topology of symplectomorphism groups of rational ruled surfaces let m be either s 2 s 2 or the one point blow up cp 2 cp 2 of cp 2 in both cases m carries a family of symplectic forms where 1 determines the cohomology class this paper calculates the rational co homology of the group g of symplectomorphisms of m as well as the rational homotopy type of its classifying space bg it turns out that each group g contains a nite collection kk k 0 of nite dimensional lie subgroups that generate its homotopy we show that these subgroups asymptotically commute i e all the higher whitehead products that they generate vanish as 1 however for each xed there is essentially one nonvanishing product that gives rise to a jumping generator w in h g and to a single relation in the rational cohomology ring h bg an analog of this generator w was also seen by kronheimer in his study of families of symplectic forms on 4 manifolds using seiberg witten theory our methods involve a close study of the space of compatible almost complex structures on m

the cyclic and simplicial cohomology of the cuntz semigroup algebra abstract the main objective of this paper is to determine the simplicial and cyclic cohomology groups of the cuntz semigroup algebra l 1 s m we also determine the simplicial and cyclic cohomology of the tensor algebra of a banach space a class which includes the algebra on the free semigroup on m generators l 1 fs m in order to do so we first establish some general results which can be used when studying simplicial and cyclic cohomology of banach algebras in general we then turn our attention to l 1 s m showing that the cyclic cohomology groups of degree n vanish when n is odd and are one dimensional when n is even n 2 using the connes tzygan exact sequence these results are used to show that the simplicial cohomology groups of degree n vanish for n 1 a similar strategy is used for the tensor algebra of a banach space

residual finite dimensionality and representations of amenable operator algebras abstract we consider a version of a famous open problem formulated by kadison asking whether bounded representations of operator algebras are automatically completely bounded we investigate this question in the context of amenable operator algebras and we provide an affirmative answer for representations whose range is residually finite dimensional furthermore we show that weak closed amenable residually finite dimensional operator algebras are similar to c algebras and in particular have the property that all their bounded representations are completely bounded we prove our results for operator algebras having the so called total reduction property which is known to be weaker than amenability

some properties of wkb series we investigate some properties of the wkb series for arbitrary analytic potentials and then specifically for potentials xn n even where more explicit formulae for the wkb terms are derived our main new results are i we find the explicit functional form for the general wkb terms k where one has only to solve a general recursion relation for the rational coefficients ii we give a systematic algorithm for an essential simplification of the integrated wkb terms k dx that enter the energy eigenvalue equation and iii we derive almost explicit formulae for the wkb terms for the energy eigenvalues of the homogeneous power law potentials v x xn where n is even

amenable and weakly amenable banach algebras with compact multiplication abstract we investigate amenable and weakly amenable banach algebras with compact multiplication any amenable banach algebra with compact multiplication is biprojective as a consequence every semisimple such algebra which has the approximation property is a topological direct sum of full matrix algebras in the radical case no such structure theorem is at hand we also investigate banach algebras which have a bounded approximate identity consisting of normalized powers of an element x any such banach algebra is either unital or radical if the algebra is also generated by x it is weakly amenable we construct a radical example with compact multiplication which moreover is an integral domain this furnishes a new example of a commutative weakly amenable non amenable radical banach algebra

grothendieck s theorem past and present probably the most famous of grothendieck s contributions to banach space theory is the result that he himself described as the fundamental theorem in the metric theory of tensor products that is now commonly referred to as grothendieck s theorem gt in short or sometimes as grothendieck s inequality this had a major impact first in banach space theory roughly after 1968 then later on in c algebra theory roughly after 1978 more recently in this millennium a new version of gt has been successfully developed in the framework of operator spaces or non commutative banach spaces in addition gt independently surfaced in several quite unrelated fields in connection with bell s inequality in quantum mechanics in graph theory where the grothendieck constant of a graph has been introduced and in computer science where the grothendieck inequality is invoked to replace certain np hard problems by others that can be treated by semidefinite programming and hence solved in polynomial time in this expository paper we present a review of all these topics starting from the original gt we concentrate on the more recent developments and merely outline those of the first banach space period since detailed accounts of that are already available for instance the author s 1986 cbms notes

the spatial isomorphism problem for close separable nuclear c algebras the kadison kastler problem asks whether close c algebras on a hilbert space must be spatially isomorphic we establish this when one of the algebras is separable and nuclear we also apply our methods to the study of near inclusions of c algebras

