bilinear restriction estimates for surfaces with curvatures of different signs recently the sharp l 2 bilinear adjoint restriction estimates for the cone and the paraboloid were established by wolff and tao respectively their results rely on the fact that for the cone and the paraboloid the nonzero principal curvatures have the same sign we generalize those bilinear restriction estimates to surfaces with curvatures of different signs

restriction theorems for a surface with negative curvature we prove a bilinear restriction theorem for a surface of negative curvature this is the analogue of the results of t wolff 19 and t tao 14 15 for cones and paraboloids as a consequence we obtain an almost sharp linear restriction theorem

a local smoothing estimate for the schrodinger equation we show that the schrodinger operator eit is bounded from to for all a 2n 1 2 1 q 2 q and q2 4 n 1 this is almost sharp with respect to the sobolev index we also show that the schrodinger maximal operator sup0 s0 if and only if it is bounded from to when s 2s0 a corollary is that sup0 3 4

asymptotic expansions in the clt in free probability we prove edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions the proofs are based on the analytic definition of free convolution

restriction estimates for some surfaces with vanishing curvatures abstract we obtain bilinear restriction estimates for surfaces with vanishing curvatures as application we also prove new linear restriction estimates for some class of conic surfaces

gruenhage compacta and strictly convex dual norms abstract we prove that if k is a gruenhage compact space then c k admits an equivalent strictly convex dual norm as a corollary we show that if x is a banach space and x span k where k is a gruenhage compact in the w topology and is equivalent to a coarser w lower semicontinuous norm on x then x admits an equivalent strictly convex dual norm we give a partial converse to the first result by showing that if is a tree then c 0 admits an equivalent strictly convex dual norm if and only if is a gruenhage space finally we present some stability properties satisfied by gruenhage spaces in particular gruenhage spaces are stable under perfect images

strictly convex norms and topology we introduce a new topological property called and the corresponding class of topological spaces which includes spaces with g diagonals and gruenhage spaces using we characterize those banach spaces which admit equivalent strictly convex norms and give an internal topological characterization of those scattered compact spaces k for which the dual banach space c k admits an equivalent strictly convex dual norm we establish some relationships between and other topological concepts and the position of several well known examples in this context for instance we show that c k admits an equivalent strictly convex dual norm where k is kunen s compact s space also under additional axioms we provide examples of compact scattered non gruenhage spaces of cardinality 1 having

sigma finite dual dentability indices abstract a characterization of banach spaces admitting uniformly g teaux smooth norms in terms of finite dual dentability indices is given some applications in the area of weak compactness are discussed we also study locally uniformly rotund dual renormings in connection with countable dual dentability indices

improved restriction estimate for hyperbolic surfaces in r3 abstract recently l guth improved the restriction estimate for the surfaces with strictly positive gaussian curvature in r 3 in this paper we extend his restriction estimate to the quadratic surfaces with strictly negative gaussian curvature in r 3

l 2 homology for von neumann algebras the aim of this paper is to introduce a notion of l homology in the context of von neumann algebras finding a suitable co homology theory for von neumann algebras has been a dream for several generations see kr71a kr71b jkr72 ss95 and references therein one s hope is to have a powerful invariant to distinguish von neumann algebras unfortunately little positive is known about the kadison ringrose cohomology h b m m except that it vanishes in many cases furthermore there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von neumann algebra our interest in developing an l cohomology theory was revived by the introduction of lcohomology invariants in the field of ergodic equivalence relations in the paper of gaboriau gab02 his results in particular imply that l betti numbers 2 i of a discrete group are the same for measure equivalent groups i e for groups that can generate isomorphic ergodic measure preserving equivalence relations parallels between the worlds of von neumann algebras and measurable equivalence relations have been noted for a long time starting with the parallel between the work of murray and von neumann mvn and that of h dye dy thus there is hope that an invariant of a group that survives measure equivalence will survive also von neumann algebra equivalence i e will be an invariant of the von neumann algebra of the group the original motivation for our construction comes from the well understood analogy between the theory of ii1 factors and that of discrete groups based on the theory of correspondences con con94 this analogy has been remarkably efficient to transpose analytic properties such as amenability or property t from the group context to the factor context con80 cj and more recently in the breakthrough work of popa popa con03 we use the theory of correspondences together with the algebraic description of l betti numbers given by luck his definition involves the computation of the algebraic group homology with coefficients in the group von neumann algebra following the guiding idea that the category of bimodules over a von neumann algebra is the analogue of the category of modules over a group we are led to the following algebraic definition of l homology of a von neumann algebra m

