Closed subspaces and some basic topological properties of noncommutative Orlicz spaces

TL;DR

This paper explores the structure and properties of noncommutative Orlicz spaces, generalizing noncommutative Lp spaces, and investigates their subspaces, monotonicity, and topological convergence under certain conditions.

Contribution

It provides a new description of subspaces within noncommutative Orlicz spaces and establishes conditions for their monotonicity and topological equivalences.

Findings

The space L_φ is a Banach space with the Fatou property.

A new description of the subspace E_φ is provided, showing its closure and density properties.

Under the Δ₂-condition, L_φ is uniformly monotone and norm and measure convergence coincide.

Abstract

In this paper, we study the noncommutative Orlicz space $L_{\varphi}(\widetilde{\mathcal{M}},\tau)$, which generalizes the concept of noncommutative $L^{p}$ space, where $\mathcal{M}$ is a von Neumann algebra, and $\varphi$ is an Orlicz function. As a modular space, the space $L_{\varphi}(\widetilde{\mathcal{M}},\tau)$ possesses the Fatou property, and consequently, it is a Banach space. In addition, a new description of the subspace $E_{\varphi}(\widetilde{\mathcal{M}},\tau)=\overline{\mathcal{M}\bigcap L_{\varphi}(\widetilde{\mathcal{M}},\tau)}$ in $L_{\varphi}(\widetilde{\mathcal{M}},\tau)$, which is closed under the norm topology and dense under the measure topology, is given. Moreover, if the Orlicz function $\varphi$ satisfies the $\Delta_{2}$-condition, then $L_{\varphi}(\widetilde{\mathcal{M}},\tau)$ is uniformly monotone, and the convergence in the norm topology and measure topology coincide on the unit sphere. Hence, $E_{\varphi}(\widetilde{\mathcal{M}},\tau)=L_{\varphi}(\widetilde{\mathcal{M}},\tau)$ if $\varphi$ satisfies the $\Delta_{2}$-condition.