now turn to the proof of 1.4(d). Recall [Ro 2] that a preordered semigroup S is
almost unperforated if for k, k' s N and s, t ~ S, ks <~ k't and k > k' implies s ~< t.
P R O P O S I T I O N 3.7 [Ro 2, 3.2]. Let S be an almost unperforated preordered semigroup, and s, t ~ S. I f t is an order unit and f(s) < f(t) for every state f on S, then s < t.
L E M M A 3.8. Let A be an approximately divisible C*-algebra. Then W(A) is ahnost
unperforated.
Proof. The proof is very similar to [Ro 2, 5.1]. Write A = [wA,] - as in 1.3. Given
x, y s M ~ ( A ) + and k, k ' E N with k > k' and k ( x ) < ~ k ' ( y ) , then we must show
( x ) ~< ( y ) (i.e. x ~ y). We may assume x, y ~ A, and as in the second half of the proof
of [Ro 2, 5.1] we may assume x, y e A, for some n. Let e > 0. As in [Ro 2, 5.1] by
increasing n if necessary we may assume that k ( f , ( x ) ) <~ k ' ( y ) in W(A,). Choose m
as in 2.10 so that the sizes of the central summands of C*(B,+ 1..... B,+,,) are l~.... , l~
APPROXIMATELY CENTRAL MATRIX UNITS
277
with each l~ >~ (1/k' - l/k)- 1 Then for each i there is a d~~ N with d~k' <~ l~ <~d~k.
Thus l,(L(x))<<, ldy) for each i, and so f ~ ( x ) ~ y in C*(A,, B,+ I,..., B , + m ) c a .
Since ~ is arbitrary, x ~ y in A.
[]
The following corollary provides a more precise and general statement for
Theorem 1.4(d).
COROLLARY 3.9. Let A be an approximately divisible C*-aIgebra.
(a) I f a, b e M~o(A)+ with b full and Ct < 6 everywhere (i.e. dda ) < ddb) for all
z ~ Q T(A)), then a ~ b.
(b) I f p, q are projections in M~(A) with q full and T(p) < ~r(q)for all ~ E QT(A), then
P-~q.
Proof. See [Ro 2, 5.2].
[]
We also have the following facts, which follow immediately from 1.4(c), 2.10,
[B14, 3.1.4], and [B1 5, §7].
PROPOSITION 3.10. Let A be an approximately divisible C*-algebra. Then A has
cancellation for full projections: if p, q are full projections in M~(A), and [p] = [qJ in
Ko(A), then p ,,~ q. I f A is simple, then p and q are homotopic.
PROPOSITION 3.11. Let A be a simple approximately divisible C*-atgebra. Then
the natural map #1: UI(A)/UI(A)o ~ K I ( A ) is an isomorphism, where UI(A) is the
unitary group of A and Ul(A)o is the connected component of the identity.
Finally, we consider 1.4(e) and (f). The next proposition follows immediately from
the Kadison Representation Theorem (cf. [Alf, II.1.8]).
PROPOSITION 3.12. Let A be a compact convex set in a locally convex topological
vector space, and Aft(A) the ordered vector space of real-valued continuous affine
functions on A.
(a) The map x ~ 2, where 2 ( f ) = f(x), is an affine homeomorphism of A onto the
state space of Aft(A).
(b) I f V is a subspace of Aft(A) containing the constant functions, and V separates
the points of A, then V is uniformly dense in Aft(A).
Now let A be a C*-algebra, and Vo be the additive subgroup generated by {/3: p a
projection in A} in Aff(QT(A)). Note that if A is finite, simple, and approximately
divisible, then by 3.9(b) every strictly positive function in Vo is of the form t3 for some
projection p e M~(A). We may also represent the equivalence classes of projections as
a n n e functions on the state space A(Ko(A)); let Vo be the additive subgroup of
ANA(Ko(A))) generated by the images of the projections of A.
PROPOSITION 3.13. Let A be an approximately divisible C*-algebra. Then the
closure of Vo is a vector subspace of Aff(QT(A)), and the closure of Vo is a vector
subspace of Aft(a(K0(A))).
278
BRUCE BLACKADAR ET AL.
