linear acoustic formulas for calculation of rotating blade noise a unified approach is used to derive many of the current formulas for calculation of discrete frequency noise of helicopter rotors and propellers both compact and noncompact results are derived the noncompact results are based on the solution of ffowcs williams hawkings fw h equation the compact formulations are obtained as the limit of noncompact source results in particular the linearized acoustic theories of hawkings and lowson farassat hanson woan and gregorek succi and jou are discussed in this paper an interesting thickness noise formula by isom and its extension by ffowcs williams are also presented

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

a connes amenable dual banach algebra need not have a normal virtual diagonal let g be a locally compact group and let wap g denote the space of weakly almost periodic functions on g we show that if g is a sinj group but not compact then the dual banach algebra wap g does not have a normal virtual diagonal consequently whenever g is an amenable non compact sin group wap g is an example of a connes amenable dual banach algebra without a normal virtual diagonal on the other hand there are amenable non compact locally compact groups g such that wap g does have a normal virtual diagonal

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

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

an example of algebraic cycles with nontrivial abel jacobi images abstract in this paper we will study a family of quintic hypersurfaces and its abel jacobi images we will construct codimension 2 algebraic cycles on the blowing up of certain dimension 3 quintic hypersurfaces defined over number fields and show that they are homologically trivial but have nontrivial images under the 3 adic abel jacobi map

kolyvagin s trace relations for siegel sixfolds abstract in his earlier preprints the author offered a program of generalization of kolyvagin s result of finiteness of sh to the case of some motives which are quotients of cohomology motives of shimura and drinfeld varieties the present paper is devoted to the first step of this program finding of an analog of kolyvagin s trace relations we find it for siegel sixfolds and for the hecke correspondences related to the matrices diag 1 1 1 p p p and diag p 1 1 p p 2 p 2 this is the first non trivial case for shimura varieties some results for other types of siegel varieties and hecke correspondences are obtained ideas and methods of the present paper open a large new area of research results given here constitute a tiny part of what can be done particularly maybe it is possible to realize the program of generalization of kolyvagin s result for the functional case moduli varieties of drinfeld modules and similar objects

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

modeling and finite element formulation for acoustic problems including rotating domains a finite element formulation for the efficient numerical simulation of sound in computational domains including rotating regions is presented the mathematical description is based on an arbitrary lagrangian eulerian framework and results in a convective wave equation for the scalar acoustic potential numerically the capability of nonconforming grids is explored by applying a nitsche type mortaring between stationary and rotating regions the formulation can be applied to classical acoustics stagnant fluid as well as moving fluids in the case of aeroacoustics the validation with the analytical solution of a rotating point source in three dimensions demonstrates the accuracy and robustness of the developed numerical scheme additionally the convergence study shows that the intersection mesh operations needed in the nonconforming setting do not deteriorate the accuracy of the numerical solution

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

torsion algebraic cycles and complex cobordism atiyah and hirzebruch gave the first counterexamples to the hodge conjecture with integral coefficients 3 that conjecture predicted that every integral cohomology class of hodge type p p on a smooth projective variety should be the class of an algebraic cycle but atiyah and hirzebruch found additional topological properties which must be satisfied by the integral cohomology class of an algebraic cycle here we provide a more systematic explanation for their results by showing that the classical cycle map from algebraic cycles modulo algebraic equivalence to integral cohomology factors naturally through a topologically defined ring which is richer than integral cohomology the new ring is based on complex cobordism a well developed topological theory which has been used only rarely in algebraic geometry since hirzebruch used it to prove the riemann roch theorem 17 this factorization of the classical cycle map implies the topological restrictions on algebraic cycles found by atiyah and hirzebruch it goes beyond their work by giving a topological method to show that the classical cycle map can be noninjective as well as nonsurjective the kernel of the classical cycle map is called the griffiths group and the topological proof here that the griffiths group can be nonzero is the first proof of this fact which does not use hodge theory the proof here gives nonzero torsion elements in the griffiths group whereas griffiths s hodge theoretic proof gives nontorsion elements 13 this topological argument also gives examples of algebraic cycles in the kernel of various related cycle maps where few or no examples were known before thus answering some questions posed by colliot thelene and schoen 8 p 14 37 p 13 colliot thelene asked in particular whether the map ch2 x n h4 x z n is injective for all smooth complex projective varieties x here ch x is the group of codimension i algebraic cycles modulo rational equivalence the first examples where colliot thelene s map is not injective were found by kollar and van geemen see 4 p 135 very recently bloch and esnault found examples defined over number fields 7 over nonalgebraically closed fields k there are other examples of smooth projective varieties xk with ch2 xk n hg4t xk z n not injective due to colliot thelene and sansuc as reinterpreted by salberger see 9 and 8 remark 7 6 1 and parimala and suresh 31 these elements of ch2 xk n are not shown to remain nonzero in ch2 xc n however here our topological method