free transportation cost inequalities via random matrix approximation by means of random matrix approximation procedure we re prove biane and voiculescu s free analog of talagrand s transportation cost inequality for measures on r in a more general setup furthermore we prove the free transportation cost inequality for measures on t as well by extending the method to special unitary random matrices

hilbert function spaces and the nevanlinna pick problem on the polydisc abstract in 4 agler and mccarthy 2005 used hilbert function spaces to study the uniqueness properties of the nevanlinna pick problem on the bidisc in this work we give a geometric procedure for constructing a nevanlinna pick problem on d n with a specified set of uniqueness on the way to establishing this procedure we prove a result about hilbert function spaces and partially answer a question posed by agler and mccarthy

stable equivalence of the weak closures of free groups convolution algebras we prove in this paper that the von neumann algebras associated to the free non commutative groups are stably isomorphic i e that they are isomorphic when tensorized by the algebra of all linear bounded operators on a separable infinite dimensional hilbert space this gives positive evidence for an old question due to r v kadison see also s sakai s book on w algebras whether the von neumann algebras associated to free groups are isomorphic or not

fourier restriction to a hyperbolic cone abstract using a bilinear restriction theorem of lee and a bilinear to linear argument of stovall we obtain the conjectured range of fourier restriction estimates for a conical hypersurface in r 4 with hyperbolic cross sections

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

addition of freely independent random variables abstract a direct proof is given of voiculescu s addition theorem for freely independent real valued random variables using resolvents of self adjoint operators in contrast to the original proof no assumption is made on the existence of moments above the second

endpoint bilinear restriction theorems for the cone and some sharp null form estimates recently wolff 25 obtained a nearly sharp l 2 bilinear restriction theorem for bounded subsets of the cone in general dimension we obtain the endpoint of wolff s estimate and generalize to the case when one of the subsets is large as a consequence we are able to deduce some nearly sharp l p null form estimates

sharp null form estimates for the wave equation null form estimates from dot h alpha 1 times dot h alpha 2 to l q t l r x for the wave equation in bbb r n 1 are studied for n ge4 we obtain the sharp null form estimates except for the endpoints for n 2 3 we obtain the estimates under the additional assumption 4 n q when 2 r

fractal entropies and dimensions for microstates spaces abstract using voiculescu s notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von neumann algebra we show that they possess properties similar to their classical predecessors we relate the new quantities to free entropy and free entropy dimension and show that a modified version of free hausdorff dimension is an algebraic invariant we compute the free hausdorff dimension in the cases where the set generates a finite dimensional algebra or where the set consists of a single selfadjoint we show that the hausdorff dimension becomes additive for such sets in the presence of freeness

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

on certain free product factors via an extended matrix model abstract a random matrix model for freeness is extended and used to investigate free products of free group factors with matrix algebras and with the hyperfinite ii 1 factor the latter is shown to be isomorphic to a free group factor having one additional generator

on trees and dual rotund norms abstract we give a necessary and sufficient condition for the existence of an equivalent dual rotund norm on c 0 l 1 where is a tree the condition is expressed succinctly in terms of the embeddability of into a particular totally ordered set and compares very well with the analogous situation for local uniform rotundity this resolves an open problem from haydon s work in asplund spaces trees and renorming theory

on finite free fisher information for eigenvectors of a modular operator abstract suppose m is a von neumann algebra equipped with a faithful normal state and generated by a finite set g g g 2 we show that if g consists of eigenvectors of the modular operator with finite free fisher information then the centralizer m is a ii 1 factor and m is either a type ii 1 factor or a type iii factor 0 1 depending on the eigenvalues of g furthermore m m c m does not have property and m is full provided it is type iii 0 1

linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces abstract we obtain bilinear estimates for oscillatory integral operators which are variable coefficient generalizations of bilinear restriction estimates for hypersurfaces as applications we improve the known estimates for oscillatory integrals

properties of beurling type submodules via agler decompositions abstract in this paper we study operator theoretic properties of the compressed shift operators s z 1 and s z 2 on complements of submodules of the hardy space over the bidisk h 2 d 2 specifically we study beurling type submodules namely submodules of the form h 2 d 2 for inner using properties of agler decompositions of to deduce properties of s z 1 and s z 2 on model spaces h 2 d 2 h 2 d 2 results include characterizations in terms of of when a commutator s z j s z j has rank n and when subspaces associated to agler decompositions are reducing for s z 1 and s z 2 we include several open questions

on weakly locally uniformly rotund banach spaces abstract we show that every normed space e with a weakly locally uniformly rotund norm has an equivalent locally uniformly rotund norm after obtaining a discrete network of the unit sphere s e for the weak topology we deduce that the space e must have a countable cover by sets of small local diameter which in turn implies the renorming conclusion this solves a question posed by deville godefroy haydon and zizler for a weakly uniformly rotund norm we prove that the unit sphere is always metrizable for the weak topology despite the fact that it may not have the kadec property moreover banach spaces having a countable cover by sets of small local diameter coincide with the descriptive banach spaces studied by hansell so we present here some new characterizations of them

on non semisplit extensions tensor products and exactness of group c algebras we show the existence of a block diagonal extensionb of the suspensions a of the reduced groupc algebraa c sl 2 such that there is only onec norm on the algebraic tensor productb op b butb is not nuclear even not exact thus the class of exactc algebras is not closed under extensions the existence comes from a new established tensorial duality between the weak expectation property wep of lance and the local variant llp of the lifting property we characterize the local lifting property of separable unitalc algebrasa as follows a has the local lifting property if and only if ext s a is a group wheres a is the suspension ofa if moreovera is the quotient algebra of ac algebra withwep for brevity a isqwep but does not satisfyllp then there exists a quasidiagonal extensionb of the suspensions a by the compact operators such that on the algebraic tensor productb op b there is only onec norm the question if everyc algebra isqwep remains open but we obtain the following results onqwep ac algebrac isqwep if and only ifc isqwep a von neumannii 1 factorn with separable predualn isqwep if and only ifn is a von neumann subfactor of the ultrapower of the hyperfiniteii 1 factor ifg is a maximally almost periodic discrete non amenable group with haagerup s herz schur multiplier constant g 1 then the universal groupc algebrac g is not exact but the reduced groupc albegrac g is exact and isqwep but does not satisfywep andllp we study functiorial properties of the classes ofc algebras satisfyingwep llp resp beingqwep as applications we obtain some unexpected relations between some open questions onc algebras

weak locally uniformly rotund norms and descriptive compact spaces we prove that a dual banach space x has an equivalent w lur norm if and only if the weak topology has a isolated network we give sufficient conditions for the existence of equivalent norms with various kinds of differentiability

a refinement of the strichartz inequality on the saddle and applications abstract we consider the nonlinear nonelliptic schrodinger equation defined by i t u x 2 y 2 u u 2 u 0 with initial datum in l 2 r 2 we show that if the solution blows up in finite time then there is a mass concentration phenomenon near the blow up time the key ingredient is a refinement of the strichartz inequality on the saddle

free fisher information with respect to a completely positive map and cost of equivalence relations given a family of isometries in a tracial von neumann algebra m a unital subalgebra b m and a completely positive map we define the free fisher information of relative to b and using this notion we define the free dimension of relative to b id