Proof. The proofs of the two statements are identical; we prove the first. It suffices
to show that if p is a projection in A and 0 < 2 < 1, then 2/~ is in the closure of Vo.
Write A = [wA,]- as in 1.3. We may assume p e A, for some n. Fix e > 0 and
k > l/e, and choose m so that the summands of B = C*(B,+I .... , B,+,,) have sizes
kl .... , ks with all ki ~> k. Then there are rl,..., rs ~ N such that ](ri/kf) - 2] < e for all i.
Let e~ be a projection of rank r~ in the ith summand of B, and set q = Z~= t pet. Then q
is a projection in A with I[q - , ~ It < a
[]
P R O P O S I T I O N 3.14. Let A be an approximately divisible C*-algebra. Then
(a) ~'o is (uniformly) dense in Aff(A(Ko(A))),
(b) Vo is (uniformly) dense in Aff(QT(A)) if and only if the projections of A
distinguish quasitraces.
Proof. Follows immediately from 3.13 and 3.12(b).
[]
C O R O L L A R Y 3.15. Let A be a finite simple approximately divisible C*-algebra. If
A(Ko(A)) is a simplex (in particular, if the projections of A distinguish quasitraces), then
the ordered group Ko(A) has the Riesz interpolation property.
Proof. Ko(A) has the strict ordering from its states by 3.9(b), and if A is a simplex,
then any dense additive subgroup of Aft(A) with the strict ordering has the Reisz
interpolation property [Alf, 11.3.11]. A(Ko(A)) is a quotient of QT(A) by [BR], and
can be identified with QT(A) if the projections of A distinguish quasitraces. QT(A) is
a simplex by [BH, 11.4.4].
[]
Note, however, that A(Ko(A)) is not a simplex in general, even if A is approximately
divisible (tensor the example of [B1 3, 6.10.4] with a U H F algebra; a modification of
this example, due to Elliott, gives a simple approximately divisible C*-algebra A for
which A(K0(A)) is a square).
Proof of Theorem 1.4(e) and (f). We may assume A is finite, since every purely
infinite simple C*-algebra has real rank 0 by [Zh]. Then the proof of 1.4(e) is
essentially identical to the proof of IRa 2, 7.2], since we have already proved the facts
needed for that proof in 3.9(a), 3.14(b), and 1.4(c). 1.4(f) follows from 1.4(e) and
[Ha].
[]
This completes the proof of Theorem 1.4.
4. N o n c o m m u t a t i v e Tori and Other E x a m p l e s
In this section, we describe some examples of approximately divisible C*-algebras.
We begin with some other examples before dealing with the noncommutative tori.
P R O P O S I T I O N 4.1. Let A be an AF algebra. Then A is approximately divisible if
and only no quotient of A has an abelian projection. In particular, every infinitedimensional simple (unital) AF algebra is aproximately divisible.
APPROXIMATELYCENTRALMATRIX UNITS
279
Proof. It is obvious that no commutative C*-algebra is approximately divisible, and
so by Corollary 2.9 an approximately divisible C*-algebra cannot have abelian
projections. It is also obvious that a quotient of an approximately divisible C*-algebra
is approximately divisible. Conversely, it is an easy exercise to show that an AF algebra
has the property that no quotient contains an abelian projection if and only if it has a
Bratteli diagram in which every nonzero partial embedding multiplicity is at least 2 (in
fact, any diagram has a subdiagram with this property). The corresponding sequence of
finite-dimensional subalgebras has the property of 1.3.
[]
It should be routine to prove a similar result about the C*-algebras of real rank 0
considered in [El 2]. The situation is technically more complicated because one must
simultaneously find diagrams for Ko and KI which are compatible. The following
example should illustrate the worst of the potential problems and the probable
approach necessary to solve the general case.
P R O P O S I T I O N 4.2. Every Bunce-Deddens algebra is approximately divisible.
