a class of superrigid group von neumann algebras we prove that for any group g in a fairly large class of generalized wreath product groups the associated von neumann algebra lg completely remembers the group g more precisely if lg is isomorphic to the von neumann algebra l of an arbitrary countable group then must be isomorphic to g this represents the rst superrigidity result pertaining to group von neumann algebras

some prime factorization results for type ii1 factors we prove several unique prime factorization results for tensor products of type ii1 factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected lie groups of real rank 1 in particular we show that if mathcal r overline otimes mathcal l mathbb f r 1 overline otimes cdot cdot cdot overline otimes mathcal l mathbb f r m is isomorphic to a subfactor in mathcal r overline otimes mathcal l mathbb f s 1 overline otimes cdot cdot cdot overline otimes mathcal l mathbb f s n for some 2 ri sj then m n

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

group cocycles and the ring of affiliated operators in this article we study cocycles of discrete countable groups with values in 2 g and the ring of affiliated operators ug we clarify properties of the first cohomology of a group g with coefficients in 2 g and answer several questions from de cornulier et al transform groups 13 1 125 147 2008 moreover we obtain strong results about the existence of free subgroups and the subgroup structure provided the group has a positive first 2 betti num ber we give numerous applications and examples of groups which satisfy our assumptions

unique prime factorization for infinite tensor product factors abstract in this article we investigate a unique prime factorization property for infinite tensor product factors we provide several examples of type ii and iii factors which satisfy this property including all free product factors with diffuse free product components in the type iii setting this is the first classification result for infinite tensor product non amenable factors our proof is based on popa s intertwining techniques and a characterization of relative amenability on the continuous cores

some unique group measure space decomposition results using an approach emerging from the theory of closable deriva tions on von neumann algebras we exhibit a class of groups cr satisfying the following property given any groups 1 2 2 cr then any free ergodic measure preserving action on a probability space 1 2 y x gives rise to a von neumann algebra with unique group measure space cartan subalgebra pairing this result with popa s orbit equivalence superrigidity theorem we obtain new examples of w superrigid actions

the tits and euler forms of a tame algebra since the introduction by gabriel in 1972 of the tits quadratic form associated to finite dimensional hereditary algebras quadratic forms have played an important role in the representation theory of algebras and related categories there have been quadratic forms associated to finite dimensional algebras partially ordered sets and other objects in order to determine their representation type and the classes of indecomposable representations in the corresponding grothendieck group see for example 3 9 14 21 the class of finite dimensional algebras associative with in identity over an algebraically closed field k is divided into two disjoint classes one class consists of tame algebras for which the indecomposable modules occur in each dimension in a finite number of discrete and one parameter families the second class consist of wild algebras whose representation theory is as complicated as the classification of pairs of matrices up to simultaneous conjugation in this paper we establish properties of the tits and euler forms associated to tame simply connected algebras let a be a basic connected finite dimensional algebra over k then a has a presentation a kqa i where qa q0 q1 is the ordinary quiver of a with set of vertices q0 and set of arrows q1 and i is an admissible ideal of the path algebra kqa see 9 we shall assume that qa has no oriented cycles and hence g dim a then the euler form a zq0 z is defined by

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

von neumann algebras of equivalence relations with nontrivial one cohomology abstract using popa s deformation rigidity theory we investigate prime decompositions of von neumann algebras of the form l r for countable probability measure preserving equivalence relations r we show that l r is prime whenever r is nonamenable ergodic and admits an unbounded 1 cocycle into a mixing orthogonal representation weakly contained in the regular representation this is accomplished by constructing the gaussian extension r of r and subsequently an s malleable deformation of the inclusion l r l r we go on to note a general obstruction to unique prime factorization and avoiding it we prove a unique prime factorization result for products of the form l r 1 l r 2 l r k as a corollary we get a unique factorization result in the equivalence relation setting for products of the form r 1 r 2 r k we finish with an application to the measure equivalence of groups

tame algebras and tits quadratic forms abstract we solve a long standing problem concerning the connection between the tameness of simply connected algebras and the weak nonnegativity of the associated tits integral quadratic forms and derive some consequences

