Semicrossed products of operator algebras and their C*-envelopes

Let $\A$ be a unital operator algebra and let $\alpha$ 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 $\A \times_{\alpha}^{\is} \bbZ^+$ and we show that $\cenv(\A \times_{\alpha}^{\is} \bbZ^+) \simeq \cenv (\A) \times_{\alpha} \bbZ$. In contrast, the $\ca$-envelope of the familiar contractive semicrossed product $\A \times_{\alpha} \bbZ^+$ may not equal $\cenv (\A) \times_{\alpha} \bbZ$. 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 $\alpha$ a completely isometric automorphism of $\T_{\X}^{+}$ that fixes the diagonal elementwise, then the contractive semicrossed product satisfies $\cenv(\T_{\X}^{+} \times_{\alpha} \bbZ^+)\simeq \O_{\X} \times_{\alpha} \bbZ$, where $\O_{\X}$ denotes the Cuntz-Pimsner algebra of $(\X, \fA)$.