on the multiplication of free n tuples of noncommutative random variables let a1 a n b 1 b n be random variables in a noncommutative probability space such that a1 a n is free from b1 b n we show how the joint distribution of the n tuple a1b1 a n b n can be described in terms of the joint distributions of a1 a n a nd b 1 b n by using the combinatorics of the n dimensional r transform we point out a few applications that can be easily derived from our result concerning the left and right translation with a semicircular element see sections 1 6 1 10 and the compression with a projection see sections 1 11 1 14 of an n tuple of noncommutative random variables a different approach to two of these applications is presented by dan voiculescu in an appendix to the paper introduction the theory of free random variables was developed in a se quence of papers of d voiculescu see 17 or the recent survey in 16 as an instrument for approaching free products of operator algebras its particular aspect addressed in the present paper is the one concerning the addition and multiplica tion of free random variables as shown by voiculescu in 13 14 a powerful method in the study of these operations is the use of transforms that convert them respectively into addition and multiplication of complex analytic func tions or in an algebraic framework of formal power series in an indeterminate z the precise definitions of these transforms called r transform for the addition problem and s transform for the multiplication problem will be reviewed in the sections 1 2 1 3 below in the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the r and s transforms the first idea is that the connection between the r and the s transform is closer than one might suspect at first glance a way of making this precise was pointed out in our paper 6 in the form of the equation s r for a distribution with nonvan ishing mean and where is a combinatorial object with a precise significance the fourier transform for multiplicative functions on noncrossing partitions a byproduct of our result in 6 is the remark that the multiplication of free

cout des relations d equivalence et des groupes we study a new dynamical invariant for dicrete groups the cost it is a real number in 1 1 n 1 bounded by the number of generators of the group and it is well behaved with respect to finite index subgroups namely the quantities 1 minus the cost are related by multiplying by the index the cost of every infinite amenable group equals 1 we compute it in some other situations including free products free products with amalgamation and hnn extensions over amenable groups and for direct product situations for instance the cost of the free group on n generators equals n we prove that each possible finite value of the cost is achieved by a finitely generated group it is dynamical because it relies on measure preserving free actions on probability borel spaces in most cases groups have fixed price which implies that two freely acting groups which define the same orbit partition must have the same cost it enables us to distinguish the orbit partitions of probability preserving free actions of free groups of different ranks at the end of the paper we give a mercuriale i e a list of costs of different groups the cost is in fact an invariant of ergodic measure preserving equivalence relations and is defined using graphings a treeing is a measurable way to provide every equivalence class orbit with the structure of a simplicial tree this an example of graphing not every relation admits a treeing we prove that every free action of a cost 1 non amenable group is not treeable but we prove that subrelations of treeable relations are treeable we give examples of relations which cannot be produced by an action of any finitely generated group the cost of a relation which can be decomposed as a direct product is shown to be 1 we define the notion for a relation to be a free product or an hnn extension and compute the cost for the resulting relation from the costs of the building blocks the cost is also an invariant of the pairs von neumann algebra cartan subalgebra

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

locally uniformly convex norms in banach spaces and their duals abstract it is shown that a banach space with locally uniformly convex dual admits an equivalent norm that is itself locally uniformly convex

a fourier view on the r transform and related asymptotics of spherical integrals abstract we estimate the asymptotics of spherical integrals of real symmetric or hermitian matrices when the rank of one matrix is much smaller than its dimension we show that it is given in terms of the r transform of the spectral measure of the full rank matrix and give a new proof of the fact that the r transform is additive under free convolution these asymptotics also extend to the case where one matrix has rank one but complex eigenvalue a result related with the analyticity of the corresponding spherical integrals

on the restriction of the fourier transform to a conical surface let r be the su rface of a circular cone in r3 we show that if 1 p 4 3 1 q 3 1 1 p and f e lp r3 then the fourier transform of f belongs to lq r da for a certain natural measure a on r following p tomas we also establish bounds for restrictions of fourier transforms to conic annuli at the endpoint p 4 3 with logarithmic growth of the bound as the thickness of the annulus tends to zero

