Weak Hopf algebra symmetries of C^*-algebra inclusions

TL;DR

This paper explores the symmetries of C^*-algebra inclusions using weak Hopf algebras, establishing a reconstruction theorem that characterizes such inclusions via algebra actions and categorical methods.

Contribution

It proves a reconstruction theorem linking finite index depth 2 C^*-algebra inclusions to weak Hopf algebra actions, using C^*-2-category techniques.

Findings

Reconstruction of algebra inclusions via weak Hopf algebra actions

Establishment of isomorphism between inclusions and invariant subalgebras

Application of C^*-2-category language to operator algebra symmetries

Abstract

After a summary on module algebra actions of C^*-weak Hopf algebras we outline the proof of a reconstruction theorem stating that every finite index depth 2 inclusion N < M of unital C^*-algebras with finite dimensional centers is isomorphic to the invariant subalgebra inclusion M^A < M with respect to a regular weak Hopf algebra action. The proof uses the language of C^*-2-categories.