On Haagerup noncommutative quasi $H^p(\A)$ spaces

TL;DR

This paper investigates noncommutative Hardy spaces associated with subdiagonal algebras in von Neumann algebras, establishing independence from states for 0<p<1 and extending factorization and interpolation results.

Contribution

It proves independence of $H^p(\mathcal{A})$ from the state $\varphi$ for 0<p<1 and extends Riesz factorization and interpolation theorems to this setting.

Findings

$H^p(\mathcal{A})$ is independent of $\varphi$ for 0<p<1

Extended Riesz type factorization to 0<p<1 for type 1 subdiagonal algebras

Proved interpolation theorem for $H^p(\mathcal{A})$ for $0 < p_0, p_1 \\le \\infty$

Abstract

Let $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra, equipped with a normal faithful state $\varphi$, and let $\mathcal{A}$ be a maximal subdiagonal subalgebra of $\mathcal{M}$. We have proved that for $0< p<1$, $H^p(\mathcal{A})$ is independent of $\varphi$. Furthermore, in the case that $\mathcal{A}$ is a type 1 subdiagonal subalgebra, we have extended the most recent results about the Riesz type factorization to the case $0<p<1$ and have proved an interpolation theorem for $H^p(\mathcal{A})$ in the case where $0 < p_0, p_1 \le \infty$.