Download DIMENSION AND STABLE RANK IN THE ^

DIMENSION AND STABLE RANK
IN THE ^-THEORY OF C*-ALGEBRAS
MARC A. RIEFFEL
[Received 24 March 1982]
Introduction
In topological K-theory, which can be viewed as the algebraic side of the theory of
vector bundles, some of the interesting properties which one investigates are, for
example, the conditions under which bundles must possess trivial direct summands,
or the extent to which the cancellation property for direct (Whitney) sums of bundles
holds. Such properties turn out to be controlled in part by the dimension of the base
space, and results describing the nature of this control are among what are frequently
called stability results [17].
During the past few years there has been a sudden flowering of a /^-theory for C*algebras. (See surveys [35, 36, 37].) Since C*-algebras are profitably thought of as
'non-commutative locally compact spaces', with the finitely generated projective
modules being the appropriate generalization of vector bundles according to Swan's
theorem [40], it would be natural to look for stability results for C*-algebras. But
until now the theory has been focussed on the /C-groups of C*-algebras, and there has
been little discussion of stability properties, presumably in part for lack of an
appropriate concept of dimension for C*-algebras.
One of the main objectives of this paper is to introduce a concept of dimension for
C*-algebras which directly generalizes the classical concept of dimension for compact
spaces [16, 20, 22], and to develop some techniques for calculating it, notably for C*algebras which are obtained as crossed-product algebras for an action of the group of
integers.
In algebraic K-theory, which is the generalization of topological /C-theory to rings,
a substantial number of stability properties have been discovered [2, 39, 42-48].
Within that theory the concept which has played a role most analogous to that of
dimension is the Bass stable rank. But despite its many successes, the Bass stable rank
has often been difficult to calculate even for relatively uncomplicated rings. Another
main objective of the present paper is to make accessible for application to C*algebras some of the stability results from algebraic /C-theory. This comes about from
the fact that the concept of dimension which we introduce dominates the Bass stable
rank.
I was led to seek stability results for C*-algebras by my desire to determine whether
the cancellation property for projective modules holds for the irrational rotation C*algebras discussed in [30] (which, as Elliott [11] has suggested, can appropriately be
called 2-dimensional 'non-commutative tori'). In more concrete terms, the question is
whether two projections in such an algebra which have the same trace must be
unitarily equivalent. The present paper provides the general theory which is needed
for the affirmative answer which I obtained. In a subsequent paper I will derive the
This research was supported in part by a Senior Visiting Fellowship from the Science Research Council
and in part by National Science Foundation grant MCS-8010723.
Proc. London Math. Soc. (3), 46 (1983). 301 333.
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MARC A. RIEFFEL
detailed facts about the irrational rotation C*-algebras which are needed in order to
be able to apply to them the theory presented here [51].
The subjects of the various sections of this paper are as follows. In § 1 we introduce
our concept of dimension, which for reasons given there we call 'topological stable
rank'. Then in §2 we compare this with the Bass stable rank. Section 3 is devoted to
examining the lowest-dimensional case, which has some properties special to itself.
Then in §4 we investigate the relationships between an algebra and its quotient
algebras and ideals, while in § 5 we briefly examine inductive limits. The behaviour of
topological stable rank when one passes to matrix algebras over an algebra is quite
surprising, and this is the subject of §6. The crucial section for application to the
irrational rotation C*-algebras is §7, where crossed products for actions of the group
of integers are discussed. In § 8 some topological properties of spaces of generators are
obtained, which yield some refinements to the theory. Then in §9 some of the earlier
theory is extended to modules over C*-algebras, providing clarification of a few
earlier points. Finally, § 10 is a largely expository account of some of the stability
results from algebraic K-theory which are applicable to C*-algebras.
The results reported here were obtained while I was on sabbatical leave visiting the
University of Leeds. I should like to thank the members of the mathematics
department there for their warm hospitality. The possible relevance of stable range
results in algebraic K-theory to the problems concerning irrational rotation C*algebras which I was investigating was brought to my attention by a seminar on
algebraic K-theory conducted there by John McConnell and Chris Robson, and I am
indebted to them for many stimulating conversations concerning the Bass stable
range and related matters.
1. Dimension for C*-algebras
I was led to the notion of dimension for C*-algebras by noticing the following
standard theorem from classical dimension theory (see [16, VI. 1; 20, Theorem VII. 4;
22, Proposition 3.3.2]).
1.1. THEOREM. Let X be a compact space. Then the dimension ofX is the least integer
n such that every continuous function from X into U" + 1 can be approximated arbitrarily
closely by functions which do not contain the origin in their range.
Now a map, / , from X to 1R"+1 is just an (n+l)-tuple, fi,...,fn + u of real-valued
functions, and the condition that / miss the origin is just the condition that the f
nowhere all take the value 0 simultaneously. (Compare with Zarelua's definition [49]
of'dimension rank' discussed in 10.3 of [22].) But, if we let C^(X) denote the Banach
algebra of real-valued functions on X, this last condition is equivalent by the StoneWeierstrass theorem to the condition that the ideal in CU(X) generated by all the f
together be C®{X) itself. Thus we can rephrase the above theorem as follows.
1.2. THEOREM. Let X be a compact space. Then the dimension ofX is the least integer
n such that every (n+\)-tuple of elements of CU(X) can be approximated arbitrarily
closely by (n+l)-tuples which generate CU(X) as an ideal.
It is then obvious that one can generalize this to define the dimension of any real
commutative Banach (or topological) algebra A simply by replacing CR{X) by A. For
THE K-THEORY OF C-ALGEBRAS
303
non-commutative real Banach algebras one must specify whether one is generating a
left ideal or a right ideal, and so one must distinguish between a left dimension and a
right dimension. (At present the two-sided case does not appear interesting.)
Since n-tuples which generate an algebra as an ideal will be crucial to all our
subsequent discussion, it will be convenient to introduce at this point the following
notation.
1.3. NOTATION. For any ring A with identity element we let Lg,,(A) (Rg,,(/4)) denote
the set of n-tuples of elements of A which generate A as a left (right) ideal.
We remark that in algebraic K-theory the elements of Lgn(A) (Rg,,(A)) are
traditionally called left {right) unimodular rows. Note that for a Banach algebra A with
identity element there is no distinction between generating A algebraically or
topologically as an ideal.
Now for the purposes of C*-algebras one wants to work with the algebra C(X) of
comp/ex-valued continuous functions on X. But a complex-valued function corresponds to a pair of real-valued functions, so that an n-tuple of elements of C(X)
corresponds to a map from X into U2". Then if we let n be the least integer such that
any n-tuple of elements of C(X) can be approximated arbitrarily closely by n-tuples
which generate C(X) as an ideal, we see that n is [dim(X)/2] + 1 , where [ ] denotes
'integer part of. This observation is actually encouraging, once one notices (in, for
example, [17, Chapter 8]) that in the X-theory of complex vector bundles, stability
questions often involve expressions in [dimpQ/2]. However, this observation also
suggests that it is inappropriate to call n 'dimension'. Instead, because of the close
connections with the Bass stable rank which we will discuss shortly, we will use the
term 'topological stable rank'. In the non-commutative case we must again distinguish
between left and right ideals.
1.4. DEFINITION. Let A be a Banach algebra with identity element. By the left (right)
topological stable rank of A, denoted ltsr(/4) (rtsr(/4)), we will mean the least integer n
such that Lgn(/4) (Rg,,(/4)) is dense in A" (for the product topology). If no such integer
exists we set ltsr(/4) = oo (rtsr(/4) = oo). If A does not have an identity element, then
its topological stable ranks are defined to be those for the Banach algebra A" obtained
from A by adjoining an identity element.
We remark that if ^4 already has an identity element then A is a direct summand of
/ T , and from this it is easy to see that \tsr(A) = ltsr(/4~). Note also that if Lgn(A) is
dense in An for some n then Lgm is dense in Am for any m^n. Following common
usage in algebraic /C-theory, we will say that such an m is in the (left) topological stable
range of A.
1.5. QUESTION. Does there exist a Banach algebra whose left and right topological
stable ranks are different?
This question does not arise for Banach algebras with a continuous involution, for
such an involution will interchange Lg,,(A) and Rgn(A), so that we have:
1.6. PROPOSITION. Let A be a Banach algebra with a continuous involution. Then
\tsr(A) = rtsr(/4).
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MARC A. RIEFFEL
In such cases we will just speak of the topological stable rank, denoted tsr{A).
Recapitulating some of the earlier discussion, we have:
1.7.
PROPOSITION.
/ / X is a compact space, then
Notice that with our indexing conventions, if dim(AT) = 0 then tsr(C(X)) = 1.
One can ask whether one can actually recover the dimension of X from C(X). One
way to do this is as follows. Let / denote the unit interval, [0,1], and for any Banach
algebra A let IA denote the Banach algebra of continuous functions from / into A,
with the supremum norm. Since the dimension of / x X is exactly one greater than the
dimension of X (a non-trivial fact—see [15]), we see that the dimension of AT is exactly
tsr(C(Ar)) + tsr(/C(A r ))-2.
This suggests that one could use the same formula to define the dimension of any
complex Banach algebra. But it is not clear how useful this might be. For this idea to
work smoothly one would at first thought hope that
But as Bruce Blackadar has pointed out to me, Theorem 6.1 of this paper implies that
this equality fails already for A = 1(1M2), where M 2 is the algebra of 2 x 2 matrices
with complex entries. This still leaves open:
1.8.
QUESTION.
For every complex Banach algebra (or at least every C*-algebra) is it
true that tsr(/(/A)) ^ tsr(A) +1?
We will see later (Corollary 7.2) that at least for C*-algebras it is true that
tsr(//4) < tsr(/4) + 1 , as one would expect.
2. The Bass stable rank
In this section we will recall the definition of the Bass stable rank for rings [1, 2, 43,
47] and compare it, for the case of Banach algebras, with the topological stable rank.
2.1. DEFINITION. Let A be a ring with identity element. Then by the left Bass stable
rank of A, denoted lBsr(/l) we will mean the least integer n such that for any
(a,) e Lgn+1(/4) there is a (6,) e A" such that
is in Lgn(/4). If no such integer exists, we set lBsr(/4) = oo. The right Bass stable rank
(rBsr(,4)) is defined analogously.
Actually, it turns out that lBsr(/l) = rBsr(^) for any ring A (see [43, 47]), and so we
will usually just speak of the Bass stable rank of A (Bsr(A)). We warn the reader that
in some of the early papers concerned with the Bass stable rank the indexing
conventions are different from that given above. Note that if n is an integer for which
the condition of Definition 2.1 holds, then the same will be true for any m ^ n. It is
THE K-THEORY OF C-ALGEBRAS
305
traditional to say that such an m is in the (Bass) stable range of A. Readers who are
interested in the treatment of Bass stable rank for rings without identity element may
consult [42].
