Closed subspaces and some basic topological properties of noncommutative Orlicz spaces

In this paper, we study the noncommutative Orlicz space $L_{\phi }(\tilde{\mathcal{M}},\tau )$, which generalizes the concept of noncommutative $L^p$ space, where $\mathcal{M}$ is a von Neumann algebra, and $\phi$ is an Orlicz function. As a modular space, the space $L_{\phi }(\tilde{\mathcal{M}},\tau )$ 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 $\phi$ satisfies the ${\mathrm{\Delta }}_2$-condition, then $L_{\phi }(\tilde{\mathcal{M}},\tau )$ is uniformly monotone, and the convergence in the norm topology and measure topology coincide on the unit sphere. Hence, $E_{\phi }(\tilde{\mathcal{M}},\tau )=L_{\phi }(\tilde{\mathcal{M}},\tau )$ if $\phi$ satisfies the ${\mathrm{\Delta }}_2$-condition.