An Extrapolation of Operator Valued Dyadic Paraproducts

We consider the dyadic paraproducts \pi_\f on \T associated with an \M-valued function \f. Here \T is the unit circle and \M is a tracial von Neumann algebra. We prove that their boundedness on L^p(\T,L^p(\M)) for some $1<p<\infty$ implies their boundedness on L^p(\T,L^p(\M)) for all $1<p<\infty$ provided \f is in an operator-valued BMO space. We also consider a modified version of dyadic paraproducts and their boundedness on $L^p(\T,L^p(\M))

An Extrapolation of Operator Valued Dyadic Paraproducts

We consider the dyadic paraproducts \pi_\f on \T associated with an \M-valued function \f. Here \T is the unit circle and \M is a tracial von Neumann algebra. We prove that their boundedness on L^p(\T,L^p(\M)) for some $1<p<\infty$ implies their boundedness on L^p(\T,L^p(\M)) for all $1<p<\infty$ provided \f is in an operator-valued BMO space. We also consider a modified version of dyadic paraproducts and their boundedness on $L^p(\T,L^p(\M))

Notes on Matrix Valued Paraproducts

Denote by $M_n$ the algebra of $n\times n$ matrices. We consider the dyadic paraproducts $\pi_b$ associated with $M_n$ valued functions $b$, and show that the $L^\infty (M_n)$ norm of $b$ does not dominate $||\pi_b||_{L^2(\ell _n^2)\to L^2(\ell_n^2)}$ uniformly over $n$. We also consider paraproducts associated with noncommutative martingales and prove that their boundedness on bounded noncommutative $L^p-$% martingale spaces implies their boundedness on bounded noncommutative $L^q-$% martingale spaces for all $1<p<q<\infty$.Comment: 12 page

BMO is the intersection of two translates of dyadic BMO

Let T be the unite circle on $R^2$. Denote by BMO(T) the classical BMO space and denote by BMO_D(T) the usual dyadic BMO space on T. We prove that, BMO(T) is the intersction of BMO_D(T) and a translate of BMO_D(T).Comment: 4 page

Complete boundedness of the Heat Semigroups on the von Neumann Algebra of hyperbolic groups

We prove that $(\lambda_g\mapsto e^{-t|g|^r}\lambda_g)_{t>0}$ defines a completely bounded semigroup of multipliers on the von Neuman algebra of hyperbolic groups for all real number $r$. One ingredient in the proof is the observation that a construction of Ozawa allows to characterize the radial multipliers that are bounded on every hyperbolic graph, partially generalizing results of Haagerup--Steenstrup--Szwarc and Wysocza\'nski. Another ingredient is an upper estimate of trace class norms for Hankel matrices, which is based on Peller's characterization of such norms.Comment: v2: 28 pages, with new examples, new results, motivations and hopefully a better presentatio