Semicrossed products of operator algebras and their C*-envelopes

Abstract

Let $\A$ be a unital operator algebra and let $α$ be an automorphism of $\A$ that extends to a *-automorphism of its $\ca$-envelope $\cenv (\A)$. In this paper we introduce the isometric semicrossed product $\A \times_α^{\is} \bbZ^+ $ and we show that $\cenv(\A \times_α^{\is} \bbZ^+) \simeq \cenv (\A) \times_α \bbZ$. In contrast, the $\ca$-envelope of the familiar contractive semicrossed product $\A \times_α \bbZ^+ $ may not equal $\cenv (\A) \times_α \bbZ$. Our main tool for calculating $\ca$-envelopes for semicrossed products is the concept of a relative semicrossed product of an operator algebra, which we explore in the more general context of injective endomorphisms. As an application, we extend a recent result of Davidson and Katsoulis to tensor algebras of $\ca$-correspondences. We show that if $\T_{\X}^{+}$ is the tensor algebra of a $\ca$-correspondence $(\X, \fA)$ and $α$ a completely isometric automorphism of $\T_{\X}^{+}$ that fixes the diagonal elementwise, then the contractive semicrossed product satisfies $ \cenv(\T_{\X}^{+} \times_α \bbZ^+)\simeq Ø_{\X} \times_α \bbZ$, where $Ø_{\X}$ denotes the Cuntz-Pimsner algebra of $(\X, \fA)$.