Closed subspaces and some basic topological properties of noncommutative Orlicz spaces

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.