For the purpose of comparing the Bass stable rank with the topological stable rank
the following alternative description of the Bass stable rank given by Warfield [47] is
useful:
2.2. PROPOSITION. Let A be a ring with identity element. Then lBsr(/4) is the least
integer n such that for every (a,.) e Lgn+i{A) there is a (c,) e An+l such that J ] C A = 1
and(c!,...,cH) is in Kgn{A).
Proof [47, Lemma 1.7]. If (a, + />,an + 1) e Lgn(A) for a (bt) e A", then there is a
(c.) € A" such that
so that(c,) e Rgn(/4)and the condition above holds if we let cn + l = Y,c^i- Conversely,
if (c,) e An+l, Yuciai = 1. a n d (c l5 ...,c n ) e Rg n (4), so that 0 ^ + ... +cndn = 1 for
suitable (rf,) e A", and if we let bt = d,-cn+1> then we find that
We will give later (Proposition 9.2) a module-theoretic interpretation of this
description.
2.3.
THEOREM.
Let A be a Banach algebra with identity element. Then
rtsr(/4) ^ lBsr(/4)
(=
Proof. Let n = rtsr(/l), and let (a,) e Lg n + 1 (/4), so that there is a (b{) e / T + 1 with
closely enough by
^ 6 , a , = 1. Using the definition of rtsr(/4), approximate (blt...,bn)
(b'i) e Rgn(/4) so that, if we set
d = b\a1+ ... +b'nan + bn + 1an + 1,
then d is still close enough to 1 to be invertible. If we multiply on the left by d'1 we
obtain
1 =(d-ib\)al+...
+(d-xb'n)an + (d-ibn
+ 1)an + 1,
l
and it is easily seen that the n-tuple (d' b'i) G Rgn(A). Thus Warfield's condition for
the Bass stable rank is satisfied.
Later (Theorem 9.6) we will give a module-theoretic interpretation of why it is the
right tsr(A) which compares easily with the left Bsr(A). Of course by symmetry we
obtain:
2.4. COROLLARY. Bsr(A) ^ min(ltsr(/4),rtsr(/4)).
2.5.
QUESTION. IS
there a Banach algebra A for which Bsr(A) ^ rtsr(/4)?
It seems to me quite possible that an example exists among algebras of analytic
functions. The simplest such may be the disk algebra, A(D), consisting of the functions
5388.3.46
T
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MARC A. RIEFFEL
which are continuous on the closed unit disk in the complex plane and analytic in the
interior. It is easily seen that tsr(A(D)) = 2. But it does not seem so easy to answer:
2.6. QUESTION.! IS Bsr(A(D)) = 1?
That the answer may be affirmative is suggested by the fact that Rubel has shown
[38] that if H is the algebra of all entire functions on the complex plane then
Bsr(//)= 1.
But such an example would still leave open:
2.7.
QUESTION.
If A is a C*-algebra with identity element, is tsr(^) = Bsr(/4)?
For commutative C*-algebras the answer is affirmative, for Vaserstein [43] has
shown:
2.8.
THEOREM.
If X is a compact space, then
= [dim(Ar)/2] + l.
We remark that in the non-metrizable case one must be careful about which
definition of 'dimpQ' one uses.
3. The case where tsr(A) = 1
We now examine Banach algebras with the lowest tsr, namely 1. This case is easier
than the cases of higher tsr. It is also somewhat special in its behaviour.
The following result was suggested to me by a remark which A. G. Robertson made
to me about his paper [33].
3.1. PROPOSITION. For a Banach algebra A with identity element the following are
equivalent:
1. ltsr(/4)= 1;
2. rtsr(,4) = 1 ;
3. the invertible elements of A are dense in A.
These conditions all imply that any left (or right) invertible element is, in fact, invertible.
Proof. Note that Lgx(/4) (Rg^/i)) consists of the left (right) invertible elements of A.
Then it is clear that Condition 3 implies Conditions 1 and 2. Suppose conversely that
ltsr(/4) = 1, and let a be a left-invertible element with left-inverse b. Then by
hypothesis b can be approximated closely enough by a left-invertible element c so that
ca is still close enough to 1 to be invertible. But this means that c, and hence a, is in
fact invertible. The case in which rtsr(/4) = 1 is treated similarly.
We remark that Robertson [33], building on work of Handelman [14], showed
that for a C*-algebra, A, the conditions of the above proposition are equivalent to the
property that A have unitary \-stable range, in the sense that if (a, b) e Lg2(/1) then
there is a unitary u such that a + ub is invertible. A few further comments about this
can be found in [4, §111.2].
11 have just heard that Peter Jones, Donald Marshall, and Thomas Wolff believe that they have a proof
that the answer is affirmative.
THE K-THEORY OF C*-ALGEBRAS
307
3.2. QUESTION. For C*-algebras is there a useful analogue of unitary 1-stable range
which is equivalent to tsr(y4) = n when n ^ 2?
Before considering some examples it will be useful to have the following theorem,
one direction of which was obtained by Robertson [33]. For any algebra A we will let
Mn(A) denote the algebra of n x n-matrices with entries in A. If A is a C*-algebra, then
so is Mn(A) in a natural way.
3.3. THEOREM. Let A be a C*'-algebra with identity element. Then invertible elements
of A are dense in A if and only if invertible elements of Mn(A) are dense in Mn(A).
Proof. The proof in one direction follows immediately from the next lemma, which
we will need again later.
Let Abe a C*-algebra with identity element, and let t e A. If for some n
It 0\
and some e, where 1 > e > 0, the element
I of Mn+1(A) can be approximated
,0 I,,
within £ by an invertible element of Mn+l(A), then t can be approximated within
E(\ — e)" 1 by an invertible element of A.
3.4.
LEMMA.
Proof. Of course, /„ denotes the n x n identity matrix. Let I
ft
element of Mn+l(A) which approximates I
) be an invertible
0\
I within e, where a e A, D e Mn(A),
and B and C are 1 x n and n x 1 matrices. Then it is easily seen that || D — In || ^ e, so
that D is invertible and || D " 1 || ^ (1 - e ) " 1 . But
-BD~x\fa
,0
B\f
1
0\_/a-BD~iC
l
IJ~[
/„ AC Dj[-D- C
0
0'
D,
and since the three matrices on the left are all invertible, so is a — BD~lC. However,
\\t-(a-BD-lC)\\
^ \\t-a\\
+ \\BD~XC\\
^ e+ e^l-e)"1 = e ( l - £ ) - 1 ,
where to obtain the last inequality it is convenient to view B and C as elements of the
/1-rigged space A" and its dual, as in [27, §4] (see also the linking algebra of [6,31]).
The fact that the bound e(l — s)" 1 is independent of n will be useful later.
For completeness we include the proof of the converse direction of Theorem 3.3, as
given by Robertson [33]. The proof is by induction on n. Let I
I be any element
of M,,+ ,(/!), as above. Given e > 0 we can choose an invertible a0 e A which is within
e of a. Since by the induction hypothesis the invertible elements of Mn(A) are dense, we
can find Do e M,,(A) within e of D such that DQ — CCIQ~1B is invertible. But this last
condition implies that I °
1 is invertible, as is easily seen by elementary row and
column operations.
We now consider some examples. It is easily seen directly that invertible elements
are dense in any M,,(C), and so in any finite direct sum of such, that is, in any finitedimensional C*-algebra. But it is clear that this property persists under inductive
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MARC A. RIEFFEL
limits, even non-unital ones as is seen by adjoining an identity element. But the
inductive limits of finite-dimensional C*-algebras are exactly the AF-algebras, which
have received much attention in recent years [10]. Thus we see:
3.5.
PROPOSITION.
If A is an AF-C*-algebra then tsr(A) = 1.
But there are other examples. In particular, of course, tsr(C(X)) = 1 if dim(X) = 1
(or 0). Let T denote the group of complex numbers of modulus 1, as a space, so that
tsr(C(T)) = 1. If G is any finite subgroup of T then G acts by translation on T and this
gives an action, T, of G as a group of automorphisms of C(T). It is then not too
difficult to see that C(T)xTG, the corresponding crossed product algebra [23], is
isomorphic to an algebra of nxn matrices over C(T). Thus tsr(C(T) xTG) = 1 by
Theorem 3.3. Now let G be a dense torsion subgroup of T, so that it is the union of its
finite subgroups. Then C(T)xxG will be the inductive limit of the crossed products
with finite subgroups, and so will itself have invertible elements dense, that is,
tsr(C(T) xTG) = 1. But C(T)xzG is an alternate description of the algebras considered
by Bunce and Deddens [7], as was pointed out by Green [13].
We can extend Theorem 3.3 as follows:
3.6. THEOREM. Let A be a C*-algebra and let K be the algebra of compact operators
on a separable Hilbert space. Then tsr(A ® K) = 1 if and only if tsr(A) = 1.
Proof. We treat here only the case in which A has an identity element, since the case
without identity element will then follow by applying Theorems 4.4 and 4.11 in the
next section to the exact sequence
Suppose that tsr(A) = 1, so that invertible elements of A are dense. Now (A ® /C)~ is
clearly the inductive limit of the Mn(A) . But invertible elements of Mn(A) are dense
according to Theorem 3.3, and so the same is true of Mn(A) , and so of (A ® K) .
Conversely, suppose that tsr(A ® K) = 1. Let a be any element of A, let p be a
rank-1 projection in K, and consider the element
/-I ® p+a ® p
in (A ® K) (where / denotes the adjoined identity element). By hypothesis this
element can be approximated by invertible elements of (A ® K)~, and it is clear that
such invertible elements can be chosen of the form I + T where T e A ® K. But
A ® K is the inductive limit of the Mn(A), and so we can require that T e Mn(A) for
some n. If 1 ® q denotes the identity element of such an Mn(A) in A ® K, then this
means that we can approximate by invertible elements of the form 7 — 1 ® q + S
where S is in the corner of A ® K defined by 1 ® q and is invertible there. Clearly we
can always choose q large enough so that p = pq = qp. This all means that given any
g > 0 we can find a sufficiently large n such that there is an invertible S in Mn(A) such
that
| | / - I ® p + fl ® p - S || < e
where now / denotes the identity element of Mn{A) and p is a projection in Mn. But
then we can apply Lemma 3.4 to conclude that a can be approximated by invertible
elements of A (since the bound in the lemma is independent of n).
THE K-THEORY OF C*-ALGEBRAS
309
From the above theorem it follows immediately that the property of having
tsr(/4) = 1 is preserved under stable isomorphism. In view of [6] this also means that
within the category of C*-algebras with countable approximate identities the property
of having tsv(A) = 1 is preserved under strong Morita equivalence.
For additional information about the case where tsv(A) = 1 the reader is referred to
[4, 14, 32, 33].
