vanishing of vacuum states and blow up phenomena of the compressible navier stokes equations the navier stokes systems for compressible fluids with density dependent viscosities are considered in the present paper these equations in particular include the ones which are rigorously derived recently as the saint venant system for the motion of shallow water from the navier stokes system for incompressible flows with a moving free surface 14 these compressible systems are degenerate when vacuum state appears we study initial boundary value problems for such systems for both bounded spatial domains or periodic domains the dynamics of weak solutions and vacuum states are investigated rigorously

embedding c algebra extensions into af algebras abstract sufficient conditions are given for a c algebra extension to be embeddable into an af algebra when it is known that the ideal and quotient algebras can be so embedded an appliction is made to a class of type i c algebras

homotopy of unitaries in simple c algebras with tracial rank one let 0 be a positive number is there a number 0 satisfying the following given any pair of unitaries u and v in a unital simple c algebra a with v 0 in k1 a for which uv vu there is a continuous path of unitaries v t t 0 1 a such that v 0 v v 1 1and uv t v t u for allt 0 1 an answer is given to this question when a is assumed to be a unital simple c algebra with tracial rank no more than one let c be a unital separable amenable simple c algebra with tracial rank no more than one which also satisfies the uct suppose that c a is a unital monomorphism and suppose that v a is a unitary with v 0 in k1 a such that v almost commutes with it is shown that there is a continuous path of unitaries v t t 0 1 in a with v 0 v and v 1 1 such that the entire path v t almost commutes with provided that an induced bott map vanishes other versions of the so called basic homotopy lemma are also presented

asymptotic stability of the rarefaction wave for the compressible quantum navier stokes poisson equations abstract in this study we consider the large time behavior of the solution to the one dimensional isentropic compressible quantum navier stokes poisson equations the system describes a compressible particle fluid under quantum effects with the potential function of the self consistent electric field we show that if the initial data are close to a constant state with asymptotic values at far fields selected such that the riemann problem on the corresponding euler system admits a rarefaction wave with a strength that is not necessarily small then the solution exists for all time and it tends to the rarefaction wave as t the proof is based on the energy method by considering the effect of the self consistent electric field and quantum potential in the viscous compressible fluid in addition we compare the quantum compressible navier stokes poisson equations and the corresponding compressible navier stokes poisson equations based on the large time behavior of these two classes of models

k homology asymptotic representations and unsuspended e theory connes and higson defined a bivariant homology theory e a b for separable c algebras the elements of e a b are taken to be homotopy classes of asymptotic momorphisms from sa k to sb k in symbols e a b sa sb k if a is k nuclear then e theory agrees with kasparov s bivariant k theory we show that in many cases one need not take suspensions to calculate the e theory group e a b for many a we show e a b a b k for all b among the a for which this is true are c0 x pt for x with the homotopy type of a finite connected cw complex this gives a concrete realization of k homology related to the brown douglas fillmore description for example k0 x arises as asymptotic representations k0 x c0 x k other a for which our isomorphic holds include the nonunital dimension drop intervals in this case there is no distinction between homomorphisms and asymptotic morphisms so we have succeeded in classifying all homomorphisms from a dimension drop interval to a stable c algebra this was subsequently used by elliott j reine angew math 443 1993 179 219 in the classification of certain c algebras the dimension drop interval may also be used to describe k x z n in terms of paths of asymptotic representations of c0 x

random matrix ensembles associated to compact symmetric spaces we introduce random matrix ensembles that correspond to the infinite families of irreducible riemannian symmetric spaces of type i in particular we recover the circular orthogonal and symplectic ensembles of dyson and find other families of unitary orthogonal and symplectic ensembles of jacobi type we discuss the universal and weakly universal features of the global and local correlations of the levels in the bulk and at the hard edge of the spectrum i e at the central points 1 on the unit circle previously known results are extended and we find new simple formulas for the bessel kernels that describe the local correlations at a hard edge