on positive roots of pg critical algebras to each finite dimensional algebra a one can associate an integral quadratic form qa called the tits form of a we show that for some pg critical algebra b which is given below the value of its tits form on all indecomposable modules is bounded by 2 this gives a negative answer to the following problem posed by j a de la pena representation theory of algebras and related topics mexico city 1994 cms conference proceedings vol 19 ams providence ri 1996 p 159 can one characterize the polynomial growth of a tame stronly simply connected algebra by the boundedness of its tits form on the indecomposable modules in order to compute the value of qb on indecomposables we first translate the problem of classifying isomorphism classes of a modules into a linear matrix problem then we show that the matrix problem obtained this way belongs to a class of repetitive matrix problems whose dimension vectors are easy to calculate 2002 elsevier science inc all rights reserved

w superrigidity for bernoulli actions of property t groups we consider group measure space 1 factors m l x arising from bernoulli actions of icc property t groups more generally of groups containing an infinite normal subgroup with the relative property t and prove a rigidity result for homomorphisms m overbar m we deduce that the action x is w superrigid i e if y is any free ergodic measure preserving action such that the factors m l x and m l y are isomorphic then the actions x and y must be conjugate moreover we show that if p m 1 is a projection then pmp does not admit a group measure space decomposition nor a group von neumann algebra decomposition the latter under the additional assumption that is torsion free we also prove a rigidity result for homomorphisms m m this time for in a larger class of groups than above now including products of non amenable groups for certain groups e g f 2 f 2 we deduce that m does not embed into pmp for any projection p m 1 and obtain a description of the endomorphism semigroup of m

on the structural theory of ii1 factors of negatively curved groups ii actions by product groups abstract this paper contains a series of structural results for von neumann algebras arising from measure preserving actions by product groups on probability spaces expanding upon the methods used earlier by the first two authors we obtain new examples of strongly solid factors as well as von neumann algebras with unique or no cartan subalgebra for instance we show that every ii 1 factor associated with a weakly amenable group in the class s of ozawa is strongly solid there is also the following product version of this result any maximal abelian subalgebra of any ii 1 factor associated with a finite product of weakly amenable groups in the class s of ozawa has an amenable normalizing algebra finally pairing some of these results with a cocycle superrigidity result of ioana it follows that compact actions by finite products of lattices in s p n 1 n 2 are virtually w superrigid

one cohomology and the uniqueness of the group measure space decomposition of a ii1 factor we provide a unified and self contained treatment of several of the recent uniqueness theorems for the group measure space decomposition of a ii1 factor we single out a large class of groups characterized by a one cohomology property and prove that for every free ergodic probability measure preserving action of the associated ii1 factor has a unique group measure space cartan subalgebra up to unitary conjugacy our methods follow closely a recent article of chifan peterson but we replace the usage of peterson s unbounded derivations by thomas sinclair s dilation into a malleable deformation by a one parameter group of automorphisms

l 2 rigidity in von neumann algebras we introduce the notion of l2 rigidity for von neumann algebras a generalization of property t which can be viewed as an analogue for the vanishing of 1 cohomology into the left regular representation of a group we show that l2 rigidity passes to normalizers and is satisfied by nonamenable ii1 factors which are non prime have property or are weakly rigid as a consequence we obtain that if m is a free product of diffuse von neumann algebras or if m l where is a finitely generated group with 1 2 0 then any nonamenable regular subfactor of m is prime and does not have properties or t in particular this gives a new approach for showing solidity for a free group factor thus recovering a well known recent result of n ozawa

strongly solid group factors which are not interpolated free group factors we give examples of non amenable infinite conjugacy classes groups with the haagerup property weakly amenable with constant cb 1 for which we show that the associated ii1 factors l are strongly solid i e the normalizer of any diffuse amenable subalgebra p subset l gamma generates an amenable von neumann algebra nevertheless for these examples of groups l is not isomorphic to any interpolated free group factor l ft for 1 t

