Continuous fields of C∗-algebras arising from extensions of tensor C∗-categories

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.