on navier stokes korteweg and euler korteweg systems application to quantum fluids models in this paper the main objective is to generalize to the navier stokes korteweg with density dependent viscosities satisfying the bd relation and euler korteweg systems a recent relative entropy proposed by d bresch p noble and j p vila 2016 introduced for the compressible navier stokes equations with a linear density dependent shear viscosity and a zero bulk viscosity as a concrete application this helps to justify mathematically the convergence between global weak solutions of the quantum navier stokes system recently obtained by i lacroix violet and a vasseur 2016 and dissipative solutions of the quantum euler system when the viscosity coefficient tends to zero this selects a dissipative solution as the limit of a viscous system our results are based on the fact that euler korteweg systems and corresponding navier stokes korteweg systems can be reformu lated through an augmented system as the compressible navier stokes system with density dependent viscosities satisfying the bd algebraic relation this was also observed recently by d bresch f couderc p noble and j p vila 2016 for the euler korteweg system as a by product of our analysis we show that this augmented formulation helps to define relative entropy estimates for the euler korteweg systems in a simplest way compared to recent works see d donatelli e feireisl p marcati 2015 and j giesselmann c lattanzio a e tzavaras 2017 and with less hypothesis required on the capillary coefficient no concavity assumption needed in our result

a note on barotropic compressible quantum navier stokes equations abstract the global in time existence of weak solutions to the barotropic compressible quantum navier stokes equations has been proved very recently by jungel 2009 1 if the viscosity constant is smaller than the scaled plank constant this paper extends the results to the case that the viscosity constant equals the scaled plank constant by using a new estimate on the square root of the solution apparently not available in 1 the semiclassical limit for the viscous quantum euler equations which are equivalent to the barotropic compressible quantum navier stokes equations can be performed then the results of this paper are obtained easily

nonnuclear subalgebras of af algebras we show that any non type i separable unital af algebra b can be modeled from inside by a nonnuclear c algebra and from outside by a nonexact c algebra more precisely there exist unital separable quasidiagonal c algebras a b c of real rank zero stable rank one such that a is nonnuclear c is nonexact and both a and c are asymptotically homotopy equivalent to b in particular a b and c have the same ordered k theory groups hence isomorphic ideal lattices and a and b have affinely homeomorphic trace spaces

approximation by unitaries with finite spectrum in purely infinite c algebras abstract we prove that a unital purely infinite simple c algebra has c exponential rank at most 1 and therefore has the property weak fu every element of the identity component of the unitary group is norm limit of unitaries with finite spectrum we also prove that the appropriate modification of this statement for nonunital algebras

theory and multipliers of stable algebras the main theorem is that if a is a c algebra with a countable approximate identity consisting of projections then the unitary group of m a k is contractible this gives a realization via the index map of ko a as components in the set of fredholm operators on ha for a c algebra a let ha ai i ai e a and eai ai converges in norm then ha becomes a right a module under the action ai b aib with the a valued inner product ai b1 ea bi ha is a hilbert a module y ha is the c algebra of operators on ha that have an adjoint k ha is an ideal of y ha called the compact operators on ha set 11 2 definition 4 operators in ha invertible modulo 3f ha are called fredholm and have an index that takes its values in ko a the paper is divided into two parts in part 1 we show that index a ko a is an isomorphism where t denotes the set of path components of the set of fredholm operators in part 2 we show that the unitary group of m a k is contractible when a has a countable approximate identity consisting of projections as 2 ha m a k this in particular implies that the unitary group of s ha is connected when a is unital this latter result is used to prove the isomorphism mentioned above the results of this paper were announced in 14 see remark 2 5 2 this research is based on the author s doctoral work done at dalhousie university under the supervision of professor p a fillmore the author also wishes to thank w j phillips for many useful suggestions part 1 the index of a fredholm operator 1 1 we describe the index map from fredholm operators on hilbert c modules to the corresponding k groups the principal ingredients of this construction have been discussed by several authors kasparov 12 miscenko and fomenko 15 and pimsner popa and voiculescu 17 however as we shall need refinements and variations of these constructions we will give a detailed account of the index map throughout this paper k or y h will denote the c algebra of compact operators on a separable infinite dimensional hilbert space h for any two c algebras a and b a b will denote the completion of the algebraic tensor product aob in the spatial or minimal c norm see e g effros and lance 8 2 received by the editors july 28 1983 and in revised form december 27 1985 1980 mathematics subject classification 1985 revision primary 46l05 46m20 01987 american mathematical society 0002 9947 87 1 00 25 per page