unique cartan decomposition for ii1 factors arising from arbitrary actions of free groups we prove that for any free ergodic probability measure preserving action mathbb f n curvearrowright x mu of a free group on n generators mathbb f n 2 leq n leq infty the associated group measure space ii1 factor l infty x rtimes mathbb f n has l x as its unique cartan subalgebra up to unitary conjugacy we deduce that group measure space ii1 factors arising from actions of free groups with different number of generators are never isomorphic we actually prove unique cartan decomposition results for ii1 factors arising from arbitrary actions of a much larger family of groups including all free products of amenable groups and their direct products

unique prime factorization and bicentralizer problem for a class of type iii factors abstract we show that whenever m 1 and m 1 m m are nonamenable factors in a large class of von neumann algebras that we call c ao and which contains all free araki woods factors the tensor product factor m 1 m m retains the integer m and each factor m i up to stable isomorphism after permutation of the indices our approach unifies the unique prime factorization upf results from 33 25 and moreover provides new upf results in the case when m 1 m m are free araki woods factors in order to obtain the aforementioned upf results we show that connes s bicentralizer problem has a positive solution for all type i i i 1 factors in the class c ao

finite dimensional algebras with vanishing hochschild cohomology let a be a finite dimensional algebra associative with unit over an algebraically closed field k for a finitely generated bimodule x the hochschild cohomology a a i w x i groups h a x i g 0 were introduced in 10 we shall denote h a i i s h a a the vanishing of the low dimensional groups h a i f 2 has classical interpretations and plays an important role in the representaw x tion theory of algebras gernstenhaber 7 has shown some connections 2 w x 1 between h a and the deformation theory of a see also 5 h a is

a note about proving non γ under a finite non microstates free fisher information assumption we prove that if x1 xn n 1 are self adjoints in a w probability space with finite non microstates free fisher information then the von neumann algebra w x1 xn they generate doesn t have property especially is not amenable this is an analog of a well known result of voiculescu for microstates free entropy we also prove factoriality under finite non microstates entropy

maximal amenability and disjointness for the radial masa abstract we prove that the radial masa c in the free group factor is disjoint from other maximal amenable subalgebras in the following sense any distinct maximal amenable subalgebra cannot have diffuse intersection with c

who cares about integrability abstract problems that admit solitons and are completely integrable have been analyzed in detail since their discovery by zabusky and kruskal 25 years ago their study has influenced the development of both mathematics and physics during that time the questions addressed here are 1 i has this subject run its natural course 2 ii will it significantly influence the nonlinear science that develops over the next decade supporting examples will be chosen from several disciplines but especially from the study of water waves

variations in noncommutative potential theory finite energy states potentials and multipliers in this work we undertake an extension of various aspects of the potential theory of dirichlet forms from locally compact spaces to noncommu tative c algebras with trace in particular we introduce finite energy states potentials and multipliers of dirichlet spaces we prove several results among which the celebrated deny s embedding theorem and the deny s inequality the fact that the carre du champ of bounded potentials are finite energy func tionals and the relative supply of multipliers 1 introduction and description of the results in the present work we develop further the potential theory of dirichlet forms on noncommutative c algebras with trace we introduce and investigate finite energy states potentials and multipliers objects naturally associated to dirichlet spaces and which are meant to encode or reveal the geometric nature of the latter in a companion work the results here obtained will be crucial to construct on c algebras endowed with a dirichlet form the building blocks of a metric differential geometry dirac operators and spectral triples and topological invariants sum mable fredholm modules in k homology in the framework of the noncommutative geometry developed by a connes co classical potential theory studying harmonic functions on euclidean spaces r n finite energy measures and their potentials was based on the properties of kernel x y 1 the so called green function to understood as the integral kernel of the

