Crossed products and C*-covers of semi-Dirichlet operator algebras

TL;DR

This paper investigates the structure of semi-Dirichlet C*-covers of operator algebras, establishing a maximal cover and demonstrating a dilation theory that links full crossed products to these maximal covers.

Contribution

It introduces the concept of a maximal semi-Dirichlet C*-cover and proves a dilation theory connecting full crossed products with these covers.

Findings

Semi-Dirichlet C*-covers form a complete lattice.

Existence of a maximal semi-Dirichlet C*-cover.

Full crossed product is isomorphic to the relative full crossed product with respect to the maximal cover.

Abstract

In this paper, we show that the semi-Dirichlet C*-covers of a semi-Dirichlet operator algebra form a complete lattice, establishing that there is a maximal semi-Dirichlet C*-cover. Given an operator algebra dynamical system we prove a dilation theory that shows that the full crossed product is isomorphic to the relative full crossed product with respect to this maximal semi-Dirichlet cover. In this way, we can show that every semi-Dirichlet dynamical system has a semi-Dirichlet full crossed product.