4. Topological stable rank of ideals and quotients
Our discussion of topological stable rank for ideals and quotients is guided in part
by the corresponding discussion for the Bass stable rank given by Vaserstein [43].
When working with ideals the adjunction of identity elements is unavoidable, and it
will be helpful to begin by developing some ways of simplifying their use. For ease of
exposition we will discuss only ltsr, but of course analogous results hold for rtsr.
Let A be a Banach algebra with identity element, and let (a,-)e An. Then (a,) G Lgn{A)
exactly if there is a (6.) G A" such that £ b,a, = 1. It is clear that this has a convenient
rephrasing in terms of the ,4-valued bilinear form [ , ] defined on A" by
where v = (b,) and w = (a,). If T e Mn(A), and if v and w are viewed as a row vector
and column vector respectively, then
[u, Tw] = [vT, w].
Let Gl(n, A) denote the group of invertible elements of Mn(A). Then left multiplication
of elements of A" (viewed as column vectors) by an element of Gl(n, A) defines a
homeomorphism of A" (when A" is given the product topology).
4.1. PROPOSITION. Let A be a Banach algebra with identity element. The left action of
Gl(n, A) on A" carries Lgn(/4) into itself.
Proof. If v e Lgn(/4), so that there is a w G A" with [w, u] = 1, then for any
T e G\(n, A) we have [wT'1, 7V] = 1.
Now let A be a Banach algebra possibly without identity element, and let v G (A~)n,
with v = (flj + s,) for a, e A and st scalar. If all the s, = 0, then we can approximate v
arbitrarily closely by new v for which at least one of the s,- # 0. So assume for the
moment that not all s, are 0. Then we can find an invertible scalar matrix T such that
T(si) = e, ,the first standard basis vector. Viewing T as an element of Gl(n, A ), we see
that Tv is then of the special form {cx +1, c 2 ,..., cn). Since the action of T defines a
homeomorphism of {A~)n, we see that to calculate ltsr(/l) it suffices to show that
elements of (^4 )" of this special form can be approximated by elements of Lgn(/1 ).
Suppose now that an element of this special form is closely approximated by an
element (d,- + s,-) of Lgn(/4~). Then (s,) must be very close to ex, and so we can find an
invertible scalar matrix T very close to /„ such that TXs,) = el. Then T(d( + s,) will be
an element of Lgn(/4~) which is now of the above special form, and is still fairly close to
(cl + \,c2,---,cn). Thus if an element of (A )" of this special form can be approximated
by elements of Lgn(,4~), t n e n it can > ' n fact> be approximated by elements of Lgn(/4~)
which are themselves of this special form.
Now suppose that (ax + 1, a2,...,an) is in Lgn(/4~), so that there is a (6, + s,) G (A~)n
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MARC A. RIEFFEL
such that
n+
sn)an=\
If we multiply this equation on the left by —ax, if we let c, = — ^ ( 6 , + s,), so that
c, £ A, and if we then rearrange slightly, we obtain
a2+ ... +cnan = 1.
We have thus obtained:
4.2. PROPOSITION. Let A be a Banach algebra, and let A be the algebra obtained by
adjoining the identity element I to A. Then an element of (A )" of the special form
(al +
\,a2,...,an)
is in Lgn(A ) if and only if there is an element (bl + l,b2,---,bn) of (A )" such that
(by + 1 ) ^ + l) + b2a2+...
+bnan = 1.
Furthermore, \tsr(A) will be the least integer n such that every element of [A")" of the
above special form can be approximated arbitrarily closely by elements o/Lg n (/T) which
are themselves of this special form.
In the sequel when we speak of'ideals' we will always mean closed two-sided ideals
unless the contrary is indicated. The next theorem generalizes the fact that the
dimension of a closed subset of a locally compact space is no greater than the
dimension of the whole space.
4.3.
THEOREM.
Let A be a Banach algebra and let J be an ideal in A. Then
ltsr(A/J) ^ \tsr(A).
Proof. Since (A/J) ^ A /J, it suffices to consider the case in which A has an
identity element. Now it is easy to see that the evident continuous surjection of A"
onto {A/J)n carries Lgn(A) into Lgn(A/J). Thus if Lgn(A) is dense in A", then Lgn(A/J)
will be dense in (A/J)n.
The next theorem generalizes the fact that the dimension of an open subset of a
locally compact space is no greater than the dimension of the whole space.
4.4. THEOREM. Let A be a Banach algebra, let J be an ideal in A, and assume that J
has a bounded approximate identity for itself Then
ltsr(J) ^ ltsr(/4).
Proof. We will assume that A has an identity element, for otherwise we can adjoin
one without changing the situation. Let J denote J with the identity element, 1, of A
adjoined, let n = \tsr(A), let (d1 + 1, d2, ...,dn) be given, with dt e J, and let £ > 0 be
given. By hypothesis, we can find an element of Lgn(A), rewritten as (a! + l,a 2 , ...,a,,),
such that || </,- —a,-1| < ^e for all i. Let {ja} be a bounded approximate identity for J.
For ease of exposition we will assume that || ja \\ ^ 1 for all a. For sufficiently large a
we will have || d,- — d j a || < j£ for all i, so that || d , - a , j a || < e. Since a{ja e J, all we
THE K-THEORY OF C*-ALGEBRAS
311
need to show is that {a{ja+ \,a2ja,...,anja)
e Lgn(J ) for sufficiently large a. Now by
such that
assumption there is an element of A", rewritten as {bx + \,b2,...,bn),
that is,
bx+ax +blai + b2a2+ ... +bnan = 0.
If we multiply this equation on the left by jp and on the right by j a , we obtain
jpb i j* + Jpa i L+Z UpbdiaJa) = 0.
Then
2)(fl2./J+ - +Upbn)(aJa)'] ||
- y>fc i j a + a j 7a - jpa
^ WUpbJ-UpbJjJ
Ja
+ I l ^ - V i II + II <*i-*i II-
If we first choose (i so that the second term is less than ^e, and then choose a so that
the first term is less than je, we see that the entire quantity is less than e. Thus
(jpbx + \)(alja+l)
+ (jpb2)(a2ja)+
is invertible, so that (alja+l,a2ja,...,anja)
... +{jpbn)(anja)
is in Lgn(J~) as desired.
To continue our discussion along the lines indicated by Vaserstein [43] we need
some elementary facts about the relationship between Bsr and Gl(n, A), which are
closely related to the reasons for which Bass introduced his stable rank in the first
place [1,2, 3]. Let A be a ring with identity element. By an elementary matrix over A
we mean one which differs from the identity matrix by at most one off-diagonal entry.
Note that elementary matrices are in Gl(n, A), and that if A is a Banach algebra then
elementary matrices are even in Gl°(n, A), the connected component of the identity
element of Gl(n, A). We let El(n, A) denote the subgroup of Gl(n, A) generated by the
elementary matrices.
Now let m be in the Bass stable range of A, and let (a,) e Lgm + l(A), viewed as a
column vector. Then the definition of Bsr says that suitable left multiples of am+l can
be added to the other entries so that the first m entries of the resulting column form an
element of Lgm(A). But this is the same as saying that there is an element of
El(m+ 1, A) which carries (a,-) to an element (6,-) whose first m entries form an element
of Lgm(/4). But then suitable left multiples of the first m entries of (bt) add up to
1— bm + l, so that if they are all added to the (m+l)th entry it becomes 1. Again this
says that there is an element of El(m+ 1, A) which carries (a,) to a column whose last
entry is 1. But then by adding suitable multiples of this last entry to each of the others,
we can make all these other entries be 0. That is, there is an element of El(m+ 1, A)
which carries (a,) to the 'last standard basis vector'. We have thus obtained the
following basic result (see [3, Lemma 2.3]):
4.5. THEOREM. If A is a ring with identity element and ifm ^ Bsr(/4)+ 1, then El(m, A)
acts transitively on Lgm(/4).
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MARC A. RIEFFEL
Since El(m, A) ^ Gl°(m, A) if A is a Banach algebra with identity element, we
immediately obtain the following seemingly weaker corollary. But in fact we will see
later (Corollary 8.10) that the two results are equivalent.
4.6. COROLLARY. If A is a Banach algebra with identity element and ifm ^ Bsr(/4) + 1 ,
then Gl°(m, A) acts transitively on Lgm(A).
We will find it useful to define now a different type of stable rank, suggested by the
above corollary.
4.7. DEFINITION. Let A be a Banach algebra with identity element. By the left
connected stable rank of A, denoted \csr(A), we will mean the least integer n such that
Gl°(m, A) acts transitively on Lgm(A) for all m^n. If no such integer exists then we set
lcsr(/4) = oo. If A does not have an identity element, then we define lcsr(/4) to be
lcsr(/4~). The right connected stable rank of A, denoted rcsr(/4), is defined analogously.
4.8.
QUESTION. IS
there a Banach algebra A for which lcsr(/4) ^ rcsr(/l)?
It is evident that:
4.9. PROPOSITION. / / A is a Banach algebra with continuous involution, then
\csr(A) = rcsr(/l).
In such cases we will just speak of the connected stable rank, denoted csr(/4).
We remark that in contrast to the situation for tsr, it is not the case that if Gl°(n, A)
acts transitively on Lgn(A) then this will be true for all m ^ n.
Rephrasing the results obtained above, we get:
4.10.
COROLLARY.
For any Banach algebra A we have
Icsr(y4) ^ Bsr(/4) + 1 < ltsr(/4) + 1 .
That the inequality for lcsr cannot be improved can be seen by considering the case
in which A = C(T), where T is the unit circle. As seen earlier, tsr(C(T)) = 1, so that
csr(C(T)) ^ 2. But the condition that csr(A) = 1 means exactly that the left-invertible
elements of A are invertible and that the group of invertible elements of A is
connected. But this is not the case for A = C(T) since its components are distinguished
by their winding numbers. Thus csr(C(T)) = 2.
The pertinence of the lcsr to our immediate needs comes from the fact that any
element of Gl°(n, A/J) is a finite product of exponentials [25,41], and so can be lifted
to an element of Gl°(n, A).
4.11.
THEOREM.
Let A be a Banach algebra and let J be an ideal of A. Then
\tsr(A) ^ max(ltsr(J),ltsr(/4/J),lcsr(/4/J)).
Proof. Again it suffices to prove this for the case in which A has an identity element.
Let n be the above maximum, and let (a,) E A" be given, along with an open
neighbourhood, U, of (a,). Let p denote the projection from A to A/J. Since
THE K-THEORY OF C*-ALGEBRAS
313
n ^ Itsr(v4/J), we can find a (6-) in p(U)nLgn(A/J).
