Closed subspaces and some basic topological properties of noncommutative Orlicz spaces

In this paper, we study the noncommutative Orlicz space L _φ(M, τ), which generalizes the concept of noncommutative L^p space, where M is a von Neumann algebra, and φ is an Orlicz function. As a modular space, the space L _φ(M, τ) possesses the Fatou property, and consequently, it is a Banach space. In addition, a new description of the subspace E _φ( M, τ) =M ∩ L _φ(M, τ) in L _φ( 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 Δ₂-condition, then L _φ( M, τ) is uniformly monotone, and convergence in the norm topology and measure topology coincide on the unit sphere. Hence, E _φ( M, τ) = L _φ( M, τ) if φ satisfies the Δ₂-condition.