A few remarks on Pimsner-Popa bases and regular subfactors of depth 2

TL;DR

This paper characterizes certain finite index subfactors of type II_1 factors as crossed products by weak Kac algebras, and establishes the existence of Pimsner-Popa bases under specific conditions.

Contribution

It proves that regular depth 2 subfactors with commutative first relative commutant are crossed products by weak Kac algebras, and shows existence of Pimsner-Popa bases in these contexts.

Findings

Regular depth 2 subfactors with commutative first relative commutant are crossed products by weak Kac algebras.

Existence of Pimsner-Popa bases for depth 2 inclusions with simple first relative commutant.

Existence of unitary orthonormal bases for regular inclusions with certain properties.

Abstract

We prove that a finite index regular inclusion of $II_1$-factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of $II_1$-factors which is of depth $2$ and has simple first relative commutant (respectively, is regular and has commutative or simple first relative commutant) admits a two-sided Pimsner-Popa basis (respectively, a unitary orthonormal basis)