Closed subspaces and some basic topological properties of noncommutative Orlicz spaces

Abstract

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 $E_{\phi }(\tilde{\mathcal{M}},\tau )=\overline{\mathcal{M}\bigcap L_{\phi }(\tilde{\mathcal{M}},\tau )}$ in $L_{\phi }(\tilde{\mathcal{M}},\tau )$, which is closed under the norm topology and dense under the measure topology, is given. Moreover, if the Orlicz function $\phi$ satisfies the ${\Delta }_2$-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 ${\Delta }_2$-condition.

References in corpus (1)

• Banach embedding properties of non-commutative L^p-spaces