$C^*$-Envelope and Dilation Theory of Semigroup Dynamical Systems

TL;DR

This paper explores the construction and properties of operator algebras associated with semigroup dynamical systems, establishing their $C^*$-envelopes and providing new examples with applications in number theory.

Contribution

It introduces a universal construction of operator algebras for semigroup dynamical systems and proves the $C^*$-envelope of the non-self-adjoint algebra is the self-adjoint one, with new examples and applications.

Findings

The $C^*$-envelope of the non-self-adjoint algebra equals the self-adjoint algebra.

New operator algebra examples from number fields and rings.

Functorial properties of these algebras are established.

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 $C^*$-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 $C^*$-envelopes, with many from number fields and commutative rings. We further establish the functoriality of these operator algebras along with their applications.