# Noncommutative Lp spaces, Operator spaces and Applications (10w5005)

## Organizers

Marius Junge (University of Illinois, Urbana-Champaign)

Gilles Pisier (Texas A & M University)

Quanhua Xu (Universite de Franche-Comte)

## Objectives

(a) Noncommutative Lp spaces. These spaces provide links between the different research topics that will be addressed. Their linear structure has recently been studied in the settings of Banach spaces as well as operator

spaces. The remarkable classification (up to isomorphism) of noncommutative Lp spaces obtained by Haagerup-Rosenthal-Sukochev (Memoirs AMS 2003) belongs to topics (1) and (2) in Overview. They have given a complete classification of noncommutative Lp spaces relative to semifinite and hyperfinite algebras. A more recent work by Haagerup-Musat (JFA 2007), which concerns topics (1) and (3), represents another important progress. But in spite of these achievements, it is still open if type III is different from type II.

The $mathcal{OL}_p$-spaces introduced by Effros-Ruan have also been subject of many publications. They are the operator space

analogue of the classical $mathcal{L}_p$-spaces in the category of Banach spaces. For $p=1$ and $p=infty$ these spaces are closely related to nuclear C*-algebras and hyperfinite von Neumann

algebras (see, in particular the works by Effros-Ozawa-Ruan). In two articles (Adv. Math. 2004 and Math. Scand. 2005), Junge-Nielsen-Ruan-Xu provide a systematic study of these spaces. As application they have proved the existence of (even completely bounded) basis for a large class of noncommutative Lp spaces ($1

remain still many unsolved problems. Here is one of them. Is an $mathcal{OL}_p$-space necessarily $mathcal{COL}_p$?

There are many other interesting studies on the linear structure of noncommutative Lp spaces and their subspaces, which belong to topics (1), (2), and (3). They raise also many open questions: Is every symmetric subspace of a noncommutative Lp already contained in a commutative Lp? Is it possible to determine the linear structure of (completely) 1-complemented subspaces of a

noncommutative Lp space with $1

appear in TAMS) solves this problem for Schatten classes.

(b) Operator spaces. The theory of operator spaces is currently in full expansion and reaches out to applications. Recently, three books (by Effros-Ruan, Pisier and Blecher-Le Merdy, respectively) on the subject appeared. The field attracts more and more people from other areas, such as operator algebras, quantum probability, or Banach spaces.

The recent works of Pisier-Shlyakhtenko (Invent. 2002) on Grothendieck theorems for operator spaces and Junge (Invent. 2005) on complete embeddings of Pisier's operator space OH into a noncommutative $L_1$-space are beautiful examples of the

interactions between topics (1), (2), (3), and (4). Pisier-Shlyakhtenko have obtained a new version for operator spaces of Grothendieck's classical theorem in Banach space geometry. Very recently, Haagerup-Musat provides an alternate approach and solved an open problem of Pisier-Shlyakhtenko (Invent. 2008).

In another related direction, Junge proved that OH is completely isomorphic to a subspace of a noncommutative L1 space. Junge's theorem has important applications. These works of

Pisier-Shlyakhtenko and Junge have been extended to the case of noncommutative Lp spaces with $1

probabilistic nature. A common key ingredient is the Khintchine inequality for Shlyakhtenko's generalized circular variables, which belongs to topic (4). It is now evident that these methods

also allow to deal with other problems in operator space theory. The recent work of Junge-Parcet on the complete embedding of noncommutative Lp (to appear in GAFA) is another good example in this direction. But several important questions that have appeared in this context remain still unanswered, in particular a construction of a completely isometric embedding of $OH$ in

noncommutative L1. It is also open whether there is a version of Pisier-Shlyakhtenko's principal theorem for bilinear maps with values in B(H)? A similar problem arises for noncommutative Lp spaces, too. Inequalities for trilinear forms are also relevant for applications in quantum information theory. However, the underlying feature is an extension of Pisier-Shlyakhtenko's inequality to trilinear forms. Let us point out that these are not problems of simple generalization. If solved, they would have many interesting consequences.

(c) Quantum probability. The theory of quantum

probability is closely related to other fields such as classical probability, functional analysis, quantum computation and information, mathematical and theoretical physics. Noncommutative martingale theory and noncommutative ergodic theory are by no means new - the first result were already obtained in the fifties. But most results prior to the nineties concerned only the Hilbert space case and are in a sense in an embryonic stage. This field has experienced a new revival in recent years due to prolific interactions with operator algebras/spaces. Using tools and ideas from these areas, it became possible to prove a large part of the noncommutative martingale or ergodic inequalities that have resisted for a long time against great efforts of quantum probabilists. These inequalities have had in turn important

applications to operator algebras/spaces and gave rise to new questions.

