Ranks of Algebras of Continuous<i>C</i>*-Algebra Valued Functions

Abstract

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( C 0 ( 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( C 0 ( 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 K 1 ( 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.