Quanhua Xu

Noncommutative maximal ergodic theorems

This paper is devoted to the study of various maximal ergodic theorems in noncommutative -spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on and the analogue of Steins maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in von Neumann algebra theory and in quantum probability.

A reduction method for noncommutative L_p-spaces and applications

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 The main tool is an unpublished result of the first-named author which approximates any noncommutative L p -space by tracial ones.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.

Khintchine type inequalities for reduced free products and applications

Abstract We prove Khintchine type inequalities for words of a fixed length in a reduced free product of C*-algebras (or von Neumann algebras). These inequalities imply that the natural projection from a reduced free product onto the subspace generated by the words of a fixed length d is completely bounded with norm depending linearly on d. We then apply these results to various approximation properties on reduced free products. As a first application, we give a quick proof of Dykemas theorem on the stability of exactness under the reduced free product for C*-algebras. We next study the stability of the completely contractive approximation property (CCAP) under reduced free product. Our first result in this direction is that a reduced free product of finite dimensional C*-algebras has the CCAP. The second one asserts that a von Neumann reduced free product of injective von Neumann algebras has the weak*-CCAP. In the case of group C*-algebras, we show that a free product of weakly amenable groups with constant 1 is weakly amenable.

ON THE BEST CONSTANTS IN SOME NON-COMMUTATIVE MARTINGALE INEQUALITIES

We determine the optimal orders for the best constants in the non-commutative Burkholder-Gundy, Doob and Stein inequalities obtained recently in the non-commutative martingale theory.

On subspaces of non-commutative Lp-spaces

Abstract We study some structural aspects of the subspaces of the non-commutative (Haagerup) L p -spaces associated with a general (non-necessarily semi-finite) von Neumann algebra . If a subspace X of L p ( ) contains uniformly the spaces l p n , n ⩾1, it contains an almost isometric, almost 1-complemented copy of l p . If X contains uniformly the finite dimensional Schatten classes S p n , it contains their l p -direct sum too. We obtain a version of the classical Kadec–Pelczynski dichotomy theorem for L p -spaces, p ⩾2. We also give operator space versions of these results. The proofs are based on previous structural results on the ultrapowers of L p ( ) , together with a careful analysis of the elements of an ultrapower L p ( ) U which are disjoint from the subspace L p ( ) . These techniques permit to recover a recent result of N . Randrianantoanina concerning a subsequence splitting lemma for the general non-commutative L p spaces. Various notions of p -equiintegrability are studied (one of which is equivalent to Randrianantoaninas one) and some results obtained by Haagerup, Rosenthal and Sukochev for L p -spaces based on finite von Neumann algebras concerning subspaces of L p ( ) containing l p are extended to the general case.

Maximal theorems and square functions for analytic operators on Lp-spaces

Let T : Lp --> Lp be a contraction, with p strictly between 1 and infinity, and assume that T is analytic, that is, there exists a constant K such that n\norm{T^n-T^{n-1}} < K for any positive integer n. Under the assumption that T is positive (or contractively regular), we establish the boundedness of various Littlewood-Paley square functions associated with T. As a consequence we show maximal inequalities of the form

Interpolation between vector-valued Hardy spaces

\norm{\sup_{n\geq 0}\, (n+1)^m\bigl |T^n(T-I)^m(x) \bigr |}_p\,\lesssim\, \norm{x}_p

Rosenthal type inequalities for free chaos

, for any nonnegative integer m. We prove similar results in the context of noncommutative Lp-spaces. We also give analogs of these maximal inequalities for bounded analytic semigroups, as well as applications to R-boundedness properties.

Harmonic Analysis on Quantum Tori

Abstract Let 0 p ⩽ ∞ and let H p h (X) denote the space of X -valued harmonic functions on the half-space with boundary values almost everywhere and Poisson maximal function in L p (R n ), and H p (X) the closure of the X -valued analytic polynomials on the disc under the norm given by sup 0 r f r ∥ p . It is shown that if 0 p 0 , p 1 θ 1 p = (1 − θ) p 0 + θ p 1 , then ( H p0 h (X 0 ), H p1 h (X 1 )) 0 = H p h (X 0 ) . With the restriction p 1 we prove ( H p0 h (X 0 ), H p1 h (X 1 )) 0,p = H h p (X 0, p ) . A counterexample for the case p = 1 is given for the case of real interpolation. It is also proved that H p0 (X 0 ), H p1 (X 1 )) 0 is, in general, smaller than H p (X 0 ) . Finally BMO ( X ) is also considered as the end point for interpolation.

Strong

Let A denote the reduced amalgamated free product of a family A 1 , A 2 An of von Neumann algebras over a von Neumann subalgebra B with respect to normal faithful conditional expectations E k : A k → B. We investigate the norm in Lp (A) of homogeneous polynomials of a given degree d. We first generalize Voiculescus inequality to arbitrary degree d ≥ 1 and indices 1 ≤ p ≤ ∞. This can be regarded as a free analogue of the classical Rosenthal inequality. Our second result is a length-reduction formula from which we generalize recent results of Pisier, Ricard and the authors. All constants in our estimates are independent of n so that we may consider infinitely many free factors. As applications, we study square functions of free martingales. More precisely, we show that, in contrast with the Khintchine and Rosenthal inequalities, the free analogue of the Burkholder-Gundy inequalities does not hold in L ∞ (A). At the end of the paper we also consider Khintchine type inequalities for Shlyakhtenkos generalized circular systems.