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

the analogues of entropy and of fisher s information measure in free probability theory i analogues of the entropy and fisher information measure for random variables in the context of free probability theory are introduced monotonicity properties and an analogue of the cramer rao inequality are proved

operator algebras of functions abstract we present some general theorems about operator algebras that are algebras of functions on sets including theories of local algebras residually finite dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector valued reproducing kernel hilbert space we use these to further develop a quantized function theory for various domains that extends and unifies agler s theory of commuting contractions and the arveson drury popescu theory of commuting row contractions we obtain analogous factorization theorems prove that the algebras that we obtain are dual operator algebras and show that for many domains supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices

topological free entropy dimension in unital c algebras abstract the notion of topological free entropy dimension of n tuple of elements in a unital c algebra was introduced by voiculescu in the paper we compute topological free entropy dimension of one self adjoint element and topological free orbit dimension of one self adjoint element in a unital c algebra we also calculate the values of topological free entropy dimensions of any families of self adjoint generators of some unital c algebras including irrational rotation c algebra uhf algebra and minimal tensor product of two reduced c algebras of free groups

an endline bilinear cone restriction estimate for mixed norms we prove an l 2 times l 2 rightarrow l t ql x p bilinear fourier extension estimate for the cone when p q are on the critical line 1 q frac n 1 2 1 1 p this extends previous results by wolff tao and lee vargas

compact spaces of szlenk index ω abstract the szlenk index has found many applications in the isomorphic theory of banach spaces its definition is based in some kind of interplay between a weak topology and the norm metric with not much care on the linear structure there is no obstacle to consider the notion of szlenk index in more general settings in this paper we study the compact spaces of szlenk index at most with respect to an associated metric we include new applications to banach spaces of the these methods where the estimations of the growth speed of the finite szlenk indices play a fundamental role

multi scale bilinear restriction estimates for general phases we prove adjoint bilinear restriction estimates for general phases at different scales in the full non endpoint mixed norm range and give bounds with a sharp and explicit dependence on the phases these estimates have applications to high low frequency interactions for solutions to partial differential equations as well as to the linear restriction problem for surfaces with degenerate curvature as a consequence we obtain new bilinear restriction estimates for elliptic phases and wave klein gordon interactions in the full bilinear range and give a refined strichartz inequality for the klein gordon equation in addition we extend these bilinear estimates to hold in adapted function spaces by using a transference type principle which holds for vector valued waves

on weakly countably determined banach spaces for a topological space x let ci x denote the banach space of all bounded functions x r such that for every e 0 the set x x f x e is closed and discrete in x endowed with the supremum norm the main theorem is the following let l be a weakly countably determined subset of a banach space then there exist a subset e of the baire space e a compact space k and a bounded linear one to one operator t c l ci e x k that is pointwise to pointwise continuous in the case where l is weakly analytic e can be replaced by e this theorem is connected with the basic result of amir lindenstrauss on wcg banach spaces and has corresponding consequences such as the representation of gulko resp ta lagrand compact spaces as pointwise compact subsets of ci e x k resp ci e x k a compact space f is called gulko or talagrand compact if c o is wcd or a weakly k analytic banach space the characterization of wcd resp weakly k analytic banach spaces e using one to one operators from e into ci e x k resp ci e x k and the existence of equivalent good norms on e and e simultaneously

matricially free random variables we show that the framework developed by voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication the associated notion of independence called matricial freeness can be viewed as a concept which not only leads to a natural generalization of freeness but also underlies other fundamental types of noncommutative independence such as monotone independence and boolean independence at the same time the sums of matricially free random variables called random pseudomatrices are closely related to random matrices the main results presented in this paper concern the standard and tracial central limit theorems for random pseudomatrices and the corresponding limit distributions which can be viewed as matricial semicircle laws

problemes de renormages we show that for every scattered compact space k such that k 1 o where 1 design the first uncountable ordinal there exists on b k an equivalent norm such that the dual norm is lur moreover if k 0 o where 0 design the first infinite ordinal the norm on b k can be chosen itself lur we deduce from this result that if x and y are banach spaces such that x and y are wcg then there exists on x e y an equivalent lur norm such that the dual norm is also lur

a sobolev estimate for the adjoint restriction operator in this note we consider the adjoint restriction estimate for hypersurface under additional regularity assumption we obtain the optimal h s l q estimates and their mixed norm generalizations as applications we prove some weighted strichartz estimates for the propagator varphi rightarrow e it delta alpha 2 varphi alpha 0