equivariant novikov conjecture for groups acting on euclidean buildings we prove the equivariant novikov conjecture for groups acting on euclidean buildings by using an equivariant f tilsum skandalis methocl we also obtain an equivariant version of the connes gromov moscovic th eorem for almost flat c algebra bundles

localizing gauge theories from noncommutative geometry abstract we recall the emergence within noncommutative geometry of a generalized gauge theory from a real spectral triple for a unital c algebra a this includes a gauge group determined by the unitaries in a and gauge fields arising from a so called perturbation semigroup which is associated to a our main new result is the construction of an upper semi continuous c bundle on a hausdorff base space x whose space of continuous sections is isomorphic to a the gauge group acts by vertical automorphisms on this c bundle and can under some conditions be identified with the space of continuous sections of a group bundle on x moreover in some cases the gauge group is found to coincide with the group of inner automorphisms of a we discuss two classes of examples of our construction yang mills theory and toric noncommutative manifolds and we show that they actually give rise to continuous c bundles moreover in these examples the gauge groups coincide with the inner automorphism groups and can be realized as spaces of sections of group bundles that we explicitly determine

the c algebra of a vector bundle and fields of cuntz algebras abstract we study the cuntz pimsner algebra associated with the module of continuous sections of a hilbert bundle and prove that it is a continuous bundle of cuntz algebras furthermore we assign to bundles of cuntz algebras carrying a global circle action a class in the representable kk group of the zero grade bundle we explicitly compute such class for the cuntz pimsner algebra of a vector bundle

centre and representations of at roots of unity quantum groups at the roots of unity have the property that their centre is enlarged polynomial equations relate the standard deformed casimir operators and the new central elements these relations are important from a physical point of view since they correspond to relations between quantum expectation values of observables that have to be satisfied on all physical states in this paper we establish these relations in the case of the quantum lie superalgebra in the course of the argument we find and use a set of representations such that any relation satisfied on all the representations of the set is true in this set is a subset of the set of all the finite dimensional irreducible representations of which we classify and describe explicitly

graded tensor calculus we develop a graded tensor calculus corresponding to arbitrary abelian groups of degrees and arbitrary commutation factors the standard basic constructions and definitions like tensor products spaces of multilinear mappings contractions symmetrization symmetric algebra as well as the transpose adjoint and trace of a linear mapping are generalized to the graded case and a multitude of canonical isomorphisms is presented moreover the graded versions of the classical lie algebras are introduced and some of their basic properties are described

on a nonlinear hammerstein integral equation with a parameter abstract this paper following theories for asymptotically linear operators the schaefer fixed point theorem decomposition of operators and critical point theory is mainly concerned with the existence and multiplicity of solutions to a nonlinear hammerstein integral equation with a parameter the results show that when the nonlinearity satisfies certain conditions different parametric intervals lead to different existence results however in some cases only the sign of the parameter makes a contribution to the existence of solutions for the problem our results can be applied to some well known boundary value problems and some examples are given

extension of the leray schauder degree for abstract hammerstein type mappings abstract we introduce a new extension of the classical leray schauder topological degree in a real separable reflexive banach space the new class of mappings for which the degree will be constructed is obtained essentially by replacing the compact perturbation by a composition of mappings of monotone type it turns out that the class contains the leray schauder type maps as a proper subclass the new class is not convex thus preventing the free application of affine homotopies however there exists a large class of admissible homotopies including subclass of affine ones so that the degree can be effectively used we shall construct the degree and prove that it is unique we shall generalize the borsuk theorem of the degree for odd mappings and show that the principle of omitted rays remains valid to illuminate the use of the new degree we shall briefly consider the solvability of abstract hammerstein type equations and variational inequalities

