The Dirichlet property for tensor algebras

Abstract

We prove that the tensor algebra of a C*-correspondence $X$ is Dirichlet if and only if $X$ is a Hilbert bimodule. As a consequence, we point out and fix an error appearing in the proof of a famous result of Duncan. Secondly we answer a question raised by Davidson and Katsoulis concerning tensor algebras and semi-Dirichlet algebras, by giving an example of a Dirichlet algebra that cannot be described as the tensor algebra of any C*-correspondence. Furthermore we show that the adding tail technique, as extended by the author and Katsoulis, applies in a unique way to preserve the class of Hilbert bimodules. The exploitation of these ideas implies that the tensor algebra of row-finite graphs, the tensor algebra of multivariable automorphic $\ca$- dynamics and Peters' semicrossed product of an injective $\ca$-dynamical system have the unique extension property. The two latter provide examples of non-separable operator algebras that admit a Choquet boundary in the sense of Arveson.