Noncommutative Orlicz spaces over W*-algebras

TL;DR

This paper constructs a new family of noncommutative Orlicz spaces over W*-algebras using advanced integration theory, establishing their functorial properties and isometric relations to existing spaces.

Contribution

It introduces a weight-independent construction of noncommutative Orlicz spaces and proves their functoriality and isometric isomorphism to known spaces under certain representations.

Findings

Constructed noncommutative Orlicz spaces over W*-algebras.

Proved functoriality of the construction over *-isomorphisms.

Established isometric isomorphism to Kunze's spaces under specific representations.

Abstract

Using the Falcone--Takesaki theory of noncommutative integration and Kosaki's canonical representation, we construct a family of noncommutative Orlicz spaces that are associated to an arbitrary W*-algebra without any choice of weight involved, and we show that this construction is functorial over the category of W*-algebras with *-isomorphisms as arrows. Under a choice of representation, these spaces are isometrically isomorphic to Kunze's noncommutative Orlicz spaces over crossed products.