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Let A be a C*-algebra. We prove that every absolutely summing operator from A into ${\ell }_2$ factors through a Hilbert space operator that belongs to the 4-Schatten-von Neumann class. We also provide finite-dimensional examples that show that one cannot replace the 4-Schatten-von Neumann class by the p-Schatten-von Neumann class for any p < 4. As an application, we show that there exists a modulus of capacity ε → N(ε) so that if A is a C*-algebra and $T\in {\Pi }_1(A,{\ell }_2)$ with ${\pi }_1\left(T\right)\le 1$, then for every ε >0, the ε-capacity of...

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 study Banach space properties of non-commutative martingale VMO-spaces associated with general von Neumann algebras. More precisely, we obtain a version of the classical Kadets-Pełczyński dichotomy theorem for subspaces of non-commutative martingale VMO-spaces. As application we prove that if ℳ is hyperfinite then the non-commutative martingale VMO-space associated with a filtration of finite-dimensional von Neumannn subalgebras of ℳ has property (u).

Let ℳ be a hyperfinite finite von Nemann algebra and ${\left({ℳ}_k\right)}_{k\ge 1}$ be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration ${\left({ℳ}_k\right)}_{k\ge 1}$. For a finite noncommutative martingale $x={\left(x_k\right)}_{1\le k\le n}\subseteq L₁\left(ℳ\right)$ adapted to ${\left({ℳ}_k\right)}_{k\ge 1}$ and 0 < α < 1, the fractional integral of x of order α is defined by setting $I^{\alpha }x={\sum }_{k=1}^n{\zeta }_k^{\alpha }d{x}_k$ for an appropriate sequence ${\left({\zeta }_k\right)}_{k\ge 1}$ of scalars. For the case of a noncommutative dyadic martingale in L₁() where is the type II₁ hyperfinite factor equipped...