Since n ^ \csr(A/J) we can find
S e G l > , A/J) such that S(b't) = (1,0,..., 0). Lift S to a T e G l > , /I), and lift (6J) to a
(6.) e £/. Then T(6,) is a lift of (1,0,..., 0), and so is of the form {di + 1, d2,...,dn) where
(d,-)eJ n . Since n ^ ltsr(J), we can find (c,) e T{U)n Lgn(J~). But then
T~ *(<:,•) E U n Lgn{A), as desired.
4.12.
COROLLARY.
Let A be a Banach algebra and let J be an ideal in A. Then
\tsr(A) < max(ltsr(J), \tsr(A/J) + \).
This corollary is the exact analogue of Vaserstein's Theorem 4 of [43] for the Bass
stable rank. He comments there that he does not know a counter-example to
improving the theorem by replacing ' B s r ( / l / J ) + r by 'Bsr(/4/J)\ But the following
examples provide such a counter-example, both for his Theorem 4, and for a similar
improvement of the above corollary.
4.13. EXAMPLES. Let S denote the unilateral shift operator with respect to a basis
{e,} for an infinite-dimensional Hilbert space, so that Se, = eI + 1 . Let A = C*(S)
denote the C*-algebra generated by S. Then it is well known [9, Theorem 7.23] that A
contains the algebra K of compact operators as an ideal, and that A/K is isomorphic
to C(T), the algebra of continuous functions on the circle, which is the essential
spectrum of S. Thus we have the exact sequence
0 -> K -> C*(S) -* C(T) -+ 0.
Now tsr(K) = 1 as seen earlier, and tsr(C(T)) = 1 since T is one-dimensional. Thus
tsr(C*(S)) is either 1 or 2 by the above corollary. Now 5 is a Fredholm operator of
index 1 in C*(S), and the Fredholm operators of a given index form an open subset in
a C*-algebra [9]. Thus the invertible operators cannot be dense in C*(S) and so
tsr(C*(S)) = 2. Notice that in view of Theorem 4.11 this is possible only because
csr(C(T)) = 2, not 1.
For a closely related example let Z u { + oo} denote the integers with a point
adjoined at + oo, and let the group Z act on Z u {+ oo} by translation, leaving the
point -I- oo fixed. Let A be the corresponding transformation group C*-algebra, that
is, the crossed product [23] of Z with C^iZ u {-I- oo}). Then again A will contain the
algebra K of compact operators as an ideal, corresponding to the invariant ideal
CJ^Z) of C 0 0 (Zu { + oo}) [23, 26], and the quotient will be isomorphic to the C*group algebra of Z, which is isomorphic to C{T). Thus again tsr(/4) is 1 or 2. But, as
shown in the proof of Lemma 5 of [12], A contains an isometry of index 1, so that
tsr(y4) = 2. We remark that C*(S) can be viewed as a corner of A.
A very closely related example [12] is obtained by everywhere replacing Z by U in
the above.
By very much the same reasoning one can show that if G is the 'ax + 6' group (either
the connected or the disconnected version), then tsr(C*(G)) = 2. (See [50, 34, 12].)
4.14. QUESTION. If G is a Lie group, how does one compute tsr(C*(G)) in terms of the
structure of G?
We remark that if G is the Heisenberg group (the upper-triangular 3 x 3 matrices
with l's on the diagonal), then one can use Theorem 4.11 to show that tsr(C*(G)) = 2,
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MARC A. RIEFFEL
if one first uses Corollary 8.5 and elementary properties of spheres to show that
csr(C(S2)) = 2 where S2 is the 2-sphere. We also remark that if G is any compact
group, then tsr(C*(G)) = 1.
4.15. QUESTION. If one puts some kind of finiteness condition on a C*-algebra A,
such as that left-invertible elements must be invertible, can one improve Corollary
4.12 by replacing 'ltsr(/4/J)+ 1' by Ttsr(/4/J)'?
There seems to be no relationship between the tsr of a Banach algebra and the tsr of
its subalgebras. For example, if / denotes the unit interval and if / : / -> / 2 is a curve
with dense image, then / embeds C(/2) as a subalgebra of C(/). Also, any C{X) for X
separable can occur as the centre of an AF-C*-algebra [5].
A standard result of classical dimension theory (see [22, 10.2.4 and 10.3.2] is that if
X is a locally compact space, then its dimension is the same as the dimension of its
Stone-tech compactification. Now the formation of Stone-Cech compactifications
corresponds at the C*-algebra level to associating to a C*-algebra A its algebra M(A)
of double-centralizers (i.e. multipliers) [23]. Thus if A is a commutative C*-algebra,
then tsr(M(/4)) = tsr{A). But this relation does not persist for non-commutative C*algebras. For example, if A = K, then M(A) = B(H), the algebra of all bounded
operators, and we will see shortly (Proposition 6.5) that tsr(B{H)) = oo. But again one
can ask:
4.16. QUESTION. IS there some appropriate finiteness condition which one can
impose on M(A) (or A) which will ensure that tsr(M(A)) = tsr(A)1
5. Inductive limits
We saw in § 3 that the condition tsr(A) = 1 is preserved by inductive limits. We now
show similar nice behaviour for higher tsr.
5.1.
Then
THEOREM.
Let the C*-algebra A be the inductive limit of the C*-algebras An.
tsr(/4) ^ liminf(tsr(/4n)).
Proof. Without changing the tsr we can adjoin an identity element to A, and then
adjoin the same identity element to each of the An. Thus we can assume from now on
that A and all the An have an identity element, and that the mappings of the inductive
system are all unital. Now let s be the indicated lim inf. Then for any element v of As
and any e > 0 one can find a k such that tsr(/4fc) = s and that there is an element w of
(Ak)s whose image in As is within ^e of v. But w can be approximated within je by an
element u of Lgs(Ak). Now it is easily seen that the Lg are preserved under unital
homomorphisms. Thus the image of u in As will be in Lgs(/4), and will be within £ of v.
We remark that it can easily happen that tsr(lim/4J is strictly less than
liminf(tsr(/4n)). For example, one can let An be obtained from C(/ 2 ), where I2 is the
unit square, by adjoining increasing finite collections of characteristic functions of
open subsets of/2 in such a way that the maximal ideal space of the limit algebra is 0dimensional. Another example is implicit in Theorem 6.4.
THE K-TIIEORY OF C*-ALGEBRAS
315
For algebras with identity element it is easily seen that the tsr of the direct sum of
two algebras will be the maximum of their tsr. This is also true for algebras without
identity element, as is easily seen using Proposition 4.2.
By the co-direct sum of a family {An} of Banach algebras we will mean the
subalgebra of the direct sum consisting of the elements (a,) which 'vanish at infinity' on
the index set, with the supremum norm. Since the co-direct sum will be the inductive
limit of the various finite direct sums, we rapidly obtain from Theorem 5.1, Theorem
4.4, and the comments above:
5.2. THEOREM. Let A be the co-direct sum of the family {An} of Banach algebras. Then
tsr(/4) = supremum{tsr(/4J}.
6. Matrix algebras
The relationship between Bsr and the formation of matrix algebras is quite
surprising, as first discovered by Vaserstein [43]. (See [47] for another proof.) The
following theorem is the analogue for tsr of Vaserstein's result. It also generalizes
Theorem 3.3 above.
6.1.
THEOREM.
Let A be a C*-algebra. Then for any positive integer m,
tsr(Mm(A))={(tsv(A)-\)/m}
+ \,
where here { } denotes 'least integer greater than\
Proof. If A does not have an identity element, then we have the exact sequence
0 -+ MJ[A) -> Mn{A~) -> Mn -> 0.
Since tsr(MJ = 1 = csr(MJ, it follows, from Theorems 4.4 and 4.11, that
tsr(Mn(A)) = tsr(Mn(A~)). Thus it suffices to consider the case in which A has an
identity element.
Let MmXn(A) denote the space of mxn matrices with entries in A. For any
S € MmXn(A) we will say that S is left-invertible if there is a T e MnXm(A) such that
TS = /„, the n x n identity matrix.
6.2. DEFINITION. We will say that the C*-algebra A satisfies Condition Ln(/c) if the
left-invertible (n + k)xk matrices are dense in M(n+k)xk(A).
Thus A satisfies Ln(l) exactly if n+ 1 ^ tsr(A).
6.3.
LEMMA.
If A satisfies L n (l), then it satisfies Ln{k)for all k^
1.
Proof. We argue by induction on k, with the case in which k — 1 holding by
hypothesis. Notice that if S is an m x n left-invertible matrix, and if R e Gl(m, A), then
RS is left-invertible. Now suppose that we also know that A satisfies Condition
L,,(/c— 1), where k ^ 2. Let T be any element of M{n+k)xk, and let U be any open
neighbourhood of T. According to Ln(l) we can approximate the last n+ 1 elements of
the first column of T by a column in Lgn + 1(A) closely enough so that the resulting
matrix T is still in U. But then, according to Theorem 4.5, there is an R e E\(n + k, A)
316
MARC A. RIEFFEL
such that the first column of S = RT is the first standard n + k basis vector. Let Q be
the (n + k — 1) x (k — 1) matrix obtained by removing the first row and column of S. By
the induction hypothesis we can approximate Q by a left-invertible matrix Q' closely
enough so that if S' is the matrix obtained from S by replacing Q by Q', then S' e R{U).
But it is easily seen that S' must be left-invertible. Then /? - 1 S' will be a left-invertible
matrix in U.
Proof of Theorem 6.1 continued. A column of length p of elements of Mm(A) can be
viewed a s a p m x m matrix. Let t = tsr(/4), so that A satisfies property Ln(l) if n ^ t— 1
(which we take as vacuous if t = oo). Then by Lemma 6.3, A also satisfies Ln(/c) for all
k. That is, left-invertible (n + k) xfematrices are dense. In particular, if pm = n + m for
some n^ t — \ then the left-invertible (pm) x m matrices are dense. But it is easily seen
that if a pm x m matrix is left-invertible, then the corresponding column of length p
in Mm(A) is in Lgp(MJ,4)). Thus Lgp(Mm(A)) is dense in (Mm(A)Y if
pm = n + m^ t-\ +m, that is, ifp ^ (t- \)/m+ \. Thus tsr(Mm{A)) ^ {(t-l)/m} + l.
We must now show the reverse inequality. For this we use a device which is similar
to a device for proving the analogous fact for Bsr which Chris Robson showed me. Let
s = tsr(Mm(/4)), and let q = ( s - l)m + 1 . Let v = (a,) be any element of A", viewed as a
column vector, and let U be any open neighbourhood of v in Aq. Let V be an open
neighbourhood of 0 in Aq such that v+V—V^U.