c algebras generated by stable relations abstract presentations of certain c algebras such as c0 0 1 mn by generators and relations are proven to be stable by the weakest definition of stability a set p1 pn of star polynomials is stable if operators a1 an which almost satisfy these relations meaning pi a1 an can be perturbed within c a1 an to satisfy these relations exactly homotopy and perturbation results for these c algebras are corollaries in particular we prove some new results about the dimension drop c algebras a n c 0 1 m n 0 1 c i we show that an is semiprojective answering a question of effros and kaminker also we show that whenever an b is a homomorphism approximately contained in a sub c algebra b0 b there is a nearby homomorphism of an into b0 this answers a question raised by elliott we prove technical results regarding mn c0 0 1 which should be useful elsewhere first we show this is projective meaning a star homomorphism mn c0 0 1 b j always lifts to a star homomorphism mn c0 0 1 b secondly we show that for a finitely generated m n a m n c 0 0 1 c 0 0 1 a if the amalgamation is done properly

random matrix theory and l functions at s 1 2 recent results of katz and sarnak 8 9 suggest that the low lying zeros of families of l functions display the statistics of the eigenvalues of one of the compact groups of matrices u n o n or usp 2n we here explore the link between the value distributions of the l functions within these families at the central point s 1 2 and those of the characteristic polynomials z u of matrices u with respect to averages over so 2n and usp 2n at the corresponding point 0 using techniques previously developed for u n in 10 for any matrix size n we find exact expressions for the moments of z u 0 for each ensemble and hence calculate the asymptotic large n value distributions for z u 0 and log z u 0 the asymptotic results for the integer moments agree precisely with the few corresponding values known for l functions the value distributions suggest consequences for the non vanishing of l functions at the central point

effective velocity in compressible navier stokes equations with third order derivatives abstract a formulation of certain barotropic compressible navier stokes equations with third order derivatives as a viscous euler system is proposed by using an effective velocity variable the equations model for instance viscous korteweg or quantum navier stokes flows the formulation in the new variable allows for the derivation of an entropy identity which is known as the bd bresch desjardins entropy equation as a consequence of this estimate a new global in time existence result for the one dimensional quantum navier stokes equations with strictly positive particle densities is proved

about the barotropic compressible quantum navier stokes equations abstract in this paper we consider the barotropic compressible quantum navier stokes equations with a linear density dependent viscosity and its limit when the scaled planck constant vanishes following recent works on degenerate compressible navier stokes equations we prove the global existence of weak solutions by the use of a singular pressure close to vacuum with such singular pressure we can use the standard definition of global weak solutions which also allows to justify the limit when the scaled planck constant denoted by e tends to 0

af embeddability of crossed products of af algebras by the integers ifais an af algebra and aut a it is shown that af embeddability of the crossed product a z is equivalent toa z being stably finite this equivalence follows from a simple k theoretic characterization of af embeddability it is then shown that ifa z is af embeddable then the af embedding can be chosen in such a way as to induce a rationally injective map onk0 a z

on the averages of characteristic polynomials from classical groups we provide an elementary and self contained derivation of formulae for averages of products and ratios of characteristic polynomials of random matrices from classical groups using classical results due to weyl and littlewood

equivariant kirchberg phillips type absorption for amenable group actions we show an equivariant kirchberg phillips type absorption theorem for pointwise outer actions of discrete amenable groups on kirchberg algebras with respect to natural model actions on the cuntz algebras mathcal o infty and mathcal o 2 this generalizes results known for finite groups and poly mathbb z groups the model actions are shown to be determined up to strong cocycle conjugacy by natural abstract properties which are verified for some examples of actions arising from tensorial shifts we also show the following homotopy rigidity result which may be understood as a precursor to a general kirchberg phillips type classification theory if two outer actions of an amenable group on a unital kirchberg algebra are equivariantly homotopy equivalent then they are conjugate this marks the first c dynamical classification result up to cocycle conjugacy that is applicable to actions of all amenable groups

the fine structure of the kasparov groups i continuity of the kk pairing in this paper it is demonstrated that the kasparov pairing is continuous with respect to the natural topology on the kasparov groups so that a kk equivalence is an isomorphism of topological groups in addition we demonstrate that the groups have a natural pseudopolonais structure and we prove that various kk structural maps are continuous

classification of certain infinite simple c algebras the class of c algebras that arise as the crossed product of a stable simple af algebra with a z action determined by an automorphism which maps a projection in the algebra onto a proper subprojection is proved to consist of simple purely infinite c algebras and a specific subclass of it is proved to be classified by k theory this subclass is large enough to exhaust all possible k groups if g0 and g1 are countable abelian groups with g1 torsion free as it must be then there is a c algebras a in the classified subclass with k0 a g0 and k 1 a g1 the subclass contains the cuntz algebras o with n even and the cuntz krieger algebras o a with k0 o a of finite old order and it is closed under forming inductive limits the c algebras in the classified subclass can be viewed as classifiable models in a strong sense of general simple purely infinite c algebras with the same k theory

