On the stable rank of simple C*-algebras

Abstract

In [16] M. A. Rieffel introduced the notion of stable rank for C*-algebras. For A a unital C*-algebra the stable rank, denoted by sr(A), is the least integer n such that the set of n-tuples over A which generate A as a left ideal is dense in An if no such n exists we set sr(A) = oo. For unital C*-algebras the stable rank was shown to coincide with the Bass stable rank, which is defined for rings; see [12]. The stable rank can also be defined for nonunital C*-algebras, but we shall only consider the unital case. Stable rank is related to nonstable K-theory: In [16, Theorem 10.12] and [3, Corollary 7.14] it was shown that if A is a simple unital C*-algebra of stable rank one, then the natural map U(A)/U(A)o -4 Ki(A) is an isomorphism. See also [17], [2] and [18]. It is a natural question which values occur as the stable rank of simple C*algebras. Suppose that A is simple and infinite. Then, by [7, Proposition 1.5], A contains two isometries with orthogonal ranges and so, by [16, Proposition 6.5], sr(A) = oo. In the case of finite, simple C*-algebras the following is known: Whenever A is simple and stably finite and B is a UHF-algebra, the tensor product A0 B has stable rank one; see [19]. This is also the case for every simple approximately homogeneous (AH) algebra with slow dimension growth; see [4] and [8]. So, in short, sr = oo for every infinite simple C*-algebra and sr = 1 for various classes of finite simple C*-algebras. In this paper we answer the question above by showing that every n C N U {oo} is the stable rank of some simple, finite C*-algebra. More precisely, it is shown that for every n = 2, 3, 4,... ., oo there exists a simple, separable and unital AH-algebra A of stable rank n. In particular, the group of invertible elements in A is not dense. And, like the examples in [20], A does not have slow dimension growth and is not approximately divisible; see [5]. I thank Marius Dadarlat for suggesting that I study AH-algebras with connecting homomorphisms as described in the following section, and Juliana Erlijman for drawing my attention to some of the powerful tools of differential topology, and I thank George Elliott for my stay at The Fields Institute, where this paper was written.