Let Vlt V2 and K3 be open
neighbourhoods of 0, / and 0, in Mqx(m_i){A), Mm-i{A) and M (m _ 1)xl (/1) respectively, such that every element of V2 is invertible and Kt K2 ~ i K3 ^ K. Consider the
s elements of Mm(A),
0/
\ ais.l)m
0/
\0
viewed as a column in (Mm(/4))s, and let T be the corresponding smxm matrix, so
fv 0
\
T =I
1. By the choice of s the matrix T can be approximated by a left-
invertible matrix I
I sufficiently closely that (v-w) e V, B e Vu D 6 V2 and
\C DJ
C G K3. Now if a left-invertible matrix is multiplied on either the left or the right by an
invertible matrix, one obtains a left-invertible matrix. We calculate now, just as in the
proof of Lemma 3.4, to obtain
/, -BD-'Ww
0
B\f
1
l
/m_t A c Dj\-D' C
0 \_/w-BD~iC
Im-J~\
0
0
D
It follows from this that the matrix on the right is left-invertible. But then it is easily
seen that w-BD~lC must be in Lgq(A). Now by hypothesis BD'^C e V, and so
w — BD~lC is in the neighbourhood U of v. We have thus shown that Lgq(A) is dense
in A", so that tsr(,4) ^ q = (s-l)m+ 1. That is,
(tsr(A)-l)/m+l^tsr(Mm(A)),
as desired.
Notice that the above theorem implies that if tsr(A) is finite, then tsr(Mm(/l)) is either
THE K-THEORY OF C'-ALGEBRAS
317
1 or 2 as soon as m ^ tsr(/4)— 1, and that it must be 2 unless tsr(A) = 1 by Theorem
3.3. It also implies that if tsv(A) = oo then tsr(Mm(A)) = oo for every m. For the case in
which tsr(/4) = oo it is then natural to ask what happens if instead of forming
M,,{A) = A ® Mn, one forms A ® K, where K is the algebra of compact operators.
What happens is that K wins:
6.4. THEOREM. Let A be a C*-algebra. Then tsr(/l ® K) = 2 unless tsr(A) = 1, in
which case tsr(A ® K) = 1.
Proof. By Theorem 3.6 it suffices to show that tsr(A ® K) ^ 2. If A does not have an
identity element then A ® K is an ideal in A ® K, so that by Theorem 4.4 it suffices
to treat the case in which A has an identity element.
According to Proposition 4.2 it suffices for us to consider elements of the special
form (I + bl,b2) in ((A ® K)~)2. Let e > 0 be given. Then bx and b2 can be
approximated within e by finite tensors cx and c2 whose terms in K are of finite rank,
so that there is a projection, p, in K such that 1 ® p acts as the identity operator on cx
and c2. Let g be a projection in /C orthogonal to p but equivalent to p, so that there
are partial isometries u and v in K such that uv = p, vu = q, pu = u = «<?,
qv = v = vp. Let c'2 = c 2 + e(l ® i>), so that || b2 — c'2 \\ ^ 2e. Then
(/ +
C l ,c' 2 )eLg 2 (O4(g)Kf),
for if dy = — 1 ® p and d2 = (1 ® u)/e, then
In particular, for stable C*-algebras (ones such that A ® /C = /I), there are only two
possible values of tsr, namely 1 and 2.
It is appropriate at this point to give examples of C*-algebras whose tsr is +oo.
Now if A = C(X) where X is an infinite-dimensional compact set, such as /°°, then
from Theorem 2.8 we see that tsr(C(X)) = oo. But in a different direction, the algebra
B(H) has tsr = oo, as mentioned earlier, as do the algebras On introduced by Cuntz
[8]. This follows immediately from:
6.5. PROPOSITION. If A is a C*-algebra with identity element which contains two
isometries with orthogonal ranges, then tsr(A) = oo = Bsr(/4).
Proof. In view of Theorem 2.3, it suffices to show that Bsr(/4) = oo. Now for any n
one can take products of the two isometries to produce n isometries S l 5 ..., Sn which
have mutually orthogonal ranges, and such that if we let Po = I — £S,-S* then Po # 0.
Then (S*,...,S*,P 0 )eLg,, + 1 (/l), since
But this element of Lgn + 1(A) cannot be contracted according to the definition of Bsr,
for if it could then there would be Tlt..., Tn in A such that
is in Lg,,04). But this means that there would be /? l 5 ...,/? n in A such that
Multiplying this equation on the right by Sh one finds that Rt = S, for 1 ^ i'^ n. But
then multiplying this equation on the left by Po, one finds that 0 = P o , which is a
318
MARC A. RIEFFEL
contradiction. (I am indebted to Joel H. Anderson for pointing out to me a mistake in
my original proof.)
However, let me remark that in the absence of either a compact space of known
dimension closely related to the situation, or of isometries, I do not know techniques
for getting lower bounds for the tsr, or even for showing that invertible elements are
not dense. In particular, I do not know how to obtain lower bounds for the tsr of
simple C*-algebras which are finite in some sense, such as the reduced group C*algebras of various discrete groups. For example:
6.6. QUESTION. What is tsr(C*(F2)), where C*(F2) is the reduced group C*-algebra of
the free group on two generators?
Presumably the answer is 1, 2, or oo.
Joel Anderson sent me the following result and proof.
6.7.
THEOREM.
Let C*(Fn) denote the full group C*-algebra of the free group on n
generators (2 ^ n ^ oo). Then tsr(C*(Fn)) = oo.
Earlier M.-D. Choi had mentioned to me that he thought he could prove the above
result by using the Cuntz algebras On, and I imagine that the following proof is also
what he had in mind.
6.8. LEMMA. If a unital C*-algebra A is generated by n elements, then M2n + \{A) is
singly generated.
Proof. Suppose that t l 9 ..., tn generate A, arranged so that || t,-1| < 1 for all j . For
each j let ai and bj denote the real and imaginary parts of t,, and let h be the diagonal
matrix in M2n + l(A) with diagonal entries
\,ax,...,an,bl,...,bn.
Note that the positive powers ofh converge to the matrix with 1 in the upper left-hand
corner and 0 elsewhere. Let p denote the permutation matrix in M2n + 1(A) corresponding to the cyclic permutation on 2n + 1 places, and let g be a self-adjoint matrix which
generates the same C*-algebra as the unitary p. Then it is a straightforward matter to
check that the two self-adjoint matrices g and h together generate M2n + l(A), and
hence so does g + ih.
Proof of Theorem 6.7. Let A be a unital C*-algebra generated by two isometries
with orthogonal ranges, so that tsr(A) = oo by Proposition 6.5. Let B = M5(A), so
that B is singly generated, by Lemma 6.8, and tsr(B) = oo, by Theorem 6.1. Let t be a
generator for B and let a and b be its real and imaginary parts. Then we can find
unitaries u and v in B which individually generate the same C*-algebras as a and b
respectively, and so together generate B. If we then send two of the generators of F,, to
u and v and the rest (if any) to 1, we see that B is a quotient algebra of C*(FJ, so that
tsr(C*(FJ) = oo, by Theorem 4.3.
THE K-THEORY OF C*-ALGEBRAS
319
As Joel Anderson remarked to me, Choi has shown {Pacific J. Math., 87 (1980), 4046) that C*{F,,) has a faithful trace, so that left-invertible elements are invertible, even
though tsr(C*(FJ) = oo.
7. Crossed products
The main reason that the concept of topological stable rank is useful in connection
with the irrational rotation C*-algebras is that it behaves nicely with respect to
forming crossed product C*-algebras for actions of the group of integers. I have not
found in the literature any analogous property for the Bass stable rank.
7.1. THEOREM. Let A be a C*-algebra, let a. be an action of the group Z of integers on
A by automorphisms, and let A x a Z be the corresponding crossed product C*-algebra.
Then
In fact, ifm ^ tsr(/4)+ 1, if A has an identity element, and ife{ denotes the first 'standard
basis vector' in (A x a Z) m , then E\(m, A xaZ)el is dense in (A x a Z) m .
We remark that we will see later (Corollary 8.6) that the last statement above
actually implies that Gl°(m, AxaZ)
acts transitively on Lgm(AxaZ)
for all
m ^ tsr(/4) + 1, so that csr(/l x a Z) ^ tsr(/4) + 1. In fact, even El(m, AxaZ) will act
transitively (Corollary 8.10).
Proof. If A does not have an identity element, then A xaZ is an ideal in A~ x Z
(where Z acts trivially on the adjoined identity element), so that Theorem 4.4 is
applicable. We will thus assume throughout that A has an identity element. For
brevity let B = AxaZ. Let m^tsv(A)+\,
let (b,) e Bm, and let U be an open
neighbourhood of (6,). Now B is defined [23] to be the norm closure of the aconvolution algebra, D, of /4-valued functions on Z of finite support. Thus U n Dm is
not empty. We wish to carry out an approximate Euclidean algorithm on elements of
U n D'n. For this purpose it is convenient to think of the elements of D as being
'twisted Laurent polynomials' of form £a k t' £ , in the indeterminant t, where k can be a
negative as well as positive integer. By the support interval of such an element, / , of
D, we will mean the smallest interval [p, q] in Z which contains the support of/, so
that / = aptp + ...+aqtq, a sum of terms of ascending degree, with ap ^ 0 # aq. Then
we will let L(/) = q — p + 1 and call it the length of / (arranged so that if / is
supported on one point, then L(f) = 1). For any (/•) e Dm we let L((/)) = £ L ( / ) .
Then for any (/•) € Dm we let size((/)) denote the smallest integer n such that there is a
T e El(m, D) such that L(T(ft)) = n. Since U contains elements of Dm, we can look in
U for an element of Dm of smallest size. Let (/•) be such an element, and let s be its size.
We will show that s = 1.
By the definition ofs there is a T e El(m, D) such that (#,) = T(f) has length s. Note
that TU is a neighbourhood of (#,). Assume first that none of the g-x are 0, so that
L((yi)) ^ w ^ 2. From among the gx- choose one of maximal length. For simplicity of
notation we assume that this g,- is gm. (In fact, we could apply an element of El(m, D) to
make it gm.) For each i let attn(i) denote the non-zero term of highest degree of g,-. Now
on A'"~l the map (/?,) I—* (a,,(m)_„(,-)(/?,)) is a homeomorphism (which need not preserve
Lgn,-{(A)). Since m—\ ^ tsr(/4), the element (ax,...,am..x)
can be approximated
320
MARC A. RIEFFEL
arbitrarily closely by (bl,...,bm-i)
such that (an(m)_n(I)(b,)) is in Lgm.1(A). Make such
an approximation close enough so that if h{ is obtained from g{ by replacing a{ by bf for
1 < / < m— 1, and if hm = gm, then (ht) £ TU. For simplicity of notation we now
replace (/)) by T~x(h) (and (#,) by (hi)). That is, we have shown that we can choose the
(/,) above such that, with (#,) = T(f{) and a( as above, we have the added property that
Now by the definition of Lgm_1(/1), there must exist (c.) e Am~l such that
C a
l n ( m ) - n ( l ) ( a l ) + --- + Cm- l a n ( m ) - n ( m - l)( f l rn- l )
=
fl
m!
so that, operating in A x a Z , we have
Notice now that since #m is of maximal length among the gh we do not increase the
length of gm if we subtract from it all the (c,^" 0 "" 0 ^,- for 1 ^ i! < m - 1 , and that in
fact we actually decrease its length because the coefficient of tn(m) becomes 0. But the
subtraction of all these elements can be performed by applying to (g() a suitable
element, say S, of El(m, D). That is, ST(/) is an element of strictly smaller length than
T(fi), contradicting the choice of s. It follows that, for the original situation, at least
one of the gt is zero. Then by applying a suitable element of El(m, D), we can assume
that this element is gx.