the first fundamental theorem of invariant theory for the orthosymplectic super group we give an elementary and explicit proof of the first fundamental theorem of invariant theory for the orthosymplectic supergroup by generalising the geometric method of atiyah bott and patodi to the supergroup context we use methods from super algebraic geometry to convert invariants of the orthosymplectic supergroup into invariants of the corresponding general linear supergroup on a different space in this way super schur weyl brauer duality is established between the orthosymplectic supergroup of superdimension m 2n and the brauer algebra with parameter m 2n the result may be interpreted either in terms of the group scheme osp v over mathbb c where v is a finite dimensional super space or as a statement about the orthosymplectic lie supergroup over the infinite dimensional grassmann algebra lambda we take the latter point of view here and also state a corresponding theorem for the orthosymplectic lie superalgebra which involves an extra invariant generator the super pfaffian

eigenvalues of casimir invariants for uq osp m n for each quantum superalgebra uq osp m n with m 2 an infinite family of casimir invariants is constructed this is achieved by using an explicit form for the lax operator the eigenvalue of each casimir invariant on an arbitrary irreducible highest weight module is also calculated

casimir elements of e lie algebras the classical framework for investigating the casimir elements of a lie algebra is generalized to the case of an e lie algebra l we construct the standard l module isomorphism of the e symmetric algebra of l onto its enveloping algebra and we introduce the harish chandra homomorphism in case the generators of l can be written in a canonical two index form we construct the associated standard sequence of casimir elements and derive a formula for their eigenvalues in an arbitrary highest weight module

groupoids and an index theorem for conical pseudo manifolds we define an analytical index map and a topological index map for conical pseudomanifolds these constructions generalize the analogous constructions used by atiyah and singer in the proof of their topological index theorem for a smooth compact manifold m a main ingredient is a non commutative algebra that plays in our setting the role of c 0 t m we prove a thom isomorphism between non commutative algebras which gives a new example of wrong way functoriality in k theory we then give a new proof of the atiyah singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds we thus prove a topological index theorem for conical pseudomanifolds

the c0 x algebra of a net and index theory abstract given a connected and locally compact hausdorff space x with a good base k we assign in a functorial way a c 0 x algebra to any precosheaf of c algebras a defined over k afterwards we consider the representation theory and the kasparov k homology of a and interpret them in terms respectively of the representation theory and the k homology of the associated c 0 x algebra when a is an observable net over the spacetime x in the sense of algebraic quantum field theory this yields a geometric description of the recently discovered representations affected by the topology of x

a characterization of 0 epi maps with a degree abstract a map of the form i f with a strictly condensing operator f is 0 epi if and only if it has nonzero degree on a component of its domain of definition this proves a conjecture of the second author in m vath j math anal appl 257 2001 223 237

fixed point theory and nonlinear problems introduction among the most original and far reaching of the contributions made by henri poincare to mathematics was his introduction of the use of topological or qualitative methods in the study of nonlinear problems in analysis his starting point was the study of the differential equations of celestial mechanics and in particular of their periodic solutions his work on this topic began with his thesis in 1879 and was developed in detail in his great three volume work methodes nouvelles de la mechanique celeste which appeared in the early 1890s and summarized his many memoirs of the intervening period it continued until his memoir shortly before his death in 1912 in which he put forward the unproved fixed point result usually referred to as poincare s last geometric theorem the ideas introduced by poincare include the use of fixed point theorems the continuation method and the general concept of global analysis the writer s acquaintance with poincare s influence came through contact with solomon lefschetz and marston morse both of whom were very explicit as to the role of poincare as an initiator in this direction of mathematical development in 1934 in the foreword to his colloquium volume on the calculus of variations in the large morse had put this forward very forcefully in the first paragraph

