Closed subspaces and some basic topological properties of noncommutative Orlicz spaces

In this paper, we study the noncommutative Orlicz space L-phi((M) over tilde, tau), which generalizes the concept of noncommutative L-p space, where M is a von Neumann algebra, and phi is an Orlicz function. As a modular space, the space L-phi((M) over tilde, tau) possesses the Fatou property, and consequently, it is a Banach space. In addition, a new description of the subspace in E-phi((M) over tilde, tau) = <(M boolean AND L-phi(<(M)over tilde>, tau)$)over bar> in L-phi((M) over tilde, 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((M) over tilde, tau) is uniformly monotone, and convergence in the norm topology and measure topology coincide on the unit sphere. Hence, if E-phi((M) over tilde, tau) = L-phi((M) over tilde, tau) if phi satisfies the Delta(2)-condition.