covariance and geometrical invariance in quantization several notions of invariance and covariance for products with respect to lie algebras and lie groups are investigated some examples including the poincare group are given the passage from the lie algebra invariance to the lie group covariance is performed the compact and nilpotent cases are treated

rigid symmetries and conservation laws in non lagrangian field theory making use of the lagrange anchor construction introduced earlier to quantize non lagrangian field theories we extend the noether theorem beyond the class of variational dynamics

hochschild cohomology and atiyah classes abstract in this paper we prove that on a smooth algebraic variety the hkr morphism twisted by the square root of the todd genus gives an isomorphism between the sheaf of poly vector fields and the sheaf of poly differential operators both considered as derived gerstenhaber algebras in particular we obtain an isomorphism between hochschild cohomology and the cohomology of poly vector fields which is compatible with the lie bracket and the cupproduct the latter compatibility is an unpublished result by kontsevich our proof is set in the framework of lie algebroids and so applies without modification in much more general settings as well

normal forms and gauge symmetry of local dynamics a systematic procedure is proposed for deriving all the gauge symmetries of the general not necessarily variational equations of motion for the variational equations this procedure reduces to the dirac bergmann algorithm for the constrained hamiltonian systems with certain extension it remains applicable beyond the scope of dirac s conjecture even though no pairing exists between the constraints and the gauge symmetry generators in general nonvariational dynamics certain counterparts still can be identified of the first and second class constraints without appealing to any poisson structure it is shown that the general local gauge dynamics can be equivalently reformulated in an involutive normal form the last form of dynamics always admits the brst embedding which does not require the classical equations to follow from any variational principle

a differential complex for poisson manifolds construction du complexe canonique d une variete de poisson homologie canonique des varietes symplectiques application a l homologie de hochschild des algebres non commutatives

moment maps at the quantum level we introduce the notion of moment maps for quantum groups acting on their module algebras when the module algebras are quantizations of poisson manifolds we prove that the construction at the quantum level is a quantization of that at the semi classical level we also prove that the corresponding smashed product algebras are quantizations of the semi direct product poisson structures

deformations of coisotropic submanifolds and strong homotopy lie algebroids in this paper we study deformations of coisotropic submanifolds in a symplectic manifold first we derive the equation that governs c deformations of coisotropic submanifolds and define the corresponding c moduli space of coisotropic submanifolds modulo the hamiltonian isotopies this is a non commutative and non linear generalization of the well known description of the local deformation space of lagrangian submanifolds as the set of graphs of closed one forms in the darboux weinstein chart of a given lagrangian submanifold we then introduce the notion of strong homotopy lie algebroid or l algebroid and associate a canonical isomorphism class of strong homotopy lie algebroids to each pre symplectic manifold y and identify the formal deformation space of coisotropic embeddings into a symplectic manifold in terms of this strong homotopy lie algebroid the formal moduli space then is provided by the gauge equivalence classes of solutions of a version of the maurer cartan equation or the master equation of the strong homotopy lie algebroid and plays the role of the classical part of the moduli space of quantum deformation space of coisotropic a branes we provide a criterion for the unobstructedness of the deformation problem and analyze a family of examples that illustrates that this deformation problem is obstructed in general and heavily depends on the geometry and dynamics of the null foliation

a new smoothing quasi newton method for nonlinear complementarity problems the nonlinear complementarity problem can be reformulated as a nonsmooth equation in this paper we propose a new smoothing quasi newton method for the solution of the nonlinear complementarity problems by constructing a new smoothing approximation function global and local superlinear convergence results of the proposed algorithm are established under suitable conditions some numerical results are reported in the paper

global and superlinear convergence of the smoothing newton method and its application to general box constrained variational inequalities the smoothing newton method for solving a system of nonsmooth equations f x 0 which may arise from the nonlinear complementarity problem the variational inequality problem or other problems can be regarded as a variant of the smoothing method at the kth step the nonsmooth function f is approximated by a smooth function f and the derivative of f at x k is used as the newton iterative matrix the merits of smoothing methods and smoothing newton methods are global convergence and convenience in handling in this paper we show that the smoothing newton method is also superlinearly convergent if f is semismooth at the solution and f satisfies a jacobian consistency property we show that most common smooth functions such as the gabriel more function have this property as an application we show that for box constrained variational inequalities if the involved function is p uniform the iteration sequence generated by the smoothing newton method will converge to the unique solution of the problem globally and superlinearly quadratically