on the topology of the kasparov groups and its applications abstract in this paper we establish a direct connection between stable approximate unitary equivalence for homomorphisms and the topology of the kk groups which avoids entirely c algebra extension theory and does not require nuclearity assumptions to this purpose we show that a topology on the kasparov groups can be defined in terms of approximate unitary equivalence for cuntz pairs and that this topology coincides with both pimsner s topology and the brown salinas topology we study the generalized rordam group kl a b kk a b 0 and prove that if a separable exact residually finite dimensional c algebra satisfies the universal coefficient theorem in kk theory then it embeds in the uhf algebra of type 2 in particular such an embedding exists for the c algebra of a second countable amenable locally compact maximally almost periodic group

classification of simple c algebras with unique traces we classify certain almost multiplicative morphisms up to approximate unitary equivalence and use this result to prove the following let a and b be two unital separable simple c algebras of real rank zero stable rank one with weakly unperforated k 0 groups and with unique normalized quasi traces suppose that both a and b are locally ah and k a k a 1 a k b k b 1 b then a is isomorphic to b

universal coefficient theorems for the stable ext groups let a be a unital separable nuclear c algebra and let b be a stable c algebra using k theory and kk theory we establish universal coefficient theorems for the stable ext groups of unital extensions of a by b when a and b have certain properties which generalize a result of l brown and m dadarlat for the strong ext groups the class of extensions being studied are also enlarged

a universal multicoefficient theorem for the kasparov groups that holds in the same generality as the universal coefficient theorem of rosenberg and schochet there are advantages in some circumstances to using hom k a k b in place of kk a b these advantages derive from the fact that k a can be equipped with order and scale structures similar to those on k0 a with this additional structure the module k a becomes a powerful invariant of c algebras we show that it is a complete invariant for the class of real rank zero ad algebras the ad algebras are a certain kind of approximately subhomogeneous c algebras which may have torsion in k1 ell in addition to classifying these algebras we calculate their automorphism groups up to approximately innerautomorphisms

simple nuclear c algebras of tracial topological rank one we give a classification theorem for unital separable nuclear c algebras with tracial rank no more than one let a and b be two unital separable simple nuclear c algebras with tr a tr b 1 which satisfy the universal coefficient theorem we show that a b if and only if there is an order and unit preserving isomorphism 0 1 2 k0 a k0 a 1a k1 a t a k0 b k0 b 1b k1 b t b where 2 1 x 0 x for each x k0 a and t b

approximate homogeneity is not a local property it is shown that the ah algebras satisfy a certain splitting property at the level of k theory with torsion coefficients the splitting property is used to prove the following i there are locally homogeneous c algebras which are not ah algebras ii the class of ah algebras is not closed under countable inductive limits iii there are real rank zero split quasidiagonal extensions of ah algebras which are not ah algebras

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

singularity dominated strong fluctuations for some random matrix averages the circular and jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval 0 1 of the real line respectively the averaged value of the modulus of the corresponding characteristic polynomial raised to the power 2 diverges for 2 1 at points approaching the eigenvalue support using the theory of generalized hypergeometric functions based on jack polynomials the functional form of the leading asymptotic behaviour is established rigorously in the circular ensemble case this confirms a conjecture of berry and keating

quasidiagonality of nuclear c algebras we prove that faithful traces on separable and nuclear c algebras in the uct class are quasidiagonal this has a number of consequences firstly by results of many hands the classification of unital separable simple and nuclear c algebras of finite nuclear dimension which satisfy the uct is now complete secondly our result links the finite to the general version of the toms winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension finally we confirm the rosenberg conjecture discrete amenable groups have quasidiagonal c algebras

the fine structure of the kasparov groups ii topologizing the uct the kasparov groups kk a b have a natural structure as pseudopolonais groups in this paper we analyze how this topology interacts with the terms of the universal coefficient theorem uct and the splittings of the uct constructed by rosenberg and the author as well as its canonical three term decomposition which exists under bootstrap hypotheses we show that the various topologies on extz1 k a k b and other related groups mostly coincide then we focus attention on the milnor sequence and the fine structure subgroup of kk a b an important consequence of our work is that under bootstrap hypotheses the closure of zero of kk a b is isomorphic to the group pextz1 k a k b finally we introduce new splitting obstructions for the milnor and jensen sequences and prove that these sequences split if k a or k b is torsion free