algebras whose tits form weakly controls the module category an important invariant associated to a finite dimensional algebra is its tits quadratic form this invariant has been extensively used in the representation theory of algebras to determine the representation type of w x an algebra see 4 11 16 to determine the classes of indecomposable w x modules in the grothendieck group see 9 11 20 and as a tool in a wide w x range of applications see 16 21 let a s kqri be a basic finite dimensional algebra over an algebraically closed field k we will assume that q is connected and without 4 oriented cycles we denote by q s 1 n resp q the set of vertices 0 1 resp arrows of q left a modules are considered as representations of w x q satisfying the ideal i for this point of view see 12 by s we denote i the simple representation associated with the vertex i g q observe that 0 1 dim ext s s is the number of arrows from i to j in q the tits form k a i j

exact solutions of the one dimensional quintic complex ginzburg landau equation abstract exact solitary wave solutions of the one dimensional quintic complex ginzburg landau equations are obtained using a method derived from the painleve test for integrability these solutions are expressed in terms of hyperbolic functions and include the pulses and fronts found by van saarloos and hohenberg we also find previously unknown sources and sinks the emphasis is put on the systematic character of the method which breaks away from approaches involving somewhat ad hoc ansatze

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

on the superrigidity of malleable actions with spectral gap some of the most interesting aspects of the dynamics of measure preserving actions of countable groups on probability spaces v rx x are revealed by the study of group measure space von neumann algebras l x x t mvnl and the classification of actions up to orbit equivalence oe i e up to isomorphism of probability spaces carrying the orbits of actions onto each other although one is in von neumann algebras and the other in ergodic theory the two problems are closely related as an oe of actions v rx x a rxy has been shown to implement an algebra isomorphism l x x t l y x a taking l x onto l y and vice versa si dy fm in particular the isomorphism class of l x xi t only depends on the equivalence relation 7 r t gt 1t x g t thus von neumann equivalence vne of group actions requiring isomorphism of their group measure space algebras is weaker than oe since there are exam ples of non oe actions whose associated von neumann algebras are all isomorphic cj1 it is in general strictly weaker on the other hand oe is manifestly weaker than classical conjugacy which for free actions r rx x a rx y requires isomor phism of probability spaces a x p y v satisfying ar a 1 a so in particular r a how much weaker vne and oe can be with respect to conju gacy is best seen in the amenable case where by a celebrated theorem of connes all free ergodic actions of all infinite amenable groups give rise to the same hi factor cl and by dy ow cfw they are undistinguishable under oe as well also any embedding of algebras l x xi t c l y x a with a amenable forces t to be amenable but the non amenable case is extremely complex and for many years progress has been slow mvn2 dy me c2 cw sc even after the discovery of the first rigidity phenomena by connes in von neumann algebras c3 c4 and by zimmer in oe ergodic theory zi z2 this changed dramatically over the last 7 8 years with the advent of a variety of striking rigidity results ful gl g2 mosh h hk p1 p8 see p9 for a survey also sh for a survey on oe rigidity our aim in this paper is to investigate the most extreme such phenomena called strong rigidity which show that for certain classes of source group actions

derivations as square roots of dirichlet forms abstract the aim of this work is to analyze the structure of a tracially symmetric dirichlet form on a c algebra in terms of a killing weight and a closable derivation taking values in a hilbert space with a bimodule structure it is shown that the generator of the associate markovian semigroup always appears in a natural way as the divergence of a closable derivation applications are shown to the decomposition of dirichlet forms and to the construction of differential calculus on metric spaces

asymptotic structure of free araki woods factors the purpose of this paper is to investigate the structure of shlyakhtenko s free araki woods factors using the framework of ultraproduct von neumann algebras we first prove that all the free araki woods factors gamma h mathbb r u t prime prime are omega solid in the following sense for every von neumann subalgebra q subset gamma h mathbb r u t prime prime that is the range of a faithful normal conditional expectation and such that the relative commutant q cap m omega is diffuse we have that q is amenable next we prove that the continuous cores of the free araki woods factors gamma h mathbb r u t prime prime associated with mixing orthogonal representations u mathbb r rightarrow mathcal o h mathbb r are omega solid type mathrm ii infty factors finally when the orthogonal representation u mathbb r rightarrow mathcal o h mathbb r is weakly mixing we prove a dichotomy result for all the von neumann subalgebras q subset gamma h mathbb r u t prime prime that are globally invariant under the modular automorphism group sigma t varphi u of the free quasi free state varphi u

