Tensor Algebras overC*-Correspondences: Representations, Dilations, andC*-Envelopes

Tensor algebras overC*-correspondences are noncommutative generalizations of the disk algebra. They contain, as special cases, analytic crossed products, semi-crossed products, and Popescuʹs noncommutative disk algebras. In this paper, a dilation theorem and a commutant lifting theorem for representations of tensor algebras is proved. These are used to show that for certainC*-correspondences, theC*-envelopes (in the sense of Arveson) of the tensor algebras are the Cuntz–Pimsner algebras of the correspondences.