the capelli problem for gl m n and the spectrum of invariant differential operators abstract the capelli problem for the symmetric pairs gl gl gl gl o and gl sp is closely related to the theory of jack polynomials and shifted jack polynomials for special values of the parameter see 12 15 14 18 in this paper we extend this connection to the lie superalgebra setting namely to the supersymmetric pairs g k gl m 2 n osp m 2 n and gl m n gl m n gl m n acting on w s 2 c m 2 n and c m n c m n to achieve this goal we first prove that the center of the universal enveloping algebra of the lie superalgebra g maps surjectively onto the algebra pd w g of g invariant differential operators on the superspace w thereby providing an affirmative answer to the abstract capelli problem for w our proof works more generally for gl m n acting on s 2 c m n and is new even for the ordinary cases m 0 or n 0 considered by howe and umeda in 9 we next describe a natural basis d of pd w g that we call the capelli basis using the above result on the abstract capelli problem we generalize the work of kostant and sahi 12 15 20 by showing that the spectrum of d is given by a polynomial c which is characterized uniquely by certain vanishing and symmetry properties we further show that the top homogeneous parts of the eigenvalue polynomials c coincide with the spherical polynomials d which arise as radial parts of k spherical vectors of finite dimensional g modules and which are super analogues of jack polynomials this generalizes results of knop and sahi 14 finally we make a precise connection between the polynomials c and the shifted super jack polynomials of sergeev and veselov 25 for special values of the parameter we show that the two families are related by a change of coordinates that we call the frobenius transform

positive solutions of semi positone hammerstein integral equations and applications abstract existence of solutions for semi positone integral equations of hammerstein type is obtained by using the fixed point index method many boundary value problems of differential equations can be transformed into these integral equations a specific example is given

mathcal c _ 0 x algebras stability and strongly self absorbing mathcal c algebras we study permanence properties of the classes of stable and so called mathcal d stable mathcal c algebras respectively more precisely we show that a mathcal c 0 x algebra a is stable if all its fibres are provided that the underlying compact metrizable space x has finite covering dimension or that the cuntz semigroup of a is almost unperforated a condition which is automatically satisfied for mathcal c algebras absorbing the jiang su algebra mathcal z tensorially furthermore we prove that if mathcal d is a k 1 injective strongly self absorbing mathcal c algebra then a absorbs mathcal d tensorially if and only if all its fibres do again provided that x is finite dimensional this latter statement generalizes results of blanchard and kirchberg we also show that the condition on the dimension of x cannot be dropped along the way we obtain a useful characterization of when a mathcal c algebra with weakly unperforated cuntz semigroup is stable which allows us to show that stability passes to extensions of mathcal z absorbing mathcal c algebras

groups generated by reflections and aspherical manifolds not covered by euclidean space a coxeter system r v is a group r a coxeter group together with a set of generators v such that each element of v has order two and such that all relations in r are consequences of relations of the form vw m v w 1 where v w e v and m v w denotes the order of vw the m v w s which are positive integers or oo obviously determine the coxeter system up to isomorphism the list of finite coxeter groups is short and well known however general coxeter groups are much more flexible in fact the set v and the m v w s can be specified arbitrarily at least if v is finite subject only to the conditions m v v 1 and m v w m w v 2 if v w suppose that r v is a coxeter system that x is a hausdorff space and that xv v v is a locally finite family of closed subspaces indexed by v the xv are called the panels of x 2 there is a classical method for constructing a transformation group from these data for each x e x let v x denote the set of v in v such that x e xv for each subset s of v let rs be the subgroup generated by s and let xs be the face of x defined by

on the zero in the spectrum conjecture we prove that the answer to the zero in the spectrum conjecture in the form suggested by j lott is negative namely we show that for any n ge 6 there exists a closed n dimensional smooth manifold m n so that zero does not belong to the spectrum of the laplace beltrami operator acting on the l 2 forms of all degrees on the universal covering tilde m

some remarks on kasparov theory on considere une autre preuve de l invariance d homotopie du bifoncteur ext on etudie l associativite du produit de kasparov a travers la motion de connextion

cohomology of lie superalgebras and their generalizations the cohomology groups of lie superalgebras and more generally of e lie algebras are introduced and investigated the main emphasis is on the case where the module of coefficients is nontrivial two general propositions are proved which help to calculate the cohomology groups several examples are included to show the peculiarities of the super case for l sl 1 2 the cohomology groups h1 l v and h2 l v with v a finite dimensional simple graded l module are determined and the result is used to show that h2 l u l with u l the enveloping algebra of l is trivial this implies that the superalgebra u l does not admit any nontrivial formal deformations in the sense of gerstenhaber garland s theory of universal central extensions of lie algebras is generalized to the case of e lie algebras

