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

Abstract

The notion of extension of a given $C^*$-category $C$ by a $C^*$-algebra $A$ is introduced. In the commutative case $A = C(Ω)$, the objects of the extension category are interpreted as fiber bundles over $Ω$ 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 $Ω$ 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.