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Register a new userA reduction method for noncommutative $L_p$-spaces and applications
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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 The main tool is a unpublished result of the first named author which approximates any noncommutative $L_p$-space by tracial ones.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.
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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, Szego and inner-outer type factorizations. One new tool of the paper is the contractivity of the underlying conditional expectation on $H^p(A)$ for $p<1$.
Let $mathcal{M}$ be a semifinite von Neumann algebra. We equip the associated noncommutative $L_p$-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for $1<p<infty$ let $$L_{p,p}(mathcal{M})=big(L_{infty}(mathcal{M}),,L_{1}(mathcal{M})big)_{frac1p,,p}$$ be equipped with the operator space structure via real interpolation as defined by the second named author ({em J. Funct. Anal}. 139 (1996), 500--539). We show that $L_{p,p}(mathcal{M})=L_{p}(mathcal{M})$ completely isomorphically if and only if $mathcal{M}$ is finite dimensional. This solves in the negative the three problems left open in the quoted work of the second author. We also show that for $1<p<infty$ and $1le qleinfty$ with $p eq q$ $$big(L_{infty}(mathcal{M};ell_q),,L_{1}(mathcal{M};ell_q)big)_{frac1p,,p}=L_p(mathcal{M}; ell_q)$$ with equivalent norms, i.e., at the Banach space level if and only if $mathcal{M}$ is isomorphic, as a Banach space, to a commutative von Neumann algebra. Our third result concerns the following inequality: $$ big|big(sum_ix_i^qbig)^{frac1q}big|_{L_p(mathcal{M})}lebig|big(sum_ix_i^rbig)^{frac1r}big|_{L_p(mathcal{M})} $$ for any finite sequence $(x_i)subset L_p^+(mathcal{M})$, where $0<r<q<infty$ and $0<pleinfty$. If $mathcal{M}$ is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if $pge r$.
For a closed cocompact subgroup $Gamma$ of a locally compact group $G$, given a compact abelian subgroup $K$ of $G$ and a homomorphism $rho:hat{K}to G$ satisfying certain conditions, Landstad and Raeburn constructed equivariant noncommutative deformations $C^*(hat{G}/Gamma, rho)$ of the homogeneous space $G/Gamma$, generalizing Rieffels construction of quantum Heisenberg manifolds. We show that when $G$ is a Lie group and $G/Gamma$ is connected, given any norm on the Lie algebra of $G$, the seminorm on $C^*(hat{G}/Gamma, rho)$ induced by the derivation map of the canonical $G$-action defines a compact quantum metric. Furthermore, it is shown that this compact quantum metric space depends on $rho$ continuously, with respect to quantum Gromov-Hausdorff distances.
We introduce noncommutative weak Orlicz spaces associated with a weight and study their properties. We also define noncommutative weak Orlicz-Hardy spaces and characterize their dual spaces.
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 $2<p<infty$.
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