Exact C${}^{\ast }$-algebras and $C_0(X)$-structure

Abstract

We study tensor products of a $C_0(X)$-algebra $A$ and a $C_0(Y)$-algebra $B$, and analyse the structure of their minimal tensor product $A\otimes B$ as a $C_0(X\times Y)$-algebra. We show that when $A$ and $B$ define continuous C${}^{\ast }$-bundles, that continuity of the bundle arising from the $C_0(X\times Y)$-algebra $A\otimes B$ is a strictly weaker property than continuity of the `fibrewise tensor products' studied by Kirchberg and Wassermann. For a fixed quasi-standard C${}^{\ast }$-algebra $A$, we show that $A\otimes B$ is quasi-standard for all quasi-standard $B$ precisely when $A$ is exact, and exhibit some related equivalences.