Continuous bundles of C*-algebras with discontinuous tensor products

Catterall, S. and Wassermann, S. (2006) Continuous bundles of C*-algebras with discontinuous tensor products. Bulletin of the London Mathematical Society, 38(4), pp. 647-656. (doi: 10.1112/S0024609306018509)

Abstract

For each non-exact C*-algebra A and infinite compact Hausdorff space X there exists a continuous bundle B of C*-algebras on X such that the minimal tensor product bundle A⊗B is discontinuous. The bundle B can be chosen to be unital with constant simple fibre. 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 ⊗ B is continuous for all unital continuous C*-bundles 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 B on [0, 1] such that A ⊗ 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 B 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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: UNSPECIFIED
• Authors: Catterall, S., and Wassermann, S.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Bulletin of the London Mathematical Society
• ISSN: 0024-6093
• ISSN (Online): 1469-2120
• Published Online: 24 July 2006