inverse eigenvalue problem of jacobi matrix with mixed data in this paper the inverse eigenvalue problem of reconstructing a jacobi matrix from its eigenvalues its leading principal submatrix and part of the eigenvalues of its submatrix is considered the necessary and sufficient conditions for the existence and uniqueness of the solution are derived furthermore a numerical algorithm and some numerical examples are given

an approximate universal coefficient theorem an approximate universal coefficient theorem auct for certain c algebras is established we present a proof that kirchberg phillips s classification theorem for separable nuclear purely infinite simple c algebras is valid for c algebras satisfying the auct instead of the uct it is proved that two versions of auct are in fact the same we also show that c algebras that are locally approximated by c algebras satisfying the auct satisfy the auct as an application we prove that certain simple c algebras which are locally type i are in fact isomorphic to simple ah algebras as another application we show that a sequence of residually finite dimensional c algebras which are asymptotically nuclear and which asymptotically satisfies the auct can be embedded into the same simple af algebra

on the classification of nuclear c algebras two of the most influential works on c algebras from the mid seventies brown douglas and fillmore s 6 and elliott s 21 both contain uniqueness and existence results in the now standard sense which we shall outline below these papers served as keystones for two separate theories kk theory and the classification program which for many years parted ways with only moderate interaction but with this common origin in mind it is not surprising that recent years have seen a fruitful interaction which has been one of the main engines behind rapid progress in the classification program in the present paper we take this interaction even further combining a concept of quasidiagonality for representations and a new characterization of equivalence in the cuntz picture of kk theory we achieve a general uniqueness result and from the quasidiagonality notion we derive a corresponding general existence result by comparing well known realizations of the kk groups these results are then employed to obtain new classification results for certain classes of quasidiagonal c algebras introduced by h lin an important novel feature of these classes is that they are defined by a certain local approximation property rather than by an inductive limit construction we emphasize that our fundamental uniqueness result does not depend on the universal coefficient theorem uct nor on nuclearity of the c algebras involved on the other hand when we refine the uniqueness result for use in classification we need to gradually add such conditions on the c algebras it is to be expected that requirements of this type are necessary but we have found that holding them back as long as possible leads to more conceptual proofs while clarifying the role of the uct and nuclearity in classification further since uniqueness results answer fundamental questions about kk and k theory they are of interest in themselves we have pursued this theme in 14 and defer to this paper the proof of the new characterization of kk theory which lies behind our uniqueness results our existence and uniqueness results are in the spirit of the classical ext theory from 6 the main complication overcome in the paper is to control the stabilization which is necessary when one works with finite c algebras in the infinite case where programs of this type have already been successfully carried out stabilization is unnecessary yet our methods are sufficiently versatile to allow us to reprove from a handful of basic results the classification of purely infinite nuclear c algebras of kirchberg and phillips

global existence of weak solution for quantum navier stokes poisson equations in this paper we consider the compressible quantum navier stokes poisson equations with a linear density dependent viscosity by the use of a singular pressure close to vacuum we prove the global in time existence of weak solutions in a three dimensional torus for large data in the sense of distribution

moments of characteristic polynomials for compact symmetric spaces and jack polynomials we express the averages of products of characteristic polynomials for random matrix ensembles associated with compact symmetric spaces in terms of jack polynomials or heckman and opdam s jacobi polynomials depending on the root system of the space we also give explicit expressions for the asymptotic behavior of these averages in the limit as the matrix size goes to infinity

a separable brown douglas fillmore theorem and weak stability we give a separable brown douglas fillmore theorem let a be a separable amenable c algebra which satisfies the approximate uct b be a unital separable amenable purely infinite simple c algebra and h 1 h 2 a b be two monomorphisms we show that h 1 and h 2 are approximately unitarily equivalent if and only if h 1 h 2 in kl a b we prove that for any e 0 and any finite subset f a there exist 0 and a finite subset g a satisfying the following for any amenable purely infinite simple c algebra b and for any contractive positive linear map l a b such that l ab l a l b 0 there is 0 such that in any unital amenable purely infinite simple c algebra b if uv e i vu for a pair of unitaries then there exists a pair of unitaries u 1 and v 1 in b such that u 1 v 1 e i v 1 u 1 u 1 u e and v 1 v e