Haagerup noncommutative Orlicz spaces

TL;DR

This paper introduces Haagerup noncommutative Orlicz spaces associated with von Neumann algebras, proving their key properties and extending noncommutative martingale inequalities beyond the tracial setting.

Contribution

It establishes the independence of these spaces from the state, proves fundamental theorems like Haagerup's reduction and duality, and extends martingale inequalities to this new context.

Findings

Spaces are independent of the chosen state up to isometry

Proved Haagerup's reduction and duality theorems for these spaces

Extended noncommutative martingale inequalities to the Orlicz space setting

Abstract

Let $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra equipped with a normal faithful state $\varphi$, and let $\Phi$ be a growth function. We consider Haagerup noncommutative Orlicz spaces $L^\Phi(\M,\varphi)$ associated with $\M$ and $\varphi$, which are analogues of Haagerup $L^p$-spaces. We show that $L^\Phi(\M,\varphi)$ is independent of $\varphi$ up to isometric isomorphism. We prove the Haagerup's reduction theorem and the duality theorem for this spaces. As application of these results, we extend some noncommutative martingale inequalities in the tracial case to the Haagerup noncommutative Orlicz space case.