on the uniform exponential stability of a wide class of linear time delay systems this paper deals with the global uniform exponential stability independent of delay of time delay linear and time invariant systems subject to point and distributed delays for the initial conditions being continuous real functions except possibly on a set of zero measure of bounded discontinuities it is assumed that the delay free system as well as an auxiliary one are globally uniformly exponentially stable and globally uniform exponential stability independent of delay respectively the auxiliary system is typically a part of the overall dynamics of the delayed system but not necessarily the isolated undelayed dynamics as usually assumed in the literature since there is a great freedom in setting such an auxiliary system the obtained stability conditions are very useful in a wide class of practical applications

quantum harmonic analysis and geometric invariants we develop two topics in parallel and show their inter relation the first centers on the notion of a fractional differentiable structure on a commutative or a non commutative space we call this studyquantum harmonic analysis the second concerns homotopy invariants for these spaces and is an aspect of non commutative geometry we study an algebra a which will be a banach algebra with unit represented as an algebra of operators on a hilbert space h in order to obtain a geometric interpretation of a we define a derivative on elements of a we do this in a hilbert space context takingdaas a commutatorda q a hereqis a basic self adjoint operator with discrete spectrum increasing sufficiently rapidly that exp q 2has a trace whenever 0 we can define fractional differentiability of order with 0 0 we concentrate on subalgebras a of j which have certain additional covariance properties under a group z2 g acting on h by a unitary representation u g in addition the derivativeqis assumed to be g invariant the geometric interpretation flows from the assumption that elements of a possess an arbitrarily small fractional derivative we study homotopy invariants of a in terms of equivariant entire cyclic cohomology in fact the existence of a fractional derivative on a allows the construction of the cochain jlo which plays the role of the integral of differential forms we give a simple expression for a homotopy invariant zq a g determined by pairing jlo with a g invariant elementa a such thatais a square root of the identity this invariant is zq a g 1 e t2tr u g ae q2 it da dt this representation of the pairing is reminiscent of the heat kernel representation for an index in fact this quantity is an invariant in the following sense we isolate a simple condition on a familyq of differentiations that yields a continuously differentiable family jlo of cochains since zq a g need not be an integer continuity of jlo in is insufficient to prove the constancy of the pairing however the existence of the derivative leads to the existence of the homotopy as a vanishes for a coboundary and asd jlo d is a coboundary our condition onq ensures that zq a g is independent of hence it is a homotopy invariant the theory of zq a g reduces to the study of the radon transform of sequences of certain functions the fractional differentiability properties of elements of a translate into properties of the asymptotics of the sequences of radon transforms the condition that jlofit into the framework of entire cyclic cohomology translates to the existence of some fractional derivative for functions in the algebra under study and in particular the assumption 1 thus the study of fractionally differentiable structures dove tails naturally with the theory of homotopy invariants in our study of quantum harmonic analysis we introduce spaces t of operator valued distributions these spaces are bounded linear operators between sobolev spaces the elements of the interpolation spaces the banach algebras j have derivativesdawhich belong to the spaces t for a certain range of and we extend the theory of the radon transform from products of regularized bounded operators to products of regularized operator valued distributions we sometimes wish to evaluate such an invariant at the endpoint of an interval such as 0 1 where zq a g becomes singular as 0 we discuss in brief a procedure to regularize the endpoint and a method to recover zq a g fully from certain partial information at the endpoint finally we generalize this approach to cover the case whenqcan be split into the sum of independent partsq1 q2 such that q1 q2 2 q1 2 q2 2 here we assume thatq1and q2 2are g invariant but not necessarilyq2 with further assumptions ona the most important being that q1 2 q2 2commutes witha we obtain a modified formula for an invariant namely z qj a g 1 e t2tr u g ae q21 q22 2 itd1a dt

