A crossed product approach to Orlicz spaces

TL;DR

This paper develops a new approach using crossed product techniques to construct and analyze noncommutative Orlicz spaces for different types of von Neumann algebras, including type III, and introduces a modified interpolation method.

Contribution

It introduces a crossed product framework for noncommutative Orlicz spaces and extends the theory to type III algebras, also proposing a new interpolation method.

Findings

Reconstruction of noncommutative Orlicz spaces via crossed products.

Construction of Orlicz spaces for type III von Neumann algebras.

Introduction of a modified K-method for interpolation.

Abstract

We show how the known theory of noncommutative Orlicz spaces for semifinite von Neumann algebras equipped with an fns trace, may be recovered using crossed product techniques. Then using this as a template, we construct analogues of such spaces for type III algebras. The constructed spaces naturally dovetail with and closely mimic the behaviour of Haagerup $L^p$-spaces. We then define a modified $K$-method of interpolation which seems to better fit the present context, and give a formal prescription for using this method to define what may be regarded as type III Riesz-Fischer spaces.