We consider the reduction of problems on general noncommutative $L_p$-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncom
mutative $L_p$-space by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products by a locally compact abelian group or to their noncommutative $L_p$-spaces. We present applications of these results to the theory of noncommutative martingale inequalities by reducing most recent general noncommutative martingale/ergodic inequalities to those in the tracial case.
This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $mathbb T$, respectively. For $1< q<inft
y$ and a Banach space $B$ we prove that there exists a positive constant $c$ such that $$sup_{z_0in D}int_{D}(1-|z|)^{q-1}| abla f(z)|^q P_{z_0}(z) dA(z) le c^qsup_{z_0in D}int_{T}|f(z)-f(z_0)|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ iff $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=frac{1-|z_0|^2}{|1-bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.
We show norm estimates for the sum of independent random variables in noncommutative $L_p$-spaces for $1<p<infty$ following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. Among applications,
we derive an equivalence for the $p$-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative $L_p$ for $2<p<infty$.
Let $A$ be a finite subdiagonal algebra in Arvesons sense. Let $H^p(A)$ be the associated noncommutative Hardy spaces, $0<ple8$. We extend to the case of all positive indices most recent results about these spaces, which include notably the Riesz, Sz
ego and inner-outer type factorizations. One new tool of the paper is the contractivity of the underlying conditional expectation on $H^p(A)$ for $p<1$.
