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 Lp 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.