Crossed Products of Operator Algebras

TL;DR

This paper develops a comprehensive theory of crossed products for operator algebras, extending classical results and analyzing specific cases like tensor and triangular AF algebras, with implications for C*-envelopes and duality.

Contribution

It introduces a general framework for crossed products of operator algebras and explores key properties and special cases, advancing the understanding of their structure and duality theories.

Findings

Extended classical results to non-selfadjoint operator algebras

Analyzed crossed products of tensor and triangular AF algebras

Addressed the Hao-Ng isomorphism problem and C*-envelopes

Abstract

We study crossed products of arbitrary operator algebras by locally compact groups of completely isometric automorphisms. We develop an abstract theory that allows for generalizations of many of the fundamental results from the selfadjoint theory to our context. We complement our generic results with the detailed study of many important special cases. In particular we study crossed products of tensor algebras, triangular AF algebras and various associated C*-algebras. We make contributions to the study of C*-envelopes, semisimplicity, the semi-Dirichlet property, Takai duality and the Hao-Ng isomorphism problem. We also answer questions from the pertinent literature.