Noncommutative martingale inequalities have made huge progress with the work of Pisier-Xu on Burkholder-Gundy inequalities and Junge on maximal Doob inequality. The subsequent results of Junge-Xu constitute another clear progress. They treated first the case of type III, then the Burkholder-Rosenthal inequalityies, including in most cases optimal estimates of the growth of the

best constants in $p$. The recent results of Randrianantoanina, partly in collaboration with Parcet, on martingale transformations and the Gundy decomposition for noncommutative martingales represent another big step forward. Let us also mention the remarkable paper by Biane-Speicher on stochastic analysis on the

Wigner space (Proba. Th. Related Topics 1998), which contains among others a surprising inequality for free martingales in

$L_infty$, that resembles the Burkholder-Gundy inequalities in Lp ($1

necessary and sufficient condition for uniform convergence (i.e. in $L_infty$) of free martingales in terms of their square functions is still open (see a recent work of Junge-Parcet-Xu for some related results). All these subjects belong naturally to topic (4).

Concerning topics (3), (5), and free probability, one can also cite the work s of Lust-Piquard on Khintchine inequalitie. These inequalities have been extended in various directions by

Haagerup-Pisier, Buchholz, and then very recently by Parcet-Pisier (Indiana 2005), Ricard-Xu (Crelle 2006) and Junge-Parcet-Xu (Ann. Proba. 2007). A very recent work of Haagerup-Musat (to appear in JFA) determines the best constants in some cases. These Khintchine inequalities play today an increasingly important role in the operator space theory. For example, as mentioned previously, Khintchine type inequalities for Shlyakhtenko's generalized circular systems are a crucial ingredient for recent works on

Grothendieck theorems for operator spaces and the embedding of OH into noncommutative Lp spaces. Khintchine-type inequalities for homogeneous free chaos were also used by Ricard-Xu to obtain

significant results on the stability of the CCAP (completely contractive approximation property) under the reduced free product. This work by Ricard-Xu leaves several fundamental questions unanswered. For example, it is unknown if the reduced free product of a family of nuclear C*-algebras has the CCAP.

The first significant results on noncommutative ergodic theory originate from the seventies and are due to Lance, Conze-Dang-Ngoc, Kummerer and Yeadon. These earlier works deal with individual ergodic convergence in von Neumann algebras and

noncommutative L1 spaces. Maximal ergodic inequalities in noncommutative Lp spaces for $1

proving such maximal estimates is the lack of a noncommutative analogue of the pointwise supremum of a sequence of functions. Junge-Xu overcame this problem by first establishing a Marcinkiewicz type interpolation theorem, which has independent

interest and would have other applications. Finally, let us also mention an article by Anantharaman (Proba. Th. Related Topics 2006), which contains other interesting applications of Junge-Xu's results. The following fundamental problem appears in these works. Does there exist a noncommutative analog of Rota's classical dilation theorem? Very recently, Ricard gave an affirmative answer for Schur multipliers. The problem of dilating a quantum dynamical semigroup or a Markovian operator is a vibrant research topic in

quantum probability as well as in operator algebras.

(d) Noncommutative harmonic analysis.

Noncommutative Lp spaces provide a correct framework for problems on convergence of Fourier series and norm estimates for multipliers in the noncommutative setting. This program has been

carried out in certain special cases, but is far from being completed even for basic examples such as free groups.

For example, Harcharras' thesis (Studia 1999) deals with Schur multipliers on Schatten classes and noncommutative $Lambda(p)$ sets on discrete groups - studied in the setting of Banach spaces

or operator spaces. This thesis opens a promising new direction of research, at the intersection of harmonic analysis and operator spaces, and raises a number of open questions that are linked to

our topic (5). For instance, little is known about Sidon sets for noncommutative discrete groups, except some very partial results due to Bozejko, Figa-Talamanca and Picardello.

One can also include in topic (5) the current research on harmonic analysis for matrix-valued functions carried out by Gillespie, Nazarov, Petermichl, Pisier, Pott, Treil, Volberg... One

remarkable result on this subject asserts that the norm of an $ntimes n$ matrix-valued Hankel operator is of order $log n$ times the matrix-valued BMO-norm of its symbol (the upper

estimate is due to Petermichl and the lower to

Nazarov-Pisier-Treil-Volberg). In the same direction, a series of newly finished papers by Blecher-Labuschagne and Bekjan-Xu provoke

renewed interest in noncommutative $H_p$-spaces. Several problems on noncommutative $H_infty$-spaces have just been answered

positively. This has significantly clarified the situation of the subject. It is to be expected that these discoveries will also lead to new progress on noncommutative $H_p$-spaces.

On the other hand, after recent works by Junge-Le Merdy-Xu on noncommutative functional calculus, Lust-Piquard on Riesz transforms in discrete groups and Fock spaces, and Mei on operator-valued Hardy spaces, it has become natural to see to which extent one can extend the classical Littlewood-Paley-Stein theory to the noncommutative case. The theory of singular

integrals for noncommutative discrete groups is widely unexplored. A good starting point would be an extension of the celebrated Fefferman duality between $H_1$ and BMO in this new setting. This

theorem was established by Pisier-Xu for the martingale case and by Mei for the case of the Poisson semigroup. The principal subject of Junge-Le Merdy-Xu's work is the functional calculus in McIntosh's sense in noncommutative Lp spaces, including in particular square functions and diffusion semigroups. This leads

to multiplier results for analytic functions of the generators. This new approach uses in an essential way various tools recently created in operator space theory.