Dimension and Stable Rank in the K -Theory of C* -Algebras

Abstract

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