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

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

a two dimensional solution of the fw h equation for rectilinear motion of sources abstract in this paper a subsonic solution of the two dimensional ffowcs williams and hawkings fw h equation is presented for calculation of noise generated by sources moving with constant velocity in a medium at rest or in a moving medium the solution is represented in the frequency domain and is valid for observers located far from the noise sources in order to verify the validity of the derived formula three test cases are considered namely a monopole a dipole and a quadrupole source in a medium at rest or in motion the calculated results well coincide with the analytical solutions validating the applicability of the formula to rectilinear subsonic motion problems

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

prediction of advanced propeller noise in the time domain this paper presents a brief derivation of a formula for prediction of the noise of high speed propellers in the time domain this formula is based on the solution of a linear wave equation ffowcs williams hawkings equation assuming that the sources are located on the actual blade surface an approximation of this formula using the assumption that the sources are on the mean surface of the blade is also presented which is considerably simpler than the full surface expression both the mean and full surface formulas are suitable for prediction of noise of supersonic sources on the blade they were derived to overcome some of the practical numerical difficulties associated with other acoustic formulations a discussion of coding these formulas for numerical work follows some comparison of predicted results with experimental data is also included which demonstrates the usefulness of the derived analytic expressions

complex multiplication cycles and a conjecture of beilinson and bloch a generalization of the conjecture of birch and swinnerton dyer is investigated using complex multiplication cycles on a particular kuga fiber variety a weak finiteness result consistent with the conjecture is proved the image of complex multiplication cycles under the abel jacobi map is computed explicitly the results provide numerical evidence supporting the conjecture they also give evidence for a relationship between complex multiplication cycles and a modular form of weight 5 2 and raise questions for further investigation

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

factorization of completely bounded bilinear operators and injectivity a completely bounded bilinear operator m m m on a von neumann algebra m is said to have a factorization in m if there exist completely bounded linear operators j j m m such that x y sum j j x j y x y m where convergence of the sum is made precise below the main result of the paper is that all completely bounded bilinear operators m m m have factorizations in m if and only if m is injective

on the birch and swinnerton dyer conjecture let f be a finite extension of the rational field q if e is an elliptic curve defined over f then the mordell weil group e f of points on e with coordinates in f is a finitely generated abelian group let l e f s be the hasseweil zeta function of e over f which at least in the case where e has complex multiplication is known to be an analytic function on the entire complex plane birch and swinnerton dyer have conjectured that l e f s has a zero at s l of order precisely equal to the rank of e f over z the strongest result known in support of this conjecture is the following theorem of coates and wiles 4

the grothendieck inequality for bilinear forms on c algebras abstract the following generalization of grothendieck s inequality is proved for any bounded bilinear form v on a pair of c algebras a b there exist two states 1 2 on a and two states 1 2 on b such that v x y v 1 x x 2 xx 2 1 2 1 y y 2 yy 2 1 2 for all x a and all y b an inequality of this type was proved a few years ago by pisier in the case where one of the c algebras has the bounded approximation property it follows from the above inequality that any bounded linear map t of a c algebra into the dual of a c algebra has a factorization t r s through a hilbert space such that r s 2 t

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

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

representations of completely bounded multilinear operators a deli ion of a completely bounded multilinear operator from one c algebra into another is introduced each completely bounded multilinear operator from a algebra into the algebra of bounded linear operators on a hilbert space is shown to be representable in terms of representations of the c algebra and interlacing operators this result extends wittstock s theorem that decomposes a completely bounded linear operator from a c algebra into an injective c algebra into completely positive linear operators 0 1987 academic press inc stinespring s theorem gives a useful representation for a completely positive linear operator from a c algebra into the algebra bl h for continuous linear operators on a hilbert space h 12 11 using this representation with wittstock s theorem that decomposes a completely bounded linear operator as a finite linear combination of completely positive linear operators 14 lo one obtains a representation of a completely bounded linear operator from a c algebra into bl h our main result is a representation theorem for completely bounded multilinear operators from a c algebra into bl h which generalizes this representation of completely bounded linear operators corollaries give multilinear generalizations of several results known for completely bounded linear operators we shall briefly describe the type of representations to be 151