new expressions for the eigenvalues of the invariant operators of the general linear and the orthosymplectic lie superalgebras we obtain expansions for the eigenvalues cp of the invariant operators casimir operators of the general linear and the orthosymplectic lie superalgebras in terms of products of suitably defined graded power sums pk the resulting expressions are closed and provide unified formulas for computing the cp s for those superalgebras as well as their corresponding lie algebras the formulas are remarkably simple to suggest that the power sums used in this text could play a more basic role in the understanding of the pattern of the expansion coefficients explicit illustrations are given for the various series for p 8

the generator problem for z stable c algebras the generator problem was posed by kadison in 1967 and it remains open until today we provide a solution for the class of c algebras absorbing the jiang su algebra z tensorially more precisely we show that every unital separable z stable c algebra a is singly generated which means that there exists an element x a that is not contained in any proper sub c algebra of a to give applications of our result we observe that z can be embedded into the reduced group c algebra of a discrete group that contains a non cyclic free subgroup it follows that certain tensor products with reduced group c algebras are singly generated in particular c r f c r f is singly generated

invariants and reduced matrix elements associated with the lie superalgebra gl m n we construct explicit formulae for the eigenvalues of certain invariants of the lie superalgebra gl m n using characteristic identities we discuss how such eigenvalues are related to reduced wigner coefficients and the reduced matrix elements of generators and thus provide a first step to a new algebraic derivation of matrix element formulae for all generators of the algebra

k theoretic amenability for sl2 qp and the action on the associated tree abstract it is proved that if a locally compact group g acts simplicially on a tree in such a way that the stabilizers of the vertices are amenable then g is k amenable in particular the canonical map from the full c algebra onto the reduced c algebra of g induces isomorphisms in k theory the main corollary of our result is that sl 2 q p and some other groups over local fields are k amenable

smooth dense subalgebras of reduced group c algebras schwartz cohomology of groups and cyclic cohomology abstract we introduce the notion of schwartz co homology of a discrete group with a length function which is the natural generalization of the bounded cohomology of a discrete group we prove that if the length function is chosen to be a word length on a group of polynomial growth then its schwartz cohomology is canonically isomorphic to the usual group cohomology with complex coefficients using this result we will show that the group cohomology of a discrete group of polynomial growth is of polynomial growth in the sense of connes and moscovici and that there are no nontrivial idempotents in the group c algebra of a torsion free group of polynomial growth by the method of cyclic cohomology of connes

on the k theory of groups with finite asymptotic dimension it is proved that the assembly maps in algebraic kand l theory with respect to the family of finite subgroups is injective for groups with finite asymptotic dimension that admit a finite model for e the result also applies to certain groups that admit only a finite dimensional model for e in particular it applies to discrete subgroups of virtually connected lie groups

eigenvalues of casimir operators for the general linear the special linear and the orthosymplectic lie superalgebras the generators of the algebras under consideration can be written in a canonical two index form and hence the associated standard sequence of casimir elements can be constructed following the classical approach by perelomov and popov we obtain the eigenvalues of these casimir elements in an arbitrary highest weight module by calculating the corresponding generating functions

invariant supersymmetric multilinear forms and the casimir elements of p type lie superalgebras the casimir elements of the p type lie superalgebras are investigated depending on the class of algebras under consideration either there do not exist any nontrivial casimir elements at all or else the casimir elements are highly degenerate basic to the investigation is a lemma about invariant supersymmetric multilinear forms on a finite dimensional module over a lie superalgebra some comments on the cartan subalgebras of a lie superalgebra are also included an appendix provides some information on multilinear algebra with e commuting scalars

on the relation between k and l theory of c algebras we prove the existence of a map of spectra tau a ka rightarrow ell a between connective topological k theory and connective algebraic l theory of a complex c algebra a which is natural in a and compatible with multiplicative structures we determine its effect on homotopy groups and as a consequence obtain a natural equivalence ka left tfrac 1 2 right xrightarrow simeq la left tfrac 1 2 right of periodic k and l theory spectra after inverting 2 we show that this equivalence extends to k and l theory of real c algebras using this we give a comparison between the real baum connes conjecture and the l theoretic farrell jones conjecture we conclude that these conjectures are equivalent after inverting 2 if and only if a certain completion conjecture in l theory is true

