Let $M$ be a von Neumann algebra, let $\varphi$ be a weight on $M$ and let $\Phi$ be $N$-function satisfying the $({\delta }_2,{\Delta }_2)$-condition. In this paper we study Orlicz spaces, associated with $M$, $\varphi$ and $\Phi$.

We continue our study of outer elements of the noncommutative $H^p$ spaces associated with Arveson’s subdiagonal algebras. We extend our generalized inner-outer factorization theorem, and our characterization of outer elements, to include the case of elements with zero determinant. In addition, we make several further contributions to the theory of outers. For example, we generalize the classical fact that outers in $H^p$ actually satisfy the stronger condition that there exist aₙ ∈ A with haₙ ∈ Ball(A)...

In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an $L_p$-space, then it is either an $L_p$-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non-Hilbertian complemented operator subspaces of non-commutative $L_p$-spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p < ∞ which can be considered...

Let G be a locally compact abelian group and ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. We study Hilbert transforms associated with G-flows on ℳ and closed semigroups Σ of Ĝ satisfying the condition Σ ∪ (-Σ) = Ĝ. We prove that Hilbert transforms on such closed semigroups satisfy a weak-type estimate and can be extended as linear maps from L¹(ℳ,τ) into $L^{1,\infty }(ℳ,\tau )$. As an application, we obtain a Matsaev-type result for p = 1: if x is a quasi-nilpotent compact operator...

We show that recently introduced noncommutative $L_p$-spaces can be used to constructions of Markov semigroups for quantum systems on a lattice.

Let $ℳ$ be a factor of type II${}_{\infty }$ or II${}_1$ having separable predual and let $\overline{ℳ}$ be the algebra of affiliated $\tau$-measurable operators. We characterize the commutator space $[ℐ,𝒥]$ for sub-$(ℳ,ℳ)$- bimodules $ℐ$ and $𝒥$ of $\overline{ℳ}$.