formality and kontsevich duflo type theorems for lie pairs newcommand poly operatorname poly bullet newcommand td operatorname td l a nabla frac 1 2 newcommand cx 1 operatorname tot big gamma lambda bullet a vee otimes r mathcal 1 poly big newcommand cy 1 mathbb h bullet operatorname ce a mathcal 1 poly kontsevich s formality theorem states that there exists an l infty quasi isomorphism from the dgla t poly m of polyvector fields on a smooth manifold m to the dgla d poly m of polydifferential operators on m which extends the classical hochschild kostant rosenberg map in this paper we extend kontsevich s formality theorem to lie pairs a framework which includes a range of diverse geometric contexts such as complex manifolds foliations and mathfrak g manifolds the spaces cx t and cx d associated with a lie pair l a each carry an l infty algebra structure canonical up to l infty isomorphism these two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators respectively their corresponding cohomology groups cy t and cy d admit canonical gerstenhaber algebra structures we establish the following formality theorem for lie pairs there exists an l infty quasi isomorphism from cx t to cx d whose first taylor coefficient is equal to operatorname hkr circ td here td acts on cx t by contraction furthermore we prove a kontsevich duflo type theorem for lie pairs the hochschild kostant rosenberg map twisted by the square root of the todd class of the lie pair l a is an isomorphism of gerstenhaber algebras from cy t to cy d as applications we establish formality theorems and kontsevich duflo type theorems for complex manifolds foliations and mathfrak g manifolds in the case of complex manifolds we recover the kontsevich duflo theorem of complex geometry

poisson bivectors and poisson brackets on affine derived stacks abstract let a be a commutative dg algebra concentrated in degrees m and let spec a be the associated derived stack we give two proofs of the existence of a canonical map from the moduli space of shifted poisson structures in the sense of 16 on spec a to the moduli space of homotopy shifted poisson algebra structures on a the first makes use of a more general description of the poisson operad and of its cofibrant models while the second is more computational and involves an explicit resolution of the poisson operad

star products and branes in poisson sigma models we prove that non coisotropic branes in the poisson sigma model are allowed at the quantum level when the brane is defined by second class constraints the perturbative quantization of the model yields kontsevich s star product associated to the dirac bracket on the brane finally we present the quantization for a general brane

deformation quantization of surjective submersions and principal fibre bundles in this paper we establish a notion of deformation quantization of a surjective submersion which is specialized further to the case of a principal fibre bundle the functions on the total space are deformed into a right module for the star product algebra of the functions on the base manifold in case of a principal fibre bundle we require in addition invariance under the principal action we prove existence and uniqueness of such deformations the commutant within all differential operators on the total space is computed and gives a deformation of the algebra of vertical differential operators applications to noncommutative gauge field theories and phase space reduction of star products are discussed

formal series of generalized functions and their application to deformation quantization foundations of the formal series calculus in deformation quantization are discussed several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced the notion of non negativity in formal series calculus is proposed problems with defining quantum states over the set of formal series are analyzed foundations of the formal series calculus in deformation quantization are discussed several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced the notion of non negativity in formal series calculus is proposed problems with defining quantum states over the set of formal series are analyzed

brst cohomology and phase space reduction in deformation quantization in this article we consider quantum phase space reduction when zero is a regular value of the momentum map by analogy with the classical case we define the brst cohomology in the framework of deformation quantization we compute the quantum brst cohomology in terms of a quantum chevalley eilenberg cohomology of the lie algebra on the constraint surface to prove this result we construct an explicit chain homotopy both in the classical and quantum case which is constructed out of a prolongation of functions on the constraint surface we have observed the phenomenon that the quantum brst cohomology cannot always be used for quantum reduction because generally its zero part is no longer a deformation of the space of all smooth functions on the reduced phase space but in case the group action is sufficiently nice e g proper which is the case for all compact lie group actions it is shown for a strongly invariant star product that the brst procedure always induces a star product on the reduced phase space in a rather explicit and natural way simple examples and counterexamples are discussed

superlinear convergence of smoothing quasi newton methods for nonsmooth equations abstract we study local convergence of smoothing quasi newton methods for solving a system of nonsmooth nondifferentiable equations in r n the feature of smoothing quasi newton methods is to use a smooth function to approximate the nonsmooth mapping and update the quasi newton matrix at each step convergence results are given under directional derivative consistence property without differentiability we establish a dennis more type superlinear convergence theorem for smoothing quasi newton methods and we prove linear convergence of the smoothing broyden method furthermore we propose a superlinear convergent smoothing newton broyden method without using the generalized jacobian and the semismooth assumption we illustrate the smoothing approach on box constrained variational inequalities

