Tensor algebras of C∗-correspondences and their C∗-envelopes

We show that the C∗-envelope of the tensor algebra of an arbitrary C∗-correspondence X coincides with the Cuntz–Pimsner algebra OX, as defined by Katsura [T. Katsura, On C∗-algebras associated with C∗-correspondences, J. Funct. Anal. 217 (2004) 366–401]. This improves earlier results of Muhly and Solel [P.S. Muhly, B. Solel, Tensor algebras over C∗-correspondences: Representations, dilations and C∗-envelopes, J. Funct. Anal. 158 (1998) 389–457] and Fowler, Muhly and Raeburn [N. Fowler, P. Muhly, I. Raeburn, Representations of Cuntz–Pimsner algebras, Indiana Univ. Math. J. 52 (2003) 569–605], who came to the same conclusion under the additional hypothesis that X is strict and faithful.