euler characteristics and gysin sequences for group actions on boundaries let g be a locally compact group let x be a universal proper g space and let open image in new window be a g equivariant compactification of x that is h equivariantly contractible for each compact subgroup open image in new window let open image in new window assuming the baum connes conjecture for g with coefficients open image in new window and c x we construct an exact sequence that computes the map on k theory induced by the embedding open image in new window this exact sequence involves the equivariant euler characteristic of x which we study using an abstract notion of poincare duality in bivariant k theory as a consequence if g is torsion free and the euler characteristic open image in new window is non zero then the unit element of open image in new window is a torsion element of order open image in new window furthermore we get a new proof of a theorem of luck and rosenberg concerning the class of the de rham operator in equivariant k homology

the l 2 index theorem for homogeneous spaces of lie groups the l2 index theorem for covering spaces of atiyah 3 and singer 20 asserts that given a discrete group f acting smoothly and freely on a manifold m with compact quotient m f m and an elliptic differential operator d on m which is f invariant and thus descends to an elliptic differential operator d on the compact manifold m then the p index indrd dimrkerd dimrkerd of d coincides with the ordinary index ind d dim ker d dim ker d of d here ker d is the space of l2 solutions of du 0 and dim r denotes the dimension function corresponding to the trace on the commutant of f acting on the hilbert space of l2 sections over m naturally associated to p the importance of this theorem lies in the fact that indd 0 implies the existence of nontrivial l2 solutions for the equation du 0 and as such it was used in a crucial way by atiyah and schmid 5 to construct explicit realizations of the discrete series representations for semisimple lie groups indeed if g is a lie group which possesses a discrete torsion free cocompact subgroup f and h is a compact subgroup of g then the l2 index theorem applied to the covering space m g h of m f g h combined with the index formula of atiyah singer 6 yields existence results for l2 solutions of g invariant elliptic equations on the homogeneous space g h it is relevant to note that with d denoting this time a g invariant elliptic differential operator on g h the ratio between the f index of d and the covolume of f gives a number independent of f this real number indg d can be in fact intrinsically defined as the difference of the two formal degrees corresponding to the representations of g on ker d and ker d respectively and hence makes sense for any unimodular lie group g even when it has no discrete cocompact subgroups which outside the semisimple case is the generic

the first fundamental theorem of invariant theory for the orthosymplectic supergroup we give an elementary and explicit proof of the first fundamental theorem of invariant theory for the orthosymplectic supergroup by generalising the geometric method of atiyah bott and patodi to the supergroup context we use methods from super algebraic geometry to convert invariants of the orthosymplectic supergroup into invariants of the corresponding general linear supergroup on a different space in this way super schur weyl brauer duality is established between the orthosymplectic supergroup of superdimension m 2n and the brauer algebra with parameter m 2n the result may be interpreted either in terms of the group scheme osp v over mathbb c where v is a finite dimensional super space or as a statement about the orthosymplectic lie supergroup over the infinite dimensional grassmann algebra lambda we take the latter point of view here and also state a corresponding theorem for the orthosymplectic lie superalgebra which involves an extra invariant generator the super pfaffian

geometrization of the strong novikov conjecture for residually finite groups in this paper we prove that the strong novikov conjecture for a residually finite group is essentially equivalent to the coarse geometric novikov conjecture for a certain metric space associated to the group as an application we obtain the coarse geometric novikov conjecture for a large class of sequences of expanders

quasi locality and property a abstract let x be a metric space with bounded geometry p 0 1 and let e be a banach space the main result of this paper is that either if x has yu s property a and p 1 or without any condition on x when p 0 1 then quasi local operators on l p x e belong to the appropriate variant of the roe algebra of x this generalises the existing results of this type by lange and rabinovich engel tikuisis and the first author and li wang and the second author as consequences we obtain that uniform l p roe algebras of spaces with property a are closed under taking inverses and another condition characterising property a akin to the operator norm localisation for quasi local operators