Now if (g{) = T(fi) for (ft) e U n Dm, and if gx = 0, then we can replace gx by an
£ ( = et°) so small that the resulting (#,•) is still in TU. Define new (/)) to be T~l applied
to this new (#,), so that (/)) 6 U and g^ = £, where (g() = T(g{). But then we can
successively subtract a suitable left multiple, namely git of e from the ith entry of (#,),
for i $s 2, so that all these other entries become 0. And all these subtractions can be
done by an element, say S, of El(m, D). That is, the length of ST(/j) is 1. Consequently
s = 1 as desired.
Going a bit further, we note that there is an R e El(m, D) such that if ex denotes the
first 'standard basis vector' in Dm, then K(eei) = ex (since m ^ 2). Then RST(f^) = ex.
It follows that El(m, A xa Z)el is dense in (A x a Z)m as desired.
As in the discussion preceding Question 1.8, let IA denote the C*-algebra of
continuous functions from the unit interval / into the C*-algebra A.
7.2.
COROLLARY.
For any C*-algebra A we have
tsr(/4)+l.
Proof. Let T denote the trivial action of Z on A. Then A x r Z = A ® C(T) where T
denotes the unit circle—the dual group of Z. Thus tsr(A ® C(T)) ^ tsr(/4)+ 1. But IA
is a quotient of A (g) C(T), and so the desired result follows from Theorem 4.3.
In view of the fact that tsr(C(T)) = 1, it is natural to ask:
7.3.
QUESTION.
For any two C*-algebras is it true that
tsr(A ® B) ^ tsr(/l) + tsr(B)
(at least for some suitable cross-norm if they are not nuclear)?
THE K-THEORY OF C*-ALGEBRAS
321
This will be true if both A and B are commutative, as follows from the 'product
theorem' of classical dimension theory [16,20,22]. It is clear from Theorem 6.1 that
equality can easily fail, but this failure also seems to occur in classical dimension
theory, as indicated by an example of Pontryagin [24].
One might suspect that forming even non-trivial crossed products with Z might
only raise the 'real dimension' by 1, corresponding to the comments before Question
1.8. Now AF-C*-algebras seem very zero-dimensional, in the sense that if A is an AFC*-algebra then not only is tsr(A) = 1 but also tsr(IA) = 1. So one might suspect that
for any action a of Z on an AF-algebra A one would have tsr(/4 x a Z) = 1. But
Cuntz has shown [8] that there is an AF-C*-algebra A and an action a such
that A xaZ ^ O2 ® K. Since tsr(0 2 ) = oo by Proposition 6.5, tsr(0 2 ® K) = 2 by
Theorem 6.4.
Theorem 7.1 immediately applies to the irrational rotation C*-algebras [30] to
give:
7.4. COROLLARY. Let a be an irrational number, and let it also denote the action ofZ
on A = C{T) coming from rotating by angle 2n<x. Then tsr(A x a Z) ^ 2.
This result will be crucial to the proof of the cancellation property for projective
modules over the irrational rotation C*-algebras which we will discuss in a later
paper.
7.5. QUESTION. IS tsr(A xaZ)
= 2.
8. The generator spaces
In this section we discuss the sets Lgn(A) as topological spaces, on which G\(n, A)
acts as a transformation group. But we do this in a more general setting. When we
consider A as a left ideal of itself, we are really considering A as a left /1-module,
which, of course, is free (if A has an identity element). It is then natural to consider
arbitrary finitely generated free modules, and, more generally, direct summands of
finitely generated free modules, that is, finitely generated projective modules. The
potential usefulness of considering generating families for projective modules is
suggested in, among other places, the papers of Warfield [44-48]. Throughout this
section we will tacitly assume that the Banach algebras considered always have an
identity element, and that the projective modules considered are always finitely
generated. On A" there are several equivalent norms which are commonly used, and
we will not specify one of them but rather just assume that whatever norm is used
gives A" the product topology and makes A" into a Banach A-module, that is that
|| av || ^ || a || || v || for all a e A and v e An. Any projective module will, of course,
inherit a corresponding norm from A". It is easily seen that the norms on a projective
module V coming from different realizations as summands of various A" will be
equivalent, and that V will be complete for these norms. Notice that if V is an Amodule and if the elements of Vk are viewed as 'column vectors', then Mk{A) acts on
the left on Vk in an evident way.
8.1.
NOTATION.
5388.3.46
For any left /4-module V and integer k we let Genk(K) denote the set
U
322
MARC A. RIEFFEL
of elements (u.) of Vk such that
= V,
and we will call such an element a k-generator of V.
Notice that if AA denotes A viewed as a left ,4-module, then Genk(AA) = Lgk(A).
Notice also that we are requiring algebraic generation, that is, it is not sufficient that
Av{ + ... + Avk be dense in V.
For many purposes it is useful to identify Vk with HomA{Ak, V) in the evident way.
Under this identification it is clear that Genfc(K) corresponds exactly to the
surjections. Now if V is projective, then every surjection will have a right inverse (and
conversely). But the right invertible elements of WomA{Ak, V) form an open subset.
Consequently:
8.2. PROPOSITION. Let V be a projective left A-module. Then Genk(K) is an open
subset of Vk for every k.
Let us put on Mk(A) the operator norm as operators on Ak, and let N be an open
neighbourhood of 0 in Ak such that if T e Mk(A) and if each row of T is in N then
|| T || < 1. Let x G Genfc(K) and let / be the corresponding element of UomA(Ak, V).
Since / is surjective, f(N) will be a neighbourhood of 0 in V by the open mapping
theorem. Suppose that y e (f(N)f. Then a preimage in Nk of y under/ fc can be viewed
as an element, T, of Mk{A) each row of which is in N and such that Tx = y. Then
|| T || < 1 by the choice of N, so that I + T e Gl°(/c, A). But (/ + T)x = x+y. That is,
the orbit of x under G\°(k, A) contains x + (f(N))k, a neighbourhood of x. It follows
that every orbit for the action of Gl°(/c, A) on Genk(F) is open. But the orbits are also
connected since Gl°(/c, A) is. We thus obtain:
8.3. THEOREM. Let V be a projective left A-module. Then for any k the connected
components o/Genfc(K) are exactly the orbits for the action ofG\°(k, A) on Genfc(K).
8.4. COROLLARY. Let V be a projective A-module. Then Gl°(/c, A) acts transitively on
Genfc(K) if and only //"Genk(K) is connected.
8.5. COROLLARY. If A is a Banach algebra with identity element, then \csr{A) is the
least integer n such that Lgfc(/1) is connected for all k ^ n.
Recall that we saw in Theorem 7.1 that if k ^ tsr(/4)+ 1 then El(/c, A xaZ)el is dense
in (A xaZ)k, so that, of course, Gl°(/c, A xaZ)el will also be dense. It follows that
Genfc(/4 xaZ) (— Lgfc(/4 xaZ)) is connected. We thus obtain:
8.6. COROLLARY. Let A be a C*-algebra with identity element, and let a be an action of
Z on A. Then for all k ^ tsr(A) + 1 the action of Gl°(/c, A xaZ) on hgk{A xaZ) is
transitive, that is,
csr(/4 xaZ) ^ tsr(/4)+l.
It is natural to extend Definition 4.7 to projective modules by saying that the
connected stable rank of a projective module V (denoted csr(K)) is the least integer n
such that Gl°(/c, A) acts transitively on Genk(K) for all k ^ n. Then Corollary 8.4 says
THE K-THEORY OF C*-ALGEBRAS
323
that csr(K) is the least integer n such that Genfc(K) is connected for all k ^ n.
In algebraic K-theory the subgroup El(/c, A) of G\(k, A) plays a role analogous to
that played by Gl°(/c, A) in the /(-theory of Banach algebras. In particular, the
transitivity of the action of El(/c, A) on Genfc(K) is of interest in algebraic /(-theory. To
show how this is related to the transitivity of the action of G\°(k, A), we first generalize
to the non-commutative case a result of Milnor [19, Lemma 7.4]. For this purpose we
let Hk(A) denote the subgroup of G\°(k,A) consisting of elements of the form
I where a e Gl°(l,/4). It will be convenient to let El(l,/4) consist of the
identity element alone, and Hy(A) be Gl°(l,/4).
8.7. PROPOSITION. Let A be a Banach algebra with identity element. Then every
element, T, ofG\°(k, A) is of the form RE for some R e Hk(A) and some E e E\(k, A).
Furthermore, El(/c, A) is a normal subgroup of G\°(k, A).
Proof. It is easily seen that if an elementary matrix is conjugated by an invertible
diagonal matrix, the result will again be an elementary matrix. It follows, in
particular, that El(/c, A) is normalized by the subgroup Hk(A) of G\°(k, A). Since
G\°(k, A) is connected, it thus suffices to show that there is a neighbourhood U of /
each of whose elements is a product of the desired form. This is accomplished by the
following specific analogue of Milnor's Lemma 7.4 of [19}:
8.8. LEMMA. Let Uk be the set of matrices in G\°(k, A) each of whose entries has
distance less than l/(/c— 1) from the corresponding entry oflk. IfTe Uk then T is of the
form RE for Re Hk(A) and E e El(/c, A).
Proof. We argue by induction on k, with the case where k = 1 being trivial (with
\/(k-l) = oo). So let T = (tu) e Uk. Since || 1 -tkk || < l/(/c- 1), tkk is invertible and
For each i < k let us multiply the kth row on the left by t^t^'1 and subtract it from
the /th row. This can be done by left multiplying T by an element of E\(k, A). If this is
done for all / < k, the resulting matrix will have kth row the same as T, 0 everywhere
in the last column except for the kth entry, whereas the ijth entry for i, j < k will be
tjj — tiktkk~ltkj, whose distance from the corresponding entry of / will be no greater
than
That is, T = E\
I where S e (/ k _ l5 E e El(/c, A), and v is some (k— l)-row. By
subtracting right multiples of the last column from the previous columns we can
change v to 0, without changing 5, and this can be done by right multiplying by an
element of El(/c, A). That is,
L
kk/
324
MARC A. RIEFFEL
where E, F e E\(k, A) and S e Vk.i. By the induction hypothesis S is of the form RE'
where R e H^^A) and E' e El(/c— I, A). But if we let R and £' also denote the
corresponding matrices in Hk(A) and E\(k, A) obtained by adjoining a 1 in the kkth
place, then
w
Since conjugation by diagonal matrices carries El(/c, /4) into itself, we can rewrite this
as
for a new E e El(/c, A). Now it is a standard and easily seen fact that
(
tkk'1
0
0
0
/k_2
0
0
0
tkk
is in El(/c, /4). We then have
But || 1 - t w || < 1 so that t t t e Gl°(fc, 4), and so r k k
° ) is in Hk{A).
