$C^{\ast }$-Envelope and Dilation Theory of Semigroup Dynamical Systems

Abstract

In this paper, we construct, for a certain class of semigroup dynamical systems, two operator algebras that are universal with respect to their corresponding covariance conditions: one being self-adjoint, and another being non-self-adjoint. We prove that the $C^{\ast }$-envelope of the non-self-adjoint operator algebra is precisely the self-adjoint one. This result leads to a number of new examples of operator algebras and their $C^{\ast }$-envelopes, with many from number fields and commutative rings. We further establish the functoriality of these operator algebras along with their applications.

arXiv comment34 pages

References in corpus (3)

• The Regular C*-algebra of an Integral Domain
• Representations of C*-dynamical systems implemented by Cuntz families
• C*-envelopes of semicrossed products by lattice ordered abelian semigroups

Cited by in corpus (1)

• Dilation theory for right LCM semigroup dynamical systems