Kadec-Klee property for convergence in measure of noncommutative Orlicz spaces

TL;DR

This paper investigates the Kadec-Klee property for convergence in measure in noncommutative Orlicz spaces, establishing conditions under which this property holds and exploring implications for duality and reflexivity.

Contribution

It proves that noncommutative Orlicz spaces have the Kadec-Klee property in measure when the Orlicz function satisfies the Δ₂ condition, and discusses duality and reflexivity.

Findings

Kadec-Klee property holds under Δ₂ condition

Dual space characterization provided

Reflexivity of the space established

Abstract

In this paper, we study the Kadec-Klee property for convergence in measure of noncommutative Orlicz spaces $L_{\varphi}(\widetilde{\mathcal{M}},\tau)$, where $\widetilde{\mathcal{M}}$ is a von Neumann algebra, and $\varphi$ is an Orlicz function. We show that if $\varphi\in\Delta_{2}$, $L_{\varphi}(\widetilde{\mathcal{M}},\tau)$ has the Kadec-Klee property in measure. As a corollary, the dual space and reflexivity of $L_{\varphi}(\widetilde{\mathcal{M}},\tau)$ are given.