compact actions and uniqueness of the group measure space decomposition of ii1 factors abstract we prove that any ii1 factor l x arising from a compact free ergodic probability measure preserving action of a countable group with positive first l 2 betti number 1 2 0 has a unique group measure space cartan subalgebra up to unitary conjugacy

on the additive categories of generalized standard almost cyclic coherent auslander reiten components abstract we prove geometric and homological properties of modules in the additive categories of generalized standard almost cyclic coherent components in the auslander reiten quivers of finite dimensional algebras

noncommutative riesz transforms a probabilistic approach for 2 p 0 in the classical setting this is certainly satisfied for the maps tt f f 1 t where t is a semigroup of measure preserving maps on a probability space in that case n l then the maps tt act on all noncommutative lp spaces lp n and in particular on the hilbert space l2 l2 n let a be the negative generator i e tt e ta for technical reasons we have to assume that there is a weakly dense algebra a n which lies in the domain of the unbounded operator a i e a a a the algebra a plays the role of smooth functions in the classical setting this assumption might be too restrictive for some classical applications but it is easily verified in the new noncommutative examples for elements x y a we may define the gradient form 2 x y a x y x a y a x y the correct definition of is the one given by sauvageot as a bilinear form with values in l1 again our arguments require stronger assumptions in fact in all our examples we will assume that the tt s are selfadjoint i e ttxy xtty and unital i e tt 1 1 then a is indeed a positive unbounded operator under these circumstances we can formulate p a meyer s problem is it true that x x 2 p c p a 1 2 x p 0 1 holds for all selfadjoint elements x a let us illustrate this question by considering the laplace operator a f f where i 2 x2 i then it is easily verified that

some rigidity results for non commutative bernoulli shifts abstract we introduce the outer conjugacy invariants s s s for cocycle actions of discrete groups g on type ii 1 factors n as the set of real numbers t 0 for which the amplification t of can be perturbed to an action respectively to a weakly mixing action we calculate explicitly s s s and the fundamental group of f in the case g has infinite normal subgroups with the relative property t e g when g itself has the property t of kazhdan and is an action of g on the hyperfinite ii 1 factor by connes stormer bernoulli shifts of weights t i i thus s s and f coincide with the multiplicative subgroup s of r generated by the ratios t i t j i j while s z if s 1 i e when all weights are equal and s r otherwise in fact we calculate all the 1 cohomology picture of t t 0 and classify the actions g in terms of their weights t i i in particular we show that any 1 cocycle for g vanishes modulo scalars and that two such actions are cocycle conjugate iff they are conjugate also any cocycle action obtained by reducing a bernoulli action of a group g as above on n g g m n n c tr g to the algebra pnp for p a projection in n p 0 1 cannot be perturbed to a genuine action

scattering theory for the korteweg de vries kdv equation and its hamiltonian interpretation abstract it is shown that the scattering transform for the kdv equation is not canonical in the naive sense an explanation of this phenomenon is found and a correct hamiltonian formulation of the scattering theory is proposed

strong solidity of group factors from lattices in so n 1 and su n 1 abstract we show that the group factors l where is an icc lattice in either so n 1 or su n 1 n 2 are strongly solid in the sense of ozawa and popa 2010 13 this strengthens a result of ozawa and popa 2010 14 showing that these factors do not have cartan subalgebras

a remark on t valued cohomology groups of algebraic group actions we prove that for a weakly mixing algebraic action g x the nth cohomology group hn g x t after quotienting out the natural subgroup hn g t contains hn g x as a natural subgroup for n 1 if we further assume the diagonal actions 2 4 are t cocycle superrigid and h2 g x is torsion free as an abelian group then the above also holds true for n 2 applying it for principal algebraic actions when n 1 we show that h2 g zg is torsion free as an abelian group when g has property t as a direct corollary of sorin popa s cocycle superrigidity theorem we also use it when n 2 to answer negatively a question of sorin popa on the 2nd cohomology group of bernoulli shift actions of property t groups