3 THE DIRICHLET PROPERTY FOR TENSOR ALGEBRAS
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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 C *dynamics and Peters' semicrossed product of an injective C * -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.
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