wkb approach applied to 1d time dependent nonlinear hamiltonian oscillators we present a simple wkb like approach to obtain approximate analytic solutions to a certain class of time dependent nonlinear 1d hamiltonian oscillators the case of homogeneous power law potentials is solved explicitly in a closed form in the leading order the accuracy of the approximation is surprisingly good and we illustrate it in the case of the quartic oscillator

relative yoneda cohomology for operator spaces abstract we provide two alternate presentations of the completely bounded hochschild cohomology one as a relative yoneda cohomology i e as equivalence classes of n resolutions which are relatively split and the second as a derived functor the first presentation makes clear the importance of certain relative notions of injectivity projectivity and amenability which we introduce and study we prove that every von neumann algebra is relatively injective as a bimodule over itself and consequently h n cb m m 0 for any von neumann algebra m a result obtained earlier by christensen and sinclair we prove that the relatively amenable c algebras are precisely the nuclear c algebras and hence exactly those which are amenable as banach algebras in a similar vein we prove that the only relatively projective c algebras are finite dimensional this result implies that the only c algebras that are projective as banach algebras are finite dimensional a result first obtained by selivanov and helemskii in a slightly different direction we prove that b h viewed as a bimodule over the disk algebra with the left action given by multiplication by a coisometry and the right action given by multiplication by an isometry is an injective module this result is in some sense a generalization of the sz nagy foias commutant lifting theorem or of the hypoprojectivity introduced by douglas and the author

when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional evans d g and sims a when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional preprint arxiv 1112 4549v1

systemes dynamiques non commutatifs et moyennabilite on introduit la notion de fonction de type positif relativement a un systeme dynamique c dont les proprietes sont analogues a celles des fonctions de type positif classiques on montre que dans le cas d un groupe discret g agissant sur une algebre de von neumann m la moyennabilite de l action est equivalente a l existence d une suite bornee de fonctions de type positif sur g a support fini a valeurs dans le centre z m de m qui converge vers 1 pour la topologie de la convergence simple ultrafaible

weak amenability of module extensions of banach algebras we start by discussing general necessary and sufficient conditions for a module extension banach algebra to be n weakly amenable for n 0 1 2 then we investigate various special cases all these case studies finally provide us with a way to construct an example of a weakly amenable banach algebra which is not 3 weakly amenable this answers an open question raised by h g dales f ghahramani and n gronbaek

z stability of crossed products by strongly outer actions ii we consider a crossed product of a unital simple separable nuclear stably finite z stable c algebra a by a strongly outer cocycle action of a discrete countable amenable group under the assumption that a has finitely many extremal tracial states and is elementary amenable we show that the twisted crossed product c algebra is z stable as an application we also prove that all strongly outer cocycle actions of the klein bottle group on z are cocycle conjugate to each other this is the first classification result for actions of non abelian infinite groups on stably finite c algebras

the cyclic and simplicial cohomology of l 1 mathbf n let a l 1 z be the unital semigroup algebra of n we show that the cyclic cohomology groups hc n a a vanish when n is odd and are one dimensional when n is even n 2 using connes exact sequence these results are used to show that the simplicial cohomology groups h n a a vanish for n 2 the results obtained are extended to unital algebras l 1 s for some other semigroups of r

rohlin flows on the cuntz algebra o it is shown that certain quasi free flows on the cuntz algebra o have the rohlin property and therefore are cocycle conjugate with each other this in particular shows that any unital separable nuclear purely infinite simple c algebra has a rohlin flow

affine buildings tiling systems and higher rank cuntz krieger algebras to an r dimensional subshift of finite type satisfying certain special properties we associate a c algebra a this algebra is a higher rank version of a cuntz krieger algebra in particular it is simple purely infinite and nuclear we study an example if is a group acting freely on the vertices of an a2 building with finitely many orbits and if is the boundary of that building then c o is the algebra associated to a certain two dimensional subshift

amenability for twisted covariance algebras and group c algebras abstract the basic question studied in this paper is when is a twisted covariance c algebra c g a f amenable this includes as a special case the question when is a group c algebra c g amenable we show in particular that c h a f is amenable if c g a f is amenable and h is a closed normal subgroup of g other results that we prove use the derivation and injective characterizations for amenable c algebras

generalized notions of amenability ii this paper continues the investigation of the first two authors begun in part i it is shown that approximate amenability and approximate contractibility are the same properties as are uniform approximate amenability and amenability bounded approximate contractibility and bounded approximate amenability are characterized by the existence of suitable operator bounded approximate identities for the diagonal ideal results are given on banach sequence algebras lipschitz algebras and beurling algebras and on the crucial role of approximate identities a new proof is given for a result due to n gronbaek on characterizing of amenability for beurling algebras