We remark that Proposition 8.7 has the following interesting types of consequences. Let a be an invertible element of A which is not in Gl°(l, A), but is such
that (a
faaQ
I
] is in Gl°(2,/4). Then there is an element a0 of Gl°(l,/4) such that
0\
1 is in El(2, A). We also remark that it is easily seen that Hk(A) and E\(k, A)
can have more than just the identity element in common. (For example, if p, q e A
with pq^O
but qp = 0 then the commutator of (
\0
| and (
\)
\q
) will do.) Thus
\)
Gl°(/c, A) is not, in general, the semi-direct product of these groups (unless A is
commutative, which is essentially Milnor's result). Presumably there are examples for
which E\(k,A) is not closed in G\°(k,A).
8.9. PROPOSITION. Let V be a projective A-module. Ifk^2
and i / G e n ^ ^ F ) is not
empty, then Gl°(fc, A) acts transitively on Genk(K) if and only ifE\(k, A) does.
Proof. Let (y 1 ,...,w k _ 1 ) be a fixed element of Gen k _ 1 (F), and set
x = (u 1 ,...,u k _ l J 0), so that x e Genk(V). Let y be any element of Genk(K). We
assume that G\°(k,A) acts transitively on Genk(K), so that there is a T e Gl°(fc,/4)
such that Ty = x. By Proposition 8.7, T is of the form T = RE where K e //k(/l) and
THE K-THEORY OF C*-ALGEBRAS
E E E\(k, A). Let R = I *
325
). Then REy = x s o that
Ey = R~lx =
(a~lv1,v2,...,vk-1,0).
Let
Then it is clear that FEy = x. But F e El(n, A). Thus we have shown that El(/c, A)
carries any element of Genfc(/4) to x, and so acts transitively.
That the above proposition is not true in the absence of some condition such as that
Genfc_ j(/4) be non-empty is easily seen by contemplating the case in which A = C and
V = C2.
8.10. COROLLARY. Let A be a Banach algebra with identity element. If k ^ 2 then
Gl°(/c, A) acts transitively on hgk(A) if and only ifE\(k, A) does.
9. Stable rank for modules
It seems appropriate to extend to modules the concepts of stable rank considered
previously. This has already been done for the Bass stable rank by Warfield [45,48].
Here we do it for the topological stable rank and make comparisons with the Bass
stable rank. We have no immediate applications in mind, other than some small
clarifications of a few points in earlier sections, but it seems very likely that these
concepts will eventually be useful.
Let us begin by recalling Warfield's definition. In all the following we will assume
that modules are finitely generated, and that all Banach algebras or rings have an
identity element.
9.1. DEFINITION [48, p. 455; 45, p. 16]. Let V be a left ,4-module. Then the Bass
stable rank of V, denoted Bsr(K), is the smallest integer n such that i f / 0 0 is a
surjective left ,4-module homomorphism from A" 0 A onto V, then there is a
homomorphism h from A" to A such that the homomorphism f + gh from A" to A is
surjective. If no such n exists we set Bsr(K) = oo.
If AA denotes A viewed as a left ,4-module, then it is easily seen that
Bsr{AA) = \Bsr(A).
If V is a projective module, then every surjection onto V has a right inverse, and we
can use this fact to give an alternative definition of Bsr which is the analogue of
Warfield's Condition 2 of Theorem 1.6 in [47], described above in Proposition 2.2:
9.2. PROPOSITION. Let V be a projective left A-module. Then Bsr (V) is the least integer
n such that any surjection/© gfrom A" 0 A onto Vhas a right inverse h 0 k such that h
has a left inverse.
326
MARC A. RIEFFEL
Proof. Let n = Bsr^) and l e t / © g be a surjection from A" ® A onto K. Then by
definition there is a homomorphism h' from A" to A such that f + gh' is a surjection
from /I" to V. But since K is projective, this surjection has a right inverse h (so that h
has a left inverse). Let k = h'h. Then it is easily seen that h © k is a right inverse for
Conversely, suppose n is such that the condition of the proposition is fulfilled, and
l e t / © g be any surjection from A" © A onto V. Then by hypothesis this surjection
has a right inverse h@k such that h has a left inverse, say j . This says that
= Iv = jh,
so that if we let h' = /cy we obtain
Thus f + gh' is surjective. This shows that Bsr(K) ^ n.
9.3. DEFINITION. Let A be a Banach algebra and let V be a Banach /4-module. By the
topological stable rank of K, denoted tsr(K), we mean the smallest integer k such that
Genfc(K) is dense in Vk.
One might hope to compare tsr(K) and Bsr(K) immediately, but this does not work
well. A hint that it should not comes from the fact that in Theorem 2.3 the right tsr was
compared with the left Bsr. This suggests that we need an analogue for V of the right
tsr. The way to obtain this is to pass to dual modules. For any Banach /4-module V we
let V denote Hom^K, AA) and call V the dual of V. We require the elements of V to
be continuous operators, but this will be automatic if V is finitely generated
projective. By means of the right action of A on itself V becomes a right Banach Amodule, which is easily seen to be finitely generated projective if V is. Of course, if we
let V = AA, then V is naturally identified with A as a right A-module, that is, AA.
We take it as evident how to define Bsr and tsr for right modules. Then it is
convenient to make:
9.4. DEFINITION. Let V be a left Banach /4-module. Then by the dual topological
stable rank of V, denoted dtsr(K), we will mean tsr(K').
Of course, if V = AA, then dtsr(K) = rtsr(/4).
For finitely generated projective modules these ideas can be carried further, because
in this case the process of forming duals of modules is easily seen to give a natural
realization of the dual of the category of finitely generated projective left /4-modules as
the category of finitely generated projective right /4-modules, and under this duality
right-invertible elements are interchanged with left-invertible elements. But tsr(K') is
the least integer k such that right-invertible elements of HomA(Ak, V) are dense. Thus:
9.5. PROPOSITION. Let V be a finitely generated projective (Banach) A-module. Then
dtsr(K) is the least integer k such that the left-invertible elements of HornA(V,Ak) are
dense.
9.6. THEOREM. Let V be a finitely generated projective (Banach) left A-module. Then
Bsr(K) ^ dtsr(K).
THE K-THEORY OF C*-ALGEBRAS
327
Proof. Let n = dtsr(K). We use the criterion of Proposition 9.2. Let f ®g be a
surjection from A" © A onto V. Since V is projective, f ®g has a right inverse, say
h © k. According to Proposition 9.5 we can approximate h arbitrarily closely by h'
which has a left inverse j . Make this approximation sufficiently close so that
( / ® 9)(hf © k) is still invertible, with inverse t e End/1(K). Then (h't) © (fa) is a right
inverse for / © g and h't has f ~lj as left inverse, as desired.
If A is a C*-algebra, then finitely generated projective /4-modules are self-dual in an
appropriate sense, and so the distinction between tsr(K) and dtsr(K) disappears. To be
specific, put on A the /1-valued inner product defined by
(a,b}A = ab*,
and for any k put on Ak the corresponding /4-valued inner-product, so that Ak
becomes a left /l-rigged space as discussed in [27,29,31]. Then it is easily seen that Ak
is self-dual in the sense [21,28] that if / e (Ak)' then there is a z e Ak such that
f(v) = <yv,z)>A for all v e Ak. Now if V is any finitely generated projective /4-module
then it is (isomorphic to) a summand of some Ak, and so is the range of some
idempotent in EndA{Ak) £ Mk(A). But every idempotent in a C*-algebra is equivalent
to a self-adjoint idempotent, and it is easily seen that if the range of a self-adjoint
idempotent in Mk(A) is equipped with the restriction to it of the inner-product on Ak
then it also is self-dual. Thus every finitely generated projective /4-module can be
made into a self-dual /4-rigged space.
Now if V and W are two self-dual /4-rigged spaces then the usual arguments show
that for any T e Hom/1(K, W) there corresponds a T* in Hom^W, V) such that
<Tu, w> = <u, T*w} for all v e V and w e W, and that the map T \-* T* is an
isometric bijection of Hom^K, W) onto Hom^W, V) which interchanges transformations with left inverses and those with right inverses. It then becomes clear that
the right-invertible elements of WomA{Ak, V) will be dense if and only if the leftinvertible elements of Hom/1(K, Ak) are dense. We thus obtain:
9.7. PROPOSITION. If A is a C*-algebra then for every finitely generated projective left
A-module V we have
dtsr(K) = tsr(K)
so that
Bsr(K) ^ tsr(K).
We mention that in the papers of Warfield [44-48] there are further pertinent
results concerning, for example, unique presentation of modules, and minimum
numbers of generators.
10. Stability theorems
This section is largely expository. We collect together various stability theorems
from the literature of algebraic /C-theory which can be applied to C*-algebras.
Throughout this section we will assume that all algebras have an identity element and
that all projective modules are finitely generated.
We have seen that it is useful to consider whether Gl°(/c, A) acts transitively on
Lgfc(/4). This is also true for Gl(/c, A), and so we make:
328
MARC A. RIEFFEL
10.1. DEFINITION. We define the left general stable rank of an algebra A, denoted
lgsr(/4), to be the smallest integer n such that Gl(/c, A) acts (on the left) transitively on
Lgk{A) for all k ^ n. If no such integer exists we set lgst(/4) = oo. The right general
stable rank, denoted rgsr(/4), is defined similarly.
A simple argument shows that:
10.2.
PROPOSITION.
If A has an involution then
lgsr(,4) = rgsr(X).
In this case we will just speak of the general stable rank, denoted gsr(A).
Note that if Gl(/c, A) acts transitively on Lgk(A), for some k, it does not follow that it
must act transitively for all higher k (compare with [2, Chapter V, Definition (3.1)'].
For example, the condition that Gl(l,/4) acts transitively on Lg^/4) is just the
condition that all left-invertible elements of A are invertible. This condition will be
satisfied, in particular, by any commutative C*-algebra (as will the condition that
Gl(2, A) acts transitively Lg2{A), discussed on page 7 of [3]). But there are
commutative C*-algebras for which Gl(/c, A) will not act transitively on Lgk(A) for
certain higher k, as can be seen from Theorem 10.6 below together with the fact that
over certain compact spaces there are stably free vector bundles which are not free.