Proof. We consider only the case of the Bunce-Deddens algebra A of type 2~; the
general case is similar. The difficulty is that the usual inductive system for A, in which
the n'th algebra C, is C(T)® M2- and the connecting maps are twice-around
embeddings, is not a system of the form given in 1.3; in fact, C', c~ A is just the center
of C,. However, we can write A as an inductive limit in a different way. Let
An = C(T)@M16-, and let the embedding of A, into A,+I be a diagonal sum of
three standard twice-around embeddings and two standard (-5)-times around
embeddings. Then lim
A, is isomorphic to A (by [El 2] it is only necessary to check
-->
that it has real rank zero and the same K-theory), and A; c~ An+ ~ contains a copy of
M3 @ My
[]
It is quite possible that every simple approximately homogeneous C*-algebra with
slow dimension growth [BDR] is approximately divisible. Such a C*-Mgebra is
stably finite and is known to satisfy the conclusions of Theorem 1.4.
We now consider the noncommutative tori. Recall that a noncommutative n-torus is
the universal C*-algebra generated by n unitaries ul .... , u, with (ui, u i) = p~jl, where
Pij -- e 2~i°~ is a scalar and (u, v) denotes the multiplicative commutator uvu*v*. The 0ij
define a real bicharacter O on Z", and the corresponding C*-algebra is denoted Ae. If
n = 2, then Ao is a rotation algebra with angle 0 = 012, denoted Ao. Ao is nonrational if
not all 0~j are rational. Ao is simple if and only if O is completely irrational in the sense
that for every nonzero x e Z" there is a y E Z" with ®(x, y) irrational. A rotation algebra
Ao is simple if and only if 0 is irrational; such an algebra is called an irrational rotation
algebra. References for noncommutative tori are [Rf2] and [El 1].
We need some additional approximation lemmas for the proof of Theorem 1.5.
The first is simple, and the proof is left to the reader.
P R O P O S I T I O N 4.3. Let f l,...,fr [resp. gl,..., g~] be polynomials in m noncommutin 9
variables [resp. n noneommutin9 variables], with complex coefficients. Then for any
e > 0 there is a 3 > 0 such that, whenever A is a C*-aIoebra and xt ..... xm, Yl, ..., Y,
280
B R U C E B L A C K A D A R E T AL.
are elements of A of norm 1 with I][xi, )9] II < 6 for all i, j, then IJ[fk(xl,..., X,,),
g~(Yl .... , Y,)] II < 5for all k, I.
P R O P O S I T I O N 4.4. Let 0 be an irrational number, and let ul, Uz be the generators of
Ao. Then for any e > 0 there are polynomials {f[j} in four noncommuting variables,
with complex coefficients, such that ~r f r~i(ul, u2, ui,
• u2)}
• is a set of approximate matrix
units of type M2 ~3 M3 within 5 in Ao.
Proof. W e need only show that Ao contains a C*-subalgebra isomorphic to
M2 @ M3. Since Ko(Ao) is Z + Z 0 as an ordered group, and since Ao has cancellation, we need only show that in Z + Z 0 there are x, y > 0 with 2x + 3y = 1. Let
z ~ Z + ZO with 1/2 < z < 2/3, and set x = 2 - 3z and y = 2z - 1. Then these x, y
have the desired properties. (In fact, one can explicitly write d o w n the matrix units
for M2 (3 M 3 inside Ao in m u c h the same way as the Rieffel projection.)
[]
P R O P O S I T I O N 4.5. Let B be a finite-dimensional C*-aIgebra, 0 ~ R, and 5 > O. Let
udO), Uz(0) be the generators of Ao, and let { f i~} be a set of polynomials in four
noncommuting variables, with complex coefficients, such that {f~j(ul(O),u2(O),u*(O),
u*(O))} form a set of approximate matrix units of type B in Ao within 5/2. Then there is
a 6 > 0 such that whenever 10 - 0'1 < (5, then {f~j(ul(O'), u2(0'), u*(O'), u*(0'))} form a
set of approximate matrix units of type B within e in Ao,.
Proof. This is an immediate consequence of the fact that the rotation algebras
form a continuous field [El 1].
[]
T h e next l e m m a gives the key a p p r o x i m a t i o n result.
L E M M A 4.6. Let v, 23 ..... 2,, # 3 , . - - , # , , c o E T and 0 < ~ < 1. Assume v is of infinite
order. Then there are integers p, q such that Ivp - 11, Ivq - 11, 12~ - 11, 1#q - 11, and
[vpq - col are all less than 5.
