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

Abstract

Let $\mathcal{C}$ be a C*-algebra and $\alpha :\mathcal{C}\to \mathcal{C}$ 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 ${\mathcal{C}}_{\infty }{⋊}_{{\alpha }_{\infty }}\mathbb{Z}$. 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.

27 pages, changes around theorem 4.8

References in corpus (6)

• Adding tails to C*-correspondences
• The Jacobson radical for analytic crossed products
• Representations of C*-dynamical systems implemented by Cuntz families
• The C${}^{\ast }$-envelope of a semicrossed product and nest representations
• Extensions and Dilations for $C^{\ast }$-dynamical Systems
• Semicrossed Products and Reflexivity

Cited by in corpus (3)

• Representations of C*-dynamical systems implemented by Cuntz families
• C*-envelopes of semicrossed products by lattice ordered abelian semigroups
• Conjugate Dynamical Systems on C*-algebras