It is clear from earlier comments that for a C*-algebra A the various stable ranks
are related by
gsr(y4) ^
QST(A)
= csr(A) ^ Bsr(/1)+ 1 ^ tsr{A) +1.
Here, with Corollary 8.10 in mind, we have included esr(yl), the elementary stable rank
of A, defined as above but with G\(k, A) replaced by E\(k, A), though it only works as
long as k ^ 2.
We now discuss various stability results which are governed in part by the various
stable ranks, beginning with gsr(/4). Recall that by a free module we mean one which is
isomorphic to As for some s.
10.3. DEFINITION. A projective module V is said to be stably free if V © Ar s As for
some r and s.
Our first stability result concerns finding an upper bound on the values of s which
need to be considered in the above definition.
10.4. LEMMA (see [3, p. 7]). Let v e As. Then v s Rgs(A) if and only if Av is a free
direct summand with basis v of the left A-module As.
Proof. Let v = (a,). If v e Rgs(/4), then there is a w = (b,) such that £ f l A- = 1. Let /
be the evident left /1-module homomorphism from A into As defined by v, and let g be
the evident left ,4-module homomorphism from As to A defined by w. Then g is a left
inverse for / , so that Av is a free direct summand with basis v of/4s. Conversely, if v is a
basis for a free direct summand of As, with complement W, then let / be the left Amodule homomorphism from As to A whose kernel is W and which takes v to 1. Let
ex,...,es be the standard basis for As as left /1-module. Then
fl
!=/(») =/(I «^)=E ./(e,),
THE K-THEORY OF C*-ALGEBRAS
329
so that v e Rgs(/4).
10.5. PROPOSITION. The rgsr(/l) is the least integer n such that whenever W is a
projective left A-module such that W © A = As for some s ^ n, then W = X s " 1 .
Proof. Suppose that s ^ rgsr(/4) and that W © A = As. Identify W with its image in
A , and let v be a basis for the free complement corresponding to A. Then v e Rgs(A)
by Lemma 10.4. Since s ^ rgsr(A), there must be a T e Gl(s, A) such that vT = eY, the
first standard basis vector. But T defines an automorphism of the left /4-module As,
and under this automorphism W is carried to a module which is isomorphic to the
quotient of As by Aelf which is isomorphic to /4 5 " 1 .
Conversely, let n be such that whenever W is a projective left A-module such that
W © A ^ As for some s ^ n, then W ^ As~ *. For any s ^ n let u = (a,) be any element
of Rgs(/4). Then /4u is a free direct summand of As with v as basis. Let W be a
complementary summand for Av in /4s, so that W ® Av = As. Then W ^ ,4 s "* by the
choice of s, so that we can choose a free basis, w l5 ..., w s _ l5 for W. But then
v, w l 5 ..., w s _ 1 will be a free basis for As, and so there will be an element, T, of Gl(s, A)
such that vT = ex. Thus G\(s,A) acts transitively on Rgs(/4).
s
From the above cancellation result we immediately obtain the following stability
theorem:
10.6. THEOREM. IfWis a stably free projective left A-module which is not free, then W
is isomorphic to a direct summand of Ar with free complement, where r = rgsr(/4)—1.
10.7.
free.
COROLLARY.
If rgsr(A) = 1 then every stably free projective left A-module is
The following theorem, which is very applicable to the irrational rotation C*algebras, illustrates how these results can be used.
10.8. THEOREM. Let A be a C*-algebra with identity element, and let a be an action on
A of the group, Z, of integers. lftsv(A) = 1 and if A has a faithful finite cn-invariant trace,
then every stably free finitely generated projective module over AxaZ is free.
Proof. We show that gsv(A xa Z) = 1. Now csr(A xa Z) ^ 2 according to Corollary
8.6, so that gsr(/4 x a Z) ^ 2. Thus to show that gsr(/4 x a Z) = 1 it suffices to show that
every right-invertible element of A xa Z is invertible. But this follows from the fact that
xonAxaZ
the faithful finite a-invariant trace on A, say T, defines a faithful finite trace
(by the formula ?(/) = T ( / ( 0 ) ) , for every /4-valued function / of finite support on Z).
Some assumption, such as that A have a faithful a-invariant trace, is necessary for
the above theorem to be true, as is seen by the following example. Let M be the twopoint compactification of the integers obtained by adjoining points at + oo and — oo.
Let A = C(M), and let a denote the action of Z on A by means of its evident action on
M by translation, leaving the points at + oo and — oo fixed. (Note that A has finite ainvariant traces, but no faithful such.) As seen in the proof of Lemma 5 of [12], A xaZ
will contain an isometry S whose range projection P has as complement the projection
330
MARC A. RIEFFEL
Q, corresponding to the characteristic function of the point 0, viewed as an element of
A. Then Q(A xa Z) is a stably free module since
Q(AxaZ)@P(AxaZ)
= AxaZ
and P(A xa Z) is free (since S defines a module isomorphism of A xa Z with P(A xa Z)).
But Q(AxaZ) is not free since the characteristic function of either point at oo will
annihilate it.
Actually, the general cancellation property for modules can be put into a form
analogous to Proposition 10.5, as the following proposition shows. But it is not clear
to me how useful this may be.
10.9. PROPOSITION. For a ring A with identity element the following conditions are
equivalent:
1. (cancellation) ifU, V and W are finitely generated projective A-modules such that
U © W ^ V © W, then U £ V;
2. for every finitely generated projective A-module W and every integer n ^ 1,
Gl(n, A) acts transitively on Gen,,(W).
Proof For the proof it is convenient to identify GQnn(W) with the surjections in
UomA(A", W) as was done in § 8. We show first that Condition 2 implies Condition 1.
So let U, V and W be such that U © W = V © W. Then we can find an integer n and a
finitely generated projective A-module X such that W © X s A". It follows that
U © A" = V © A". In the same way we can find an integer m and module Y such that
V © Y^ Am, so that U © Y ®An^ Am+n. Now notice that Condition 2 implies that
rgsr(/4)=l. Thus we can apply Proposition 10.5 to conclude that
C / © y ^ / 4 m ^ K © y . Let / and g be surjections from Am to Y whose kernels are
isomorphic to U and V respectively. By Condition 2 there is a t e Gl(m, A) such that
ft = g. It follows that t is an isomorphism of the kernel of g onto the kernel of/, and
so gives an isomorphism of V with U.
Suppose, conversely, that Condition 1 holds. Let W be a finitely generated
projective /4-module and let / and g be surjections of A" onto W. Since W
is projective, / and g are split, say by h and k respectively. Thus
A" = ker(/) © h{W) = ker(#) © k{W). But h{W) ^ W ^ k{W), and so by Condition 1
there is an isomorphism r of ker(/) with ker(#). Let s be the isomorphism from h(W) to
k(W) such that gs = / , and let t = r@s. Then t is in Gl(n, A) and gt = / . Thus
G\(n,A) acts transitively on Genn{W).
We now turn to situations controlled by csr(a). Recall [41] that the topological
KX{A) is defined to be the direct limit of the Gl(/c, A)/G\°(k, A) where Gl(/c, A) is
embedded in Gl(/c + 1, A) as the upper left corner.
10.10. THEOREM (see [3, Lemma 2.4]). Let A be a C*-algebra. Ifn ^ csr(A) then the
natural map ofG\(n— 1, A)/G\°(n— 1, A) into Gl(n, A)/G\°(n, A) is surjective, so that
K^A) is a quotient ofG\{n-\, A)/G\°(n-l,
A).
Proof. Let T e G\(n, A). Then the columns of T are in Lgn(/4), and so by hypothesis
there is an element, S, of Gl°(n, A) which carries the last column of T to the column en,
the last 'standard basis vector'. But this means that the last column of ST will be en.
Elementary column operations can now be performed so that the last row of ST is
THE K-THEORY OF C'-ALGEBRAS
331
also en. That is, there is an element R of El(n, A) such that STR is in the image of
G l ( n - 1, A) in G\{n, A). But E\(n, A) £ G\°(n, A), so STR represents the same element
in the quotient group as does T.
If esr(/4) is used instead of csr(A) then essentially the same result holds for the
algebraic Kx defined by using El(n, A) instead of G\°{n,A)—see [3, Lemma 2.4].
For the algebraic K{ the following theorem is due to Vaserstein [42], though a
slightly weaker version was obtained earlier by Bass [2]. Vaserstein's proof appears
difficult. Possibly a somewhat shorter proof could be extracted from [39].
10.11. THEOREM (Vaserstein [42]). / / n ^ Bsr(A) + 2, then the natural map of
G\(n, A)/E\(n, A) into Gl(n + 1, A)/E\{n + 1, A) is injective, and so is an isomorphism. In
particular, alg-K^A) ^ Gl(n, A)/E\(n, A).
Using this theorem together with Proposition 8.7 we rapidly obtain the corresponding result for the topological Kt:
10.12. THEOREM. Let A be a C*-algebra. If n^ Bsr{A) + 2, then the natural map from
G\(n, A)/G\°{n, A) to Gl(n+ 1, A)/G\°(n+ 1, A) is injective, and so is an isomorphism. In
particular, K,{A) ^ Gl(n, A)/G\°(n, A).
Proof. Let T represent an element of the kernel of the natural map, so that
T 0\
I G Gl°(n + 1, A). According to Proposition 8.7 there is an a e Gl°(l, A) and an
E e E\(n+\,A) such that
'T
,0
Let S = I
0\
1/
fa 0
10 In'E-
I, so that 5 e Gl°(n, A) and S ~i T represents the same element of the
kernel as does T. Then
and so by Vaserstein's theorem S~lT e El(n, A), so that T e Gl°(n, A). Thus the
kernel of the natural map is trivial.
Warfield [47] seems to have been the first to notice a general direct relationship
between the Bass stable rank and the cancellation property for projective modules.
Recall that two projective /1-modules V and W represent the same element of K0(A)
exactly if they are stably isomorphic, that is, if V © Ak s W © Ak for some k. A
stability result for this situation for a given A would say that one could always choose
k smaller than some number which is independent of V and W. In other words, it
would say that if k is larger than some number, then one can cancel a copy of A from
both sides of the isomorphism. More generally, the cancellation question for
projective modules asks to what extent an isomorphism V © X ^ W © X implies
that V = W. Warfield's observation is that this is in part controlled by the Bass stable
rank, not of A, but of the endomorphism ring, End^X), of X. We will content
332
MARC A. RIEFFEL
ourselves here with stating a combination of parts of Theorems 1.2 and 1.6 of [47],
and refer the reader to that paper for further details.
10.13. THEOREM. Let X be a projective A-module and let m be the Bass stable rank of
E n d ^ ) . / / V and W are projective A-modules such that (V © Xm) © X ^ W © X,
then
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Department of Mathematics
University of California
Berkeley
California 94720, U.S.A.