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 . We introduce the isometric semicrossed product and we show that . In contrast, the C⁎-envelope of the familiar contractive semicrossed product A×αZ+ may not equal . 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 is the tensor algebra of a C⁎-correspondence (X,A) and α is a ⁎-extendible automorphism of that fixes the diagonal elementwise, then the contractive semicrossed product satisfies , where OX denotes the Cuntz–Pimsner algebra of (X,A). This extends the main result of Davidson and Katsoulis (2010) [6].