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

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

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

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

p adic l functions and the coniveau filtration on chow groups we give evidence for the refined version of the bloch beilinson conjecture involving coniveau filtrations by studying several infinite families of cm motives indexed by the integers r 1 that are irreducible of hodge type 2r 1 0 0 2r 1 and whose l functions vanish at the center in each case we construct a corresponding algebraic cycle that is homologically trivial but nontrivial in the top graded piece for the coniveau filtration consequently we obtain new explicit examples of cycles that are nontorsion in the griffiths group the proof of the main result relies crucially on p adic hodge theoretic methods and in particular on the relation between images of homologically trivial cycles under the p adic abel jacobi map and the values of p adic l functions

action of hecke correspondences on heegner curves on a siegel threefold abstract we get an analog of kolyvagin s trace relations for a siegel threefold x namely let v x be a heegner curve points of v correspond to abelian surfaces with some fixed multiplication ring and let tp be a hecke correspondence on x so tp v is a codimension 2 cycle on x we describe the set of irreducible components of the support of tp v in terms of geometry of tp t where t x is a generic point tp t is a three dimensional quadric hypersurface over f p we find also some equivalence relations on this set of irreducible components the same method can be applied to other pairs v x or more generally chains xn xn 1 x1 of shimura varieties and other hecke correspondences finally we discuss the possibility of finding reductions at p of these irreducible components and of applying the birch mazur bloch method to prove that the abel jacobi image of some linear combination of v and other similar curves is not of torsion

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

generalized heegner cycles and p adic rankin l series this article studies a distinguished collection of so called generalized heegner cycles in the product of a kuga sato variety with a power of a cm elliptic curve its main result is a p adic analogue of the gross zagier formula which relates the images of generalized heegner cycles under the p adic abel jacobi map to the special values of certain p adic rankin l series at critical points that lie outside their range of classical interpolation

the similarity degree of an operator algebra ii for every integer d ge 1 there is a unital closed subalgebra a d subset b h with similarity degree equal precisely to d in the sense of our previous paper this means that for any unital homomorphism u colon a d to b h we have u cb le k u d with k 0 independent of u and the exponent d in this estimate cannot be improved the proof that the degree is larger than d 1 crucially uses an upper bound for the norms of certain gaussian random matrices due to haagerup and thorbjornsen we also include several complements to our previous publications on the same subject

an efficient two dimensional implementation of the ffowcs williams and hawkings equation abstract this paper describes a two dimensional formulation of the ffowcs williams and hawkings equation in the frequency domain by assuming subsonic rectilinear motion of all acoustic sources an efficient and easily implemented form of the equation is developed this method is capable of predicting the farfield noise from non linear nearfield flow quantities the ability to use non linear input data is a clear advantage over kirchhoff methods that are only valid in regions where the linear wave equation accurately describes the flow several example problems are used to demonstrate that the new method performs well for problems with a mean flow tonal and broadband noise signatures and non linear near fields in most practical acoustic problems three dimensionality is important and should not be neglected for these real world applications two dimensional solutions can be used to guide and augment full three dimensional calculations but not replace them

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

inner amenable locally compact groups in this paper we study the relationship between amenability inner amenability and property p of a von neumann algebra we give necessary conditions on a locally compact group g to have an inner invariant mean m such that m v 0 for some compact neighborhood v of g invariant under the inner automorphisms we also give a sufficient condition on g satisfied by the free group on two generators or an i c c discrete group with kazhdan s property t e g sl n z n 3 such that each linear form on l 2 g which is invariant under the inner automorphisms is continuous a characterization of inner amenability in terms of a fixed point property for left banach g modules is also obtained

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

hochschild cohomology of factors with property gamma the main result of this paper is that the k th continuous hochschild cohomology groups h k m m and h k m b h of a von neumann factor m b h o f type ii 1 with property are zero for all positive integers k the method of proof involves the construction of hyperfinite subfactors with special properties and a new inequality of grothendieck type for multilinear maps we prove joint continuity in the 2 norm of separately ultraweakly continuous multilinear maps and combine these results to reduce to the case of completely bounded cohomology which is already solved

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