Ranks of algebras of continuous C*-algebra valued functions

Abstract

We prove a number of results about the stable and particularly the real ranks of tensor products of C*-algebras under the assumption that one of the factors is commutative. In particular, we prove the following: (1) If X is any locally compact σ-compact Hausdorff space and A is any C*-algebra, then RR(C0(X) ⊗ A) ≤ dim(X) + RR(A). (2) If X is any locally compact Hausdorff space and A is any purely infinite simple C*-algebra, then RR(C0(X) ⊗ A) ≤ 1. (3) RR(C([0, 1]) ⊗ A) ≥ 1 for any nonzero C*-algebra A, and sr(C([0, 1]2) ⊗ A) ≥ 2 for any unital C*-algebra A. (4) If A is a unital C*-algebra such that RR(A) = 0, sr(A) = 1, and K1(A) = 0, then sr(C([0, 1]) ⊗ A) = 1. (5) There is a simple separable unital nuclear C*-algebra A such that RR(A) = 1 and sr(C([0, 1]) ⊗ A) = 1.