Tensor algebras ofC∗-correspondences and theirC∗-envelopes

Abstract

We show that the C * -envelope of the tensor algebra of an arbitrary C * -correspondence X coincides with the Cuntz-Pimsner algebra O X , 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.