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 $\overline{ℳ}$.