Proof. Relabel the 2j a n d #j so that {v, 23,..., 2~} and {~7,# > . . . , Pk} are independent, and
2~.j = v'S°~.~~ . . . 2~"J'
for j > 1,
#~j
forj>k,
=v
-sjo sj3
•
#3 ""#~J~
for r j, rj~, sj, s~i e Z with rj, sj nonzero (k or 1 could equal 2 if all the 2j or N are powers
of v). P u t
R = Irl+lrl+2""r~l,
=max
i,j
ril
1
S = ]Sk+lSk+2""Sn[ ,
,
~,J l l S j
1}
Find an integer qo ~> (2re + 1)cd/eS so that le q° - 11 and I#~° - It are less than e/flkS
for 3 ~< i ~< k, and put q = qoS. T h e n Ivq - 11 a n d I#~ - II (3 ~< i ~< k) are less than
5/Bk; and for j > k,
k
I#~ -
tl ~< I'~q*'°/*' -
11 +
Y I , g ~''¢~' /=3
11 < a
APPROXIMATELY CENTRAL MATRIX UNITS
281
Write co = e 2'~i° where 0 < 0 < 1. N o w find P o e N such that
[rp °
e2~iO/qR [
S
qR
S
I2I'° - 11 < - - ~
for 3 ~< i~< l,
and set p = poR. Then Ivpq - co I < ~ and
2re0 + s
lv p° -
11 < - -
qR
~<
2rc0s + s 2
(2re + 1)cdR
~<
s
o:lR
Hence Iv v - II and 12~' - tl (3 ~< i ~< i) are less than s/el <~ s; and for j > t,
1
[2s-l[~<
-11+ ~
-ll<s.
El
i=3
C O R O L L A R Y 4.7. Let Ao be a noncommutative torus with generators ua,...,u,.
Suppose that p = P12 = e 2"~° with 0 = 012 irrational. Then for any 6 > 0 there are
integers p, q such that vl = u p and v2 = u~ satisfy
lt[vi, uj]ll < 6 for l <<.i <. 2,1<~ j ~ n,
@1, v2) = e2~i¢1 with IO - 0'I < 6.
Proof. Set v = P12, 2j = Ply, ,a~ = P2j for 3 ~ j ~< n (ifn = 2 set 23 and #3 = 1), and
(l)2, Uj) =P2~,
q ( v t , v 2 ) = p pq, and that I][vl, uj][[=
that ( v l , u j ) = p ~ j ,
note
II ( v . u j ) -
1 II.
[]
Proof of Theorem 1.5. Let Ao be a nonrational n o n c o m m u t a t i v e torus with generators u l , . . . , u,. We m a y suppose that p = P12 = e2~i° with 0 = 012 irrational. Since
every element of Ao can be a p p r o x i m a t e d by g(Ul,...,u,,u~ .... ,u*,) for some
p o l y n o m i a l 9, by P r o p o s i t i o n 4.3 it suffices to show that for each s > 0 there is a set
of matrix units {Gj} for a copy of M 2 @ M 3 in Ae such that it[e~j, uk] tl < s for all i,j,
k,r.
Let sl = s/4. By 2.3 choose e2 so that whenever A~ is a C*-algebra and {ce~i} is a
set of a p p r o x i m a t e matrix units of type M2 @ M3 within s2 in ~E, there is a set of
exact matrix units {e~j} of the same type in ~E with tlei~ - cecil] < sl. Then by 4.4
choose a set t~f~'/, of polynomials in tbur n o n c o m m u t i n g variables, with complex
coefficients, such that { f "ij(ul, u2, ul,
* uz)}
* is a set of a p p r o x i m a t e matrix units of
type M2 @ M3 within e2/2 in C*(ul, u2)= Ao. T h e n by 4.3 choose 61 such that
whenever A is a C*-algebra and x l , . . . , x4, Yl,..., Y, are elements of A of n o r m 1 such
that It[xi, Yk] IJ < 61 for all i, k, then ll[f~j(xl,..., x4), Yk] !I < e/2 for all i, j, k, r.