perturbations of nuclear c algebras kadison and kastler introduced a natural metric on the collection of all c subalgebras of the bounded operators on a separable hilbert space they conjectured that sufficiently close algebras are unitarily conjugate we establish this conjecture when one algebra is separable and nuclear we also consider one sided versions of these notions and we obtain embeddings from certain near inclusions involving separable nuclear c algebras at the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme

ruled 4 manifolds and isotopies of symplectic surfaces we study symplectic surfaces in ruled symplectic 4 manifolds which are disjoint from a given symplectic section as a consequence in any symplectic 4 manifold two homologous symplectic surfaces which are c 0 close must be hamiltonian isotopic

algebraic connections universal bimodules and entire cyclic cohomology a new method of constructing homotopies suitable for entire cyclic cohomology is presented as a result the periodic and entire cyclic cohomology of banach algebras of finite cohomological dimension are shown to be isomorphic the same method can be used to calculate the algebraic entire cyclic cohomology of non commutative tori

amenability and uniqueness abstract the main result of this paper is a characterization of properly infinite injective von neumann algebras and of nuclear c algebras by using a uniqueness theorem based on generalizations of voiculescu s famous weyl von neumann theorem

a fixed point theorem for l1 spaces we prove a fixed point theorem for a family of banach spaces notably for l 1 applications include the optimal answer to the derivation problem for group algebras which originated in the 1960s

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

cyclic cohomology of banach algebras of quivers and their inverse semigroups abstract in this paper we consider the path semigroup l 1 algebra for a quiver and the inverse semigroup l 1 algebra of a quiver the latter of which can be used in the construction of cuntz krieger algebras the main objectives of the paper are to determine the simplicial and cyclic cohomology groups of these algebras first we determine the simplicial and cyclic cohomology of the path algebra of the quiver showing the simplicial cohomology groups of dimension n vanish for n 1 we then determine the simplicial and cyclic cohomology of the inverse semigroup algebra the work uses the connes tzygan long exact sequence

type iii representations and automorphisms of some separable nuclear c algebras abstract powers proved decades ago that if two cyclic representations 1 and 2 of a uhf algebra a satisfy that 1 a 2 a m there is an automorphism of a such that 1 and 2 are quasi equivalent this was recently extended to the class of simple separable c algebras when m is of type i in this paper we extend this result to some class of simple separable nuclear c algebras when m is of type iii in particular we show that the above result obtained by powers for the uhf algebras also holds for the class of purely infinite simple separable c algebras classified by kirchberg and phillips and for the class of approximately homogeneous simple separable unital c algebra with unique tracial state

exact energy distribution function in a time dependent harmonic oscillator following a recent work by robnik and romanovski 2006 j phys a math gen 39 l35 2006 open syst inf dyn 13 197 222 we derive an explicit formula for the universal distribution function of the final energies in a time dependent 1d harmonic oscillator whose functional form does not depend on the details of the frequency t and is closely related to the conservation of the adiabatic invariant the normalized distribution function is where e1 is the final energy is its average value and 2 is the variance of e1 and 2 can be calculated exactly using the wkb approach to all orders

regularity properties in the classification program for separable amenable c algebras we report on recent progress in the program to classify separable amenable c algebras our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank strict comparison of positive elements and z stability and on the importance of the cuntz semigroup we include a brief history of the program s successes since 1989 a more detailed look at the villadsen type algebras which have so dramatically changed the landscape and a collection of announcements on the structure and properties of the cuntz semigroup

amenability and virtual diagonals for von neumann algebras abstract it is shown that a certain universal bimodule over a von neumann algebra is automatically normal this leads to a simple direct proof of haagerup s result that a von neumann algebra is amenable or equivalently injective if and only if it has a normal virtual diagonal haagerup has used this fact to prove that nuclear c algebras are amenable

strict comparison and z absorption of nuclear c algebras for any unital separable simple infinite dimensional nuclear c algebra with finitely many extremal traces we prove that mathcal z absorption strict comparison and property si are equivalent we also show that any unital separable simple nuclear c algebra with tracial rank zero is approximately divisible and hence is mathcal z absorbing

higher dimensional virtual diagonals and ideal cohomology for triangular algebras we investigate the cohomology of non self adjoint algebras using virtual diagonals and their higher dimensional generalizations we show that infinite dimensional nest algebras always have non zero second cohomology by showing that they cannot possess 2 virtual diagonals in the case of the upper triangular atomic nest algebra we exhibit concrete modules for non vanishing cohomology

