Continuous Bundles of C*-Algebras with Discontinuous Tensor Products

For each non-exact C*-algebra A and infinite compact Hausdorffspace X there exists a continuous bundle B of C*-algebras onX such that the minimal tensor product bundle AB is discontinuous.The bundle B can be chosen to be unital with constant simplefibre. When X is metrizable, B can also be chosen to be separable.As a corollary, a C*-algebra A is exact if and only if A Bis continuous for all unital continuous C*-bundles B on a giveninfinite compact Hausdorff base space. The key to proving theseresults is showing that for a non-exact C*-algebra A there existsa separable unital continuous C*-bundle B on 0,1] such thatA B is continuous on 0,1] and discontinuous at 1, a counter-intuitiveresult. For a non-exact C*-algebra A and separable C*-bundleB on 0,1], the set of points of discontinuity of A B in 0,1]can be of positive Lebesgue measure, and even of measure 1.2000 Mathematics Subject Classification 46L06 (primary), 46L35(secondary).