Closed subspaces and some basic topological properties of noncommutative Orlicz spaces

Abstract

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