Closed subspaces and some basic topological properties of noncommutative Orlicz spaces

Abstract

In this paper, we study the noncommutative Orlicz space $L_{\varphi }(\tilde {\mathcal {M}},\tau )$, 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 $L_{\varphi }(\tilde {\mathcal {M}},\tau )$ possesses the Fatou property, and consequently, it is a Banach space. In addition, a new description of the subspace $E_{\varphi }(\tilde {\mathcal {M}},\tau )=\overline {\mathcal {M}\bigcap L_{\varphi }(\tilde {\mathcal {M}},\tau )}$ in $L_{\varphi }(\tilde {\mathcal {M}},\tau )$, which is closed under the norm topology and dense under the measure topology, is given. Moreover, if the Orlicz function φ satisfies the Δ₂-condition, then $L_{\varphi }(\tilde {\mathcal {M}},\tau )$ is uniformly monotone, and convergence in the norm topology and measure topology coincide on the unit sphere. Hence, $E_{\varphi }(\tilde {\mathcal {M}},\tau )=L_{\varphi }(\tilde {\mathcal {M}},\tau )$ if φ satisfies the Δ₂-condition.