Regular irreducible inclusions of simple $C^*$-algebras and crossed product structure

TL;DR

This paper characterizes regular irreducible inclusions of simple unital C*-algebras with a conditional expectation as reduced twisted crossed products, extending previous finite-index results.

Contribution

Introduces a generalized quasi-basis concept and shows such inclusions are isomorphic to reduced twisted crossed products, broadening the crossed product framework.

Findings

Every regular irreducible inclusion admits a unitary orthonormal generalized quasi-basis.

Such inclusions are canonically isomorphic to reduced twisted crossed products.

Extends crossed product characterizations beyond finite-index cases.

Abstract

We study regular irreducible inclusions $B\subset A$ of simple unital $C^*$-algebras admitting a conditional expectation. We introduce a generalized notion of quasi-basis extending Watatani's framework and show that such inclusions admit a unitary orthonormal generalized quasi-basis. As a consequence, we prove that every regular irreducible inclusion in this setting is canonically isomorphic to a reduced twisted crossed product of $B$ by its Weyl group. This extends earlier crossed product characterizations beyond the finite-index setting.