continuous fields of c algebras arising from extensions of tensor c categories abstract the notion of extension of a given c category c by a c algebra a is introduced in the commutative case a c the objects of the extension category are interpreted as fiber bundles over of objects belonging to the initial category it is shown that the doplicher roberts algebra dr algebra in the following associated to an object in the extension of a strict tensor c category is a continuous field of dr algebras coming from the initial one in the case of the category of the hermitian vector bundles over the general result implies that the dr algebra of a vector bundle is a continuous field of cuntz algebras some applications to pimsner c algebras are given

equivariant comparison of quantum homogeneous spaces we consider the quantum homogeneous spaces of the q deformation of simply connected simple compact lie groups and their poisson lie quantum subgroups we prove the deformation invariance in the equivariant kk theory with respect to the translation action by maximal tori this extends a result of neshveyev and tuset to the equivariant setting as applications we prove the ring isomorphism of the k homology of gq with respect to the coproduct of c gq and an analogue of the borsuk ulam theorem for quantum spheres

exotic cohomology and index theory the main result of this paper is a version of the atiyah singer index theorem for dirac type operators on noncompact complete riemannian manifolds the statement of the theorem involves a novel cohomology theory for such manifolds this theory called exotic cohomology depends on the structure at infinity of a space more precisely it depends on the way that large bounded sets fit together for each cohomology class in this theory we define a higher index of a dirac type operator enjoying the stability and vanishing properties of the usual atiyah singer index these higher indices are analogous to the novikov higher signatures our main theorem will compute these higher indices in terms of standard topological invariants applications of this index theorem include a different approach to some of the results of gromov and lawson 10 on topological obstructions to positive scalar curvature the concept of index that we will use involves the a theory functors k0 and kx for operator algebras 3 14 suppose that b is an ideal in a unital algebra c and let t e c be invertible modulo b in the classical atiyah singer index theorem one takes c to be the bounded operators on the l space of some compact manifold b the compact operators and t an elliptic pseudodifferential operator of order zero then t has an index in the theory group kq b in the classical case this is just z and one recovers the usual fredholm index now let m be a complete riemannian manifold possibly noncompact in 18 i introduced an algebra m which is defined as follows sf m consists of all bounded operators a on l m that have a kernel representation

poincare duality for cuntz pimsner algebras abstract we present a new approach to poincare duality for cuntz pimsner algebras we provide sufficient conditions under which poincare self duality for the coefficient algebra of a hilbert bimodule lifts to poincare self duality for the associated cuntz pimsner algebra with these conditions in hand we can constructively produce fundamental classes in k theory for a wide range of examples we can also produce k homology fundamental classes for the important examples of cuntz krieger algebras following kaminker putnam and crossed products of manifolds by isometries and their non commutative analogues

a remarkable connection between the representations of the lie superalgebras osp 1 2n and the lie algebras o 2n 1 we show that there is a one to one correspondence between the graded representations of osp 1 2n and the non spinorial representations of o 2n 1 the clebsch gordan series for osp 1 2n reduce to the corresponding series for o 2n 1 and the properly defined casimir operators of order at least up to four have the same eigenvalues

identifying assembly maps in k and l theory in this paper we prove the equivalence of various algebraically or geometrically defined assembly maps used in formulating the main conjectures in k and l theory and c theory

exotic crossed products and the baum connes conjecture we study general properties of exotic crossed product functors and characterise those which extend to functors on equivariant c algebra categories based on correspondences we show that every such functor allows the construction of a descent in kk theory and we use this to show that all crossed products by correspondence functors of k amenable groups are kk equivalent we also show that for second countable groups the minimal exact morita compatible crossed product functor used in the new formulation of the baum connes conjecture by baum guentner and willett 4 extends to correspondences when restricted to separable g c algebras it therefore allows a descent in kk theory for separable systems