CONTINUOUS BUNDLES OF C*-ALGEBRAS WITH DISCONTINUOUS TENSOR PRODUCTS

Abstract

For each non-exact C*-algebra $A$ and infinite compact Hausdorff space $X$ there exists a continuous bundle ${\mathcal B}$ of C*-algebras on $X$ such that the minimal tensor product bundle $A \otimes {\mathcal B}$ is discontinuous. The bundle ${\mathcal B}$ can be chosen to be unital with constant simple fibre. When $X$ is metrizable, ${\mathcal B}$ can also be chosen to be separable. As a corollary, a C*-algebra $A$ is exact if and only if $A\otimes {\mathcal B}$ is continuous for all unital continuous C*-bundles ${\mathcal B}$ on a given infinite compact Hausdorff base space. The key to proving these results is showing that for a non-exact C*-algebra $A$ there exists a separable unital continuous C*-bundle ${\mathcal B}$ on [0,1] such that $A\otimes {\mathcal B}$ is continuous on [0,1] and discontinuous at 1, a counter-intuitive result. For a non-exact C*-algebra $A$ and separable C*-bundle ${\mathcal B}$ on [0,1], the set of points of discontinuity of $A\otimes{\mathcal B}$ in [0,1] can be of positive Lebesgue measure, and even of measure 1.