Closed subspaces and some basic topological properties of noncommutative Orlicz spaces

In this paper, we study the noncommutative Orlicz space Lφ(ℳ~,τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{\varphi }(\tilde {\mathcal {M}},\tau )$\end{document}, which generalizes the concept of noncommutative Lp space, where ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {M}$\end{document} is a von Neumann algebra, and φ is an Orlicz function. As a modular space, the space Lφ(ℳ~,τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{\varphi }(\tilde {\mathcal {M}},\tau )$\end{document} possesses the Fatou property, and consequently, it is a Banach space. In addition, a new description of the subspace Eφ(ℳ~,τ)=ℳ⋂Lφ(ℳ~,τ)¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E_{\varphi }(\tilde {\mathcal {M}},\tau )=\overline {\mathcal {M}\bigcap L_{\varphi }(\tilde {\mathcal {M}},\tau )}$\end{document} in Lφ(ℳ~,τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{\varphi }(\tilde {\mathcal {M}},\tau )$\end{document}, 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 Lφ(ℳ~,τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{\varphi }(\tilde {\mathcal {M}},\tau )$\end{document} is uniformly monotone, and convergence in the norm topology and measure topology coincide on the unit sphere. Hence, Eφ(ℳ~,τ)=Lφ(ℳ~,τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E_{\varphi }(\tilde {\mathcal {M}},\tau )=L_{\varphi }(\tilde {\mathcal {M}},\tau )$\end{document} if φ satisfies the Δ2-condition.