on the holomorphicity of genus two lefschetz fibrations we prove that any genus 2 lefschetz fibration without reducible fibers and with transitive monodromy is holomorphic the latter condition comprises all cases where the number of singular fibers 10n is not congruent to 0 modulo 40 this proves a conjecture of the authors in siti1 an auxiliary statement of independent interest is the holomorphicity of symplectic surfaces in s 2 bundles over s 2 of relative degree 7 over the base and of symplectic surfaces in cp 2 of degree 17

on the weak amenability of a x and its relation with the approximation property abstract we investigate the weak amenability of the algebra a x of approximable operators on a banach space x and its relation with the bounded approximation property in particular it will be shown that the bounded approximation property is neither necessary nor sufficient for the weak amenability of a x

statistical properties of 1d time dependent hamiltonian systems from the adiabatic limit to the parametrically kicked systems we consider 1d time dependent hamiltonian systems and their statistical properties namely the time evolution of microcanonical distributions whose properties are very closely related to the existence and preservation of the adiabatic invariants we review the elements of the recent developments by robnik and romanovski during 2006 2008 for the entirely general 1d time dependent linear oscillator and try to generalize the results to the 1d nonlinear hamilton oscillators in particular the power law potentials like e g the quartic oscillator furthermore we consider the limit opposite to the adiabatic limit namely parametrically kicked 1d hamiltonian systems even for the linear oscillator interesting properties are revealed an initial kick disperses the microcanonical distribution to a spread one but an anti kick at an appropriate moment of time can annihilate it kicking it back to the microcanonical distribution the case of periodic parametric kicking is also interesting finally we propose that in the parametric kicking of a general 1d hamilton system the average value of the adiabatic invariant always increases which we prove for the power law potentials we find that the approximation of kicking is good for quite long times of the parameter variation up to the order of not much less than one period of the oscillator we also look at the behavior of the quartic oscillator for the case of the kick and anti kick and also the periodic kicking

the primitive ideals of the cuntz krieger algebra of a row finite higher rank graph with no sources abstract we catalogue the primitive ideals of the cuntz krieger algebra of a row finite higher rank graph with no sources each maximal tail in the vertex set has an abelian periodicity group of finite rank at most that of the graph the primitive ideals in the cuntz krieger algebra are indexed by pairs consisting of a maximal tail and a character of its periodicity group the cuntz krieger algebra is primitive if and only if the whole vertex set is a maximal tail and the graph is aperiodic

cohen host type idempotent theorems for representations on banach spaces and applications to figa talamanca herz algebras abstract let g be a locally compact group and let r g denote the ring of subsets of g generated by the left cosets of open subsets of g the cohen host idempotent theorem asserts that a set lies in r g if and only if its indicator function is a coefficient function of a unitary representation of g on some hilbert space we prove related results for representations of g on certain banach spaces we apply our cohen host type theorems to the study of the figa talamanca herz algebras a p g with p 1 for arbitrary g we characterize those closed ideals of a p g that have an approximate identity bounded by 1 in terms of their hulls furthermore we characterize those g such that a p g is 1 amenable for some and equivalently for all p 1 these are precisely the abelian groups

statistical properties of the energy in time dependent homogeneous power law potentials we study classical 1d hamilton systems with a homogeneous power law potential and their statistical behavior assuming a microcanonical distribution of the initial conditions and describing its change under a monotonically increasing time dependent function a t the prefactor of the potential using the nonlinear wentzel kramers brillouin like method of papamikos and robnik 2012 j phys a math theor 44 315102 and following a previous work by papamikos and robnik 2011 j phys a math theor 45 015206 we specifically analyze the mean energy the variance and the adiabatic invariant action of the systems for large time and we show that the mean energy and variance increase as powers of a t while the action oscillates and finally remains constant by means of a number of detailed case studies we show that the theoretical prediction is excellent which demonstrates the usefulness of the method in such applications

weak amenability of triangular banach algebras let a and b be unital banach algebras and let m be a banach a b module then t a m 0 b becomes a triangular banach algebra when equipped with the banach space norm a m 0 b a a m m b b a banach algebra t is said to be n weakly amenable if all derivations from t into its n th dual space t n are inner in this paper we investigate arens regularity and n weak amenability of a triangular banach algebra t in relation to that of the algebras a b and their action on the module m

exact analysis of adiabatic invariants in the time dependent harmonic oscillator the theory of adiabatic invariants has a long history and important applications in physics here we treat exactly the general time dependent 1d harmonic oscillator which cannot be solved in general we follow the time evolution of an initial ensemble of phase points with sharply defined energy e0 and calculate rigorously the distribution of energy e1 after time t and all its moments especially its average value and variance 2 using wkb theory we get the exact result for the leading asymptotic behaviour of 2