Ranks of algebras of continuous C*-algebra valued functions

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 or-compact Hausdorff space and A is any C*-algebra, then RR ( C-0 (X) circle times A) less than or equal to dim(X) + RR(A). (2) If X is any locally compact Hausdorff space and A is any purely infinite simple C*-algebra, then RR C-0(X) circle times A) less than or equal to 1. (3) RR C([0, 1]) circle times A) greater than or equal to 1 for any nonzero C*-algebra A, and sr(C([0, 1](2)) circle times A) greater than or equal to 2 for any unital C*-algebra A. (4) If A is a unital C*-algebra such that RR(A) = 0, sr(A) = 1, and K-1 (A) = 0, then sr (C([0, 1]) circle times A) = 1. (5) There is a simple separable unital nuclear C*-algebra A such that RR(A) = I and sr(C([0, 1]) circle times A) = 1.