Continuous Fields of $C^*$-Algebras Arising from Extensions of Tensor $C^*$-Categories

Details Continuous Fields of $C^*$-Algebras Arising from Extensions of Tensor $C^*$-Categories Continuous Fields of $C^*$-Algebras Arising from Extensions of Tensor $C^*$-Categories

2001-01-11

arxiv.org/abs/math/0101099v1

Journal of Functional Analysis 199(1) (2003), 122-152

Mathematics

Operator Algebras

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

Scientific paper

The notion of extension of a given $C^*$-category $C$ by a $C^*$-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 $C^*$-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 $C^*$-algebras are given.

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