Stable rank and real rank of compact transformation group C*-algebras
Let (G, X) be a transformation group, where X is a locally compact Hausdorff space and G is a compact group. We investigate the stable rank and the real rank of the transformation group C∗-algebra C0(X) ⋊ G. Explicit formulae are given in the case where X and G are second countable and X is locally of finite G-orbit type. As a consequence, we calculate the ranks of the group C∗-algebra C(R ⋊ G), where G is a connected closed subgroup of SO(n) acting on R by rotation. Introduction. For a C∗-algebra A, the stable rank sr(A) was introduced by Rieffel [24] with a view to applications in K-theory. In particular, the condition that sr(A) ≤ 2 is relevant for cancellation in K0(A) [2, 25]. The real rank RR(A), defined by Brown and Pedersen [4], is somewhat related to the stable rank but also has some different properties. There are some large classes of C∗-algebras with real rank zero (see, e.g., [5, Section V.7]). Both the stable rank and the real rank (in particular, the condition RR(A) = 0) have played a significant role in the classification theory of C∗-algebras [14, 26]. For unital A, the stable rank sr(A) is either ∞ or the smallest possible integer n such that each n-tuple in A can be approximated in norm by n-tuples (b1, . . . , bn) such that ∑n i=1 b ∗ i bi is invertible. Similarly, the real rank RR(A) is either ∞ or the smallest non-negative integer n such that each (n + 1)-tuple of self-adjoint elements in An+1 can be approximated in norm by (n+1)-tuples (b0, b1, . . . , bn) of self-adjoint elements such that ∑n i=0 b 2 i is invertible. For non-unital A, these ranks are defined to be those of the unitization of A. For the algebra of continuous functions on a compact Hausdorff 2000 Mathematics Subject Classification: 22D15, 46L35, 54H15.