Semicrossed products of operator algebras and their C∗-envelopes

Let A be a unital operator algebra and let α be an automorphism of A that extends to a ∗-automorphism of its C∗-envelope C∗env(A). We introduce the isometric semicrossed product A ×is α Z+ and we show that C∗env(A ×is α Z+) C∗env(A) ×α Z. In contrast, the C∗-envelope of the familiar contractive semicrossed productA×α Z+ may not equal C∗env(A)×α Z. Our main tool for calculating C∗-envelopes for semicrossed products is a new concept, the relative semicrossed product of an operator algebra by an endomorphism. As an application of our theory, we show that if T + X is the tensor algebra of a C∗-correspondence (X,A) and α is a ∗-extendible automorphism of T + X that fixes the diagonal elementwise, then the contractive semicrossed product satisfies C∗env(T + X ×α Z+) OX ×α Z, where OX denotes the Cuntz–Pimsner algebra of (X,A). This extends the main result of Davidson and Katsoulis (2010) [6]. © 2012 Elsevier Inc. All rights reserved