Semicrossed products of C*-algebras and their C*-envelopes

Let $\mathcal{C}$ be a C*-algebra and $\alpha:\mathcal{C} \rightarrow \mathcal{C}$ a unital *-endomorphism. There is a natural way to construct operator algebras which are called semicrossed products, using a convolution induced by the action of $\alpha$ on $\mathcal{C}$. We show that the C*-envelope of a semicrossed product is (a full corner of) a crossed product. As a consequence, we get that, when $\alpha$ is *-injective, the semicrossed products are completely isometrically isomorphic and share the same C*-envelope, the crossed product $\mathcal{C}_\infty \rtimes_{\alpha_\infty} \mathbb{Z}$. We show that minimality of the dynamical system $(\mathcal{C},\alpha)$ is equivalent to non-existence of non-trivial Fourier invariant ideals in the C*-envelope. We get sharper results for commutative dynamical systems.