moduli spaces of curves and representation theory we establish a canonical isomorphism between the second cohomology of the lie algebra of regular differential operators on x of degree 1 and the second singular cohomology of the moduli space hat f g 1 of quintuples c p z l wherec is a smooth genusg riemann surface p a point onc z a local parameter atp l a degreeg 1 line bundle onc and a class of local trivializations ofl atp which differ by a non zero factor the construction uses an interplay between various infinite dimensional manifolds based on the topological spaceh of germs of holomorphic functions in a neighborhood of 0 in x and related topological spaces the basic tool is a canonical map from hat f g 1 to the infinite dimensional grassmannian of subspaces ofh which is the orbit of the subspaceh of holomorphic functions on x vanishing at under the group auth as an application we give a lie algebraic proof of the mumford formula n 6n2 6n 1 1 where n is the determinant line bundle of the vector bundle on the moduli space of curves of genusg whose fiber overc is the space of differentials of degreen onc

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

homogeneous fedosov star products on cotangent bundles i weyl and standard ordering with differential operator representation in this paper we construct homogeneous star products of weyl type on every cotangent bundle t q by means of the fedosov procedure using a symplectic torsion free connection on t q homogeneous of degree zero with respect to the liouville vector field by a fibrewise equivalence transformation we construct a homogeneous fedosov star product of standard ordered type equivalent to the homogeneous fedosov star product of weyl type representations for both star product algebras by differential operators on functions on q are constructed leading in the case of the standard ordered product to the usual standard ordering prescription for smooth complex valued functions on t q polynomial in the momenta where an arbitrary fixed torsion free connection 0 on q is used motivated by the flat case t n another homogeneous star product of weyl type corresponding to the weyl ordering prescription is constructed the example of the cotangent bundle of an arbitrary lie group is explicitly computed and the star product given by gutt is rederived in our approach

observability and related structural results for linear hereditary systems new results on the theory of observability and deterministic observers for linear time invariant systems with delays are presented the results have been established only for the case of point delays which are commensurable however most of the results can be generalized to the situation with distributed delays and to the ease with non commensurate delays

convergence of the wick star product we construct a frechet space as a subspace of c omega mathbb c n where the wick star product converges and is continuous the resulting frechet algebra mathcal a is studied in detail including a representation of mathcal a in the bargmann fock space and a discussion of star exponentials and coherent states

formal symplectic groupoid of a deformation quantization we give a self contained algebraic description of a formal symplectic groupoid over a poisson manifold m to each natural star product on m we then associate a canonical formal symplectic groupoid over m finally we construct a unique formal symplectic groupoid with separation of variables over an arbitrary kahler poisson manifold

a formality theorem for hochschild chains abstract we prove tsygan s formality conjecture for hochschild chains of the algebra of functions on an arbitrary smooth manifold m using the fedosov resolutions proposed in math qa 0307212 and the formality quasi isomorphism for hochschild chains of r y 1 y d proposed in paper math qa 0010321 by shoikhet this result allows us to describe traces on the quantum algebra of functions on an arbitrary poisson manifold

deformation quantization and the baum connes conjecture alternative titles of this paper would have been index theory without index or the baum connes conjecture without baum

bfv complex and higher homotopy structures we present a connection between the bfv complex abbreviation for batalin fradkin vilkovisky complex and the strong homotopy lie algebroid associated to a coisotropic submanifold of a poisson manifold we prove that the latter structure can be derived from the bfv complex by means of homotopy transfer along contractions consequently the bfv complex and the strong homotopy lie algebroid structure are l quasi isomorphic and control the same formal deformation problem

