Continuous Fields of $C^{\ast }$-Algebras Arising from Extensions of Tensor $C^{\ast }$-Categories

Abstract

The notion of extension of a given $C^{\ast }$-category $C$ by a $C^{\ast }$-algebra $A$ is introduced. In the commutative case $A=C(\Omega )$, the objects of the extension category are interpreted as fiber bundles over $\Omega$ of objects belonging to the initial category. It is shown that the Doplicher-Roberts algebra (DR-algebra in the following) associated to an object in the extension of a strict tensor $C^{\ast }$-category is a continuous field of DR-algebras coming from the initial one. In the case of the category of the hermitian vector bundles over $\Omega$ the general result implies that the DR-algebra of a vector bundle is a continuous field of Cuntz algebras. Some applications to Pimsner $C^{\ast }$-algebras are given.

28 pages, uses xy.sty, submitted to Journal of Functional Analysis

References in corpus (1)

• Tensor products of C(X)-algebras over C(X)