The Dirichlet Property for Tensor Algebras

TL;DR

This paper characterizes when tensor algebras of C*-correspondences are Dirichlet, corrects a known proof error, and explores the structure and properties of various tensor algebras and semicrossed products, including their extension properties.

Contribution

It establishes a precise condition for tensor algebras to be Dirichlet, corrects a key proof error, and demonstrates the unique extension property for several classes of tensor algebras.

Findings

Tensor algebra of a C*-correspondence is Dirichlet iff it is a Hilbert bimodule.

Corrects an error in Duncan's proof regarding tensor algebras.

Shows tensor algebras of certain graphs and dynamical systems have the unique extension property.

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.