relative formality theorem and quantisation of coisotropic submanifolds we prove a relative version of kontsevich s formality theorem this theorem involves a manifold m and a submanifold c and reduces to kontsevich s theorem if c m it states that the dgla of multivector fields on an infinitesimal neighbourhood of c is l quasiisomorphic to the dgla of multidifferential operators acting on sections of the exterior algebra of the conormal bundle applications to the deformation quantisation of coisotropic submanifolds are given the proof uses a duality transformation to reduce the theorem to a version of kontsevich s theorem for supermanifolds which we also discuss in physical language the result states that there is a duality between the poisson sigma model on a manifold with a d brane and the poisson sigma model on a supermanifold without branes or more properly with a brane which extends over the whole supermanifold

hom associative algebras up to homotopy abstract a hom associative algebra is an algebra whose associativity is twisted by an algebra homomorphism in this paper we introduce a strongly homotopy version of hom associative algebras h a algebras in short on a graded vector space we describe 2 term h a algebras in detail in particular we study skeletal and strict 2 term h a algebras we also introduce hom associative 2 algebras as categorification of hom associative algebras the category of 2 term h a algebras and the category of hom associative 2 algebras are shown to be equivalent next we define a suitable hochschild cohomology theory for h a algebras which control the deformation of the structures finally we visit h l algebras introduced by sheng and chen and show that an appropriate skew symmetrization of h a algebras give rise to h l algebras

covariant and equivariant formality theorems abstract we give a proof of kontsevich s formality theorem for a general manifold using fedosov resolutions of algebras of polydifferential operators and polyvector fields the main advantage of our construction of the formality quasi isomorphism is that it is based on the use of covariant tensors unlike kontsevich s original proof which is based on jets of polydifferential operators and polyvector fields using our construction we prove that if a group g acts smoothly on a manifold m and m admits a g invariant affine connection then there exists a g equivariant quasi isomorphism of formality this result implies that if a manifold m is equipped with a smooth action of a finite or compact group g or equipped with a free action of a lie group g then m admits a g equivariant formality quasi isomorphism in particular this gives a solution of the deformation quantization problem for an arbitrary poisson orbifold

smooth approximations to nonlinear complementarity problems it is well known that a nonlinear complementarity problem ncp can be formulated as a system of nonsmooth equations chen and mangasarian comput optim appl 5 1996 pp 97 138 proposed a class of parametric smooth functions by twice integrating a probability density function as a result the nonsmooth equations can be approximated by smooth equations this paper refines the smooth functions proposed by chen and mangasarian and investigates their structural properties the refinement allows us to establish the existence uniqueness and limiting properties of the trajectory defined by the solutions of these smooth equation approximations in addition global error bounds for the ncp with a uniform p function are obtained

poisson sigma model with branes and hyperelliptic riemann surfaces we derive the explicit form of the superpropagators in the presence of general boundary conditions coisotropic branes for the poisson sigma model this generalizes the results presented by cattaneo and felder a path integral approach to the kontsevich quantization formula commun math phys 212 591 2000 and cattaneo and felder coisotropic submanifolds in poisson geometry and branes in the poisson sigma model lett math phys 69 157 2004 for kontsevich s angle function kontsevich m deformation quantization of poisson manifolds i e print arxiv hep th 0101170 used in the deformation quantization program of poisson manifolds the relevant superpropagators for n branes are defined as gauge fixed homotopy operators of a complex of differential forms on n sided polygons pn with particular alternating boundary conditions in the presence of more than three branes we use first order riemann theta functions with odd singular characteristics on the jacobian variety of a hyperelliptic ri

coisotropic embeddings in poisson manifolds we consider existence and uniqueness of two kinds of coisotropic embeddings and deduce the existence of deformation quantizations of certain poisson algebras of basic functions first we show that any submanifold of a poisson manifold satisfying a certain constant rank condition already considered by calvo and falceto 2004 sits coisotropically inside some larger cosymplectic submanifold which is naturally endowed with a poisson structure then we give conditions under which a dirac manifold can be embedded coisotropically in a poisson manifold extending a classical theorem of gotay

cyclic cocycles on deformation quantizations and higher index theorems abstract we construct a nontrivial cyclic cocycle on the weyl algebra of a symplectic vector space using this cyclic cocycle we construct an explicit local quasi isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof we give a new proof of nest tsygan s algebraic higher index theorem by computing the pairing between such cyclic cocycles and the k theory of the formal deformation quantization furthermore we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold as an application we obtain the analytic higher index theorem of connes moscovici and its extension to orbifolds