Also by 4.5 choose 62 such that whenever 10 - 0'l < 62, then {f~)(ul(O'), u2(O'), u'~(O'),
u*(0')} is a set of a p p r o x i m a t e matrix units within s2 in A0,. Set 6 = min(6~, 62).
Choose p, q, vl, v2 as in 4.7 for this 6. Then {f~)(vl, v2, v*1, v *2Js
~ is a set of
a p p r o x i m a t e matrix units of type M2 G M3 within e2 in C*(vl, v2) ~ Ao, so there
282
BRUCE BLACKADAR ET AL.
r
r
is a set {e f,ij} of exact matrix units for a copy of M2 • M 3 in Ao with lleij
-fib(v1,
v2,
v*,v*)ll < el for all i, j, r. We then have, for any i, j, k, r,
•
*
n
n
tl leT i, Uk] LI <~ II[ f b ( v l , v:, vl, v2), Uk] il + tt [eij -- f ii(vl, v2, v~, v2*),uk] II
< ~/2 + 2 IIe~j -- f rij(vx, V2, vl,v2)]l
• •
<~ e/2 + e/2 = ~.
This shows that Ao is approximately divisible.
Now suppose Ao is simple. Then Ao has stable rank 1 by Theorem 1.4(c). It is well
known that Ao has a unique trace. Ao is nuclear, hence exact, and so by Theorem
1.4(f) Ao has real rank zero.
[]
It would be interesting to characterize the dynamical conditions for a group action
on a space which insure that the crossed product C*-algebra is approximately divisible.
We now consider some negative examples. Since every approximately divisible
C*-algebra contains many projections (in fact, has no minimal projections), a simple
C*-algebra with minimal projections cannot be approximately divisible. The next
example shows that even a simple C*-algebra with real rank 0 can fail to be
approximately divisible.
EXAMPLE 4.8 Let M be a 111 factor with separable predual which does not have
property (F) (e.g. the factor generated by the regular representation of the free group
on two generators). Let ~ be the trace on M. Then no weakly dense C*-subalgebra of
M is approximately divisible; in fact, if 6 > 0 is fixed, then no such subalgebra can
have a central sequence (with respect to I1"II2 and hence with respect to I1"11)of
projections (p,) with 6 <<.z(p,) <~ 1 - 6 [Dx]. (Notice that if B is a C*-subalgebra of
M isomorphic to M2 @ M3, then B contains a projection p with 1/3 ~< z(p) ~< 2/3, so
an approximately divisible C*-subalgebra of M would contain a central sequence of
projections with trace between 1/3 and 2/3.)
A separable simple o--weakly dense C*-subalgebra of M with real rank 0 can be
constructed as follows. Let Do be any a-weakly dense separable (unital) C*subalgebra of M. Then there is a separable simple C*-subalgebra D1 of M containing
Do by [B1 1, 2.2]. Since M has real rank 0, it is easy to construct a separable
C*-subalgebra D2 of M with real rank 0, containing D1. Iterate the construction to
get an increasing sequence (D,) of separable C*-subalgebras of M with D, simple for
n odd and D, of real rank 0 for n even. Then D = [ u D , ] - is the desired algebra.
The construction can be refined to insure that D has stable rank 1, totally ordered
Ko, and trivial K1. A similar construction is possible within a type III factor [Cn],
yielding a separable simple purely infinite C*-algebra with trivial K-theory, which is
not approximately divisible.
It could, however, be true that every (separable unital) simple nuclear C*-algebra
with (SP) (or with real rank zero) is approximately divisible. It might be possible to
combine with the existence of many partial isometries the fact that in such a C*algebra the identity map can be approximated pointwise by finite-rank completely
positive maps, to prove approximate divisibility.
APPROXIMATELY CENTRAL MATRIX UNITS
283
Acknowledgements
The research for this paper was conducted while the third author was a visitor at the
University of Nevada, Reno, in October 1990, while on leave from Odense University. He thanks the Mathematics Department in Reno for its hospitality during his
visit. The second author wishes to thank Gert Pedersen and Erik Christensen for
their hospitality and comments on related topics during his visit at Copenhagen
University.
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