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Continuous Bundles of C*‐Algebras with Discontinuous Tensor Products
Author(s) -
Catterall Stephen,
Wassermann Simon
Publication year - 2006
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609306018509
Subject(s) - mathematics , hausdorff space , metrization theorem , separable space , tensor product , mathematics subject classification , bundle , pure mathematics , discrete mathematics , mathematical analysis , materials science , composite material
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. 2000 Mathematics Subject Classification 46L06 (primary), 46L35 (secondary).