the characteristic classes of morita equivalent star products on symplectic manifolds in this paper we give a complete characterization of morita equivalent star products on symplectic manifolds in terms of their characteristic classes two star products and on m are morita equivalent if and only if there exists a symplectomorphism colon m m such that the relative class t is 2 i integral for star products on cotangent bundles we show that this integrality condition is related to dirac s quantization condition for magnetic charges

cyclic cohomology of the weyl algebra abstract we give an explicit formula for sp 2 n basic representatives of the cyclic cohomology of the weyl algebra h c a 2 n as an application we prove a generalization of a theorem of nest and tsygan concerning the relation of the todd class and the cyclic cohomology of the differential operators on a complex manifold

algebraic index theorem we prove the atiyah singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold

brst reduction of quantum algebras with involutions in this paper we investigate the compatibility of the brst reduction procedure with the hermiticity of star products first we introduce the generalized notion of abstract brst algebras with corresponding involutions in this setting we define adjoint brst differentials and as a consequence one gets new brst quotients passing to the quantum brst setting we show that for compact lie groups the new quantum brst quotient and the quantum brst cohomology are isomorphic in zero degree implying that reduction is compatible with hermiticity

deformation quantization in algebraic geometry abstract we study deformation quantizations of the structure sheaf o x of a smooth algebraic variety x in characteristic 0 our main result is that when x is d affine any formal poisson structure on x determines a deformation quantization of o x canonically up to gauge equivalence this is an algebro geometric analogue of kontsevich s celebrated result

deformation quantization with generators and relations abstract in this paper we prove a conjecture of b shoikhet which claims that two quantization procedures arising from fourier dual constructions actually coincide

free products cyclic homology and the gauss manin connection abstract we use the techniques of cuntz and quillen to present a new approach to periodic cyclic homology our construction is based on a t d t i a noncommutative equivariant de rham complex of an associative algebra a here d is the karoubi de rham differential and i is an operation analogous to contraction with a vector field as a byproduct we give a simple explicit construction of the gauss manin connection introduced earlier by e getzler on the relative periodic cyclic homology of a flat family of associative algebras over a central base ring we introduce and study free product deformations of an associative algebra a new type of deformation over a not necessarily commutative base ring natural examples of free product deformations arise from preprojective algebras and group algebras for compact surface groups

discretization and fir filtering of continuous linear systems with internal and external point delays a discrete quasi autoregressive moving average qarma model and an equivalent impulse response ir description are established for linear continuous systems with internal and external point delays which are discretized through a sampler and zero order hold device the qarma model is proved to differ from the standard autoregressive moving average arma models because of its infinite input memory caused by the delays based on the ir model a finite impulse response fir filter is proposed to approximate the input output description under exponential stability of the unforced system its implementation is discussed by using fast filtering design tools

involutions and representations for reduced quantum algebras abstract in the context of deformation quantization there exist various procedures to deal with the quantization of a reduced space m red we shall be concerned here mainly with the classical marsden weinstein reduction assuming that we have a proper action of a lie group g on a poisson manifold m with a moment map j for which zero is a regular value for the quantization we follow bordemann et al 2000 6 with a simplified approach and build a star product red on m red from a strongly invariant star product on m the new questions which are addressed in this paper concern the existence of natural involutions on the reduced quantum algebra and the representation theory for such a reduced algebra we assume that is hermitian and we show that the choice of a formal series of smooth densities on the embedded coisotropic submanifold c j 1 0 with some equivariance property defines a involution for red on the reduced space looking into the question whether the corresponding involution is the complex conjugation which is a involution in the marsden weinstein context yields a new notion of quantized modular class we introduce a left c m submodule and a right c m red red submodule c cf c of c c we define on it a c m red valued inner product and we establish that this gives a strong morita equivalence bimodule between c m red and the finite rank operators on c cf c the crucial point is here to show the complete positivity of the inner product we obtain a rieffel induction functor from the strongly non degenerate representations of c m red red on pre hilbert right d modules to those of c m for any auxiliary coefficient algebra d over c

on the chern character of a theta summable fredholm module on montre qu il est possible de simplifier la theorie de connes du caractere de chern a partir d une k homologie d une algebre de banach vers une cohomologie cyclique entiere