Weak Hopf Algebras and Reducible Jones Inclusions of Depth 2. I: From Crossed products to Jones towers

TL;DR

This paper explores the structure of reducible inclusions of von Neumann algebras using weak Hopf algebra theory, establishing a link between crossed products, Jones towers, and depth 2 properties.

Contribution

It introduces a framework connecting weak C*-Hopf algebras with Jones inclusions, constructing Jones projections and showing the finite index and depth 2 properties of such inclusions.

Findings

Constructed faithful conditional expectations from left integrals.

Established Jones relations for projections in the crossed product.

Proved that the inclusion has finite index and depth 2.

Abstract

We apply the theory of finite dimensional weak C^*-Hopf algebras A as developed by G. B\"ohm, F. Nill and K. Szlach\'anyi to study reducible inclusion triples of von-Neumann algebras N \subset M \subset (M\cros\A). Here M is an A-module algebra, N is the fixed point algebra and \M\cros\A is the crossed product extension. ``Weak'' means that the coproduct \Delta on A is non-unital, requiring various modifications of the standard definitions for (co-)actions and crossed products. We show that acting with normalized positive and nondegenerate left integrals l\in\A gives rise to faithful conditional expectations E_l: M-->N, where under certain regularity conditions this correspondence is one-to-one. Associated with such left integrals we construct ``Jones projections'' e_l\in\A obeying the Jones relations as an identity in M\cros\A. Finally, we prove that N\subset M always has finite index and depth 2 and that the basic Jones construction is given by the ideal M_1:=M e_l M \subset M\cros\A, where under appropriate conditions M_1 = M\cros\A. In a subsequent paper we will show that converseley any reducible finite index and depth-2 Jones tower of von-Neumann factors (with finite dimensional centers) arises in this way.