Haagerup noncommutative Orlicz spaces

Let $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra equipped with a normal faithful state $\phi$, and let $\mathrm{\Phi }$ be a growth function. We consider Haagerup noncommutative Orlicz spaces $L^{\mathrm{\Phi }}(\text{M},\phi )$ associated with $\text{M}$ and $\phi$, which are analogues of Haagerup $L^p$-spaces. We show that $L^{\mathrm{\Phi }}(\text{M},\phi )$ is independent of $\phi$ 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.