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

On the stable rank of simple C*-algebras

“…So dim(X i ) = 2n rank(p i ) and it follows that sr(A i ) = n + 1. From Theorem 5.1 of [16] we have sr(A) ≤ n + 1.…”

“…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) = ∞. 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 A⊗B has stable rank one; see [19].…”

“…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) 0 → K 1 (A) is an isomorphism. See also [17], [2] and [18].…”

“…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 A nif no such n exists we set sr(A) = ∞.…”

Stable rank of corner rings

“…Vaserstein's formula already contains some information in this direction, namely that sr(A) ≥ sr(M n (A)) for all n ∈ N. Since A is isomorphic to a corner ring in M n (A), corresponding to the full idempotent e 11 , this suggests the inequality sr(pAp) ≥ sr(A) for any full corner pAp of A. Such a formula was conjectured by Blackadar [3, Remark A7] to hold for the topological stable rank introduced by Rieffel in [9]. It was subsequently proved by Herman and Vaserstein [5] that the Rieffel topological stable rank and the Bass stable rank agree for any C*-algebra.…”