on the kk theory and the e theory of amalgamated free products of c algebras abstract we establish six terms exact sequences relating the kk theory groups and the e theory groups of an amalgamated free product c algebra a 1 b a 2 to the respective groups of the three constituents a1 a2 and b in the kk theory case we assume that b is finite dimensional but in the e theory case only that b is nuclear or is the range of a conditional expectation in both a1 and a2 in particular we obtain the conjectured six terms exact sequence for the k theory of an amalgamated free product in this generality

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

symmetric vortices for two component ginzburg landau systems abstract we study ginzburg landau equations for a complex vector order parameter c 2 we consider symmetric vortex solutions in the plane r 2 x f r e i n with given degrees n z and prove the existence uniqueness and asymptotic behavior of solutions as r we also consider the monotonicity properties of solutions and exhibit parameter ranges in which both vortex profiles f f are monotone as well as parameter regimes where one component is non monotone the qualitative results are obtained by means of a sub and super solution construction and a comparison theorem for elliptic systems

symmetric vortices for the ginzberg landau equations of superconductivity and the nonlinear desingularization phenomenon abstract the existence of countably many distinct symmetric vortices for the nonlinear ginzberg landau equations of superconductivity with an arbitrary positive parameter is proven we then prove the existence of a linearization phenomenon as the parameter in this case the linearized equations contain an appropriate dirac delta function these equations are known as the london equations

radial solutions for the brezis nirenberg problem involving large nonlinearities abstract let us consider the problem 0 1 u a x u u p in b 1 u 0 in b 1 u 0 on b 1 where b 1 is the unit ball in r n n 3 and a x 0 is a smooth radial function under some suitable assumptions on the regular part of the green function of the operator u n 1 r u a r u we prove the existence of a radial solution to 0 1 for p large enough

towards a theory of classification abstract the well known difficulties arising in a classification which is not set theoretically trivial involving what is sometimes called a non smooth quotient have been overcome in a striking way in the theory of operator algebras by the use of what might be called a classification functor the very existence of which is already a surprise here the notion of such a functor is developed abstractly and a number of examples are considered including those which have arisen for various classes of operator algebras

bubble tower radial solutions in the slightly supercritical brezis nirenberg problem abstract in this paper we consider the brezis nirenberg problem in dimension n 4 in the supercritical case we prove that if the exponent gets close to n 2 n 2 and if simultaneously the bifurcation parameter tends to zero at the appropriate rate then there are radial solutions which behave like a superposition of bubbles namely solutions of the form j 1 k 1 1 m j 4 n 2 y 2 n 2 2 m j 1 o 1 n n 2 n 2 4 where mj and mj o mj 1 for all j these solutions lie close to turning points to the right of the associated bifurcation diagram

extensions of c x by simple c algebras of real rank zero let ext c x a be the set of unitarily equivalence classes of essential c algebra exten sions of the following form 0 a e c x 0 where a is a nonunital separable simple c algebra of real rank zero stable rank one with unique normalized trace and x is a finite cw complex we show that there is a bijection ext c x a kk c x m a a where m a is the multiplier algebra of a in particular we determine when an extension is actually splitting we also in a more general setting give a condition when an essential extension is quasidiagonal 0 introduction we study essential extensions of the following form 0 a f c x 0 where a is a j unital nonunital simple c algebra of real rank zero and stable rank one and x is finite cw complex the brown douglas fillmore bdf theory bdf1 and bdf2 gives a classification of these extensions up to unitary equivalence in the case that a k the c algebra of compact operators on an infinite dimensional separable hubert space since then the theory of c algebra extensions developed rapidly it is very difficult to give a complete list we refer the reader to bl for references while using of theory ext c x a have been computed for the most interesting cases however ext c x a does not provide enough information in general to classify extensions up to unitary equivalence in fact as shown in ln6 and lnl2 there are usually many different unitary equivalence classes of trivial extensions and an extension r with r 0 in ext c x a may not be trivial we will only consider simple c algebras that share some properties with c we consider only those with real rank zero stable rank one and unique quasi trace up to scalar multiples there is an additional property that c has namely its corona algebra m k k the calkin algebra is simple we also as sume that m a a is simple it has been studied when a nonunital but cr unital simple c algebra a has the property that m a a is simple see lnl rrl and zh2 etc it is shown in lnl that a simple c algebra a has the property that m a a is simple if and only if a has continuous scale

classifying c algebras via ordered mod p k theory we introduce an order structure on k 0 9 k0 z p this group may also be thought of as ko 7z z p we exhibit new examples of real rank zero c algebras that are inductive limits of finite dimensional and dimension drop algebras have the same ordered graded k theory with order unit and yet are not isomorphic in fact they are not even stably shape equivalent the order structure on k0 z 3 z p naturally distinguishes these algebras the same invariant is used to give an isomorphism theorem for such realrank zero inductive limits as a corollary we obtain an isomorphism theorem for all real rank zero approximately homogeneous c algebras that arise from systems of bounded dimension growth and torsion free k0 group at the 1980 kingston conference effros posed the problem of finding suitable invariants for use in studying c algebras that are limits of sequences of homogeneous c algebras these are now called almost homogeneous ah c algebras the classification of ah algebras is a rapidly developing field and we will not attempt to summarize all this activity instead we will focus on the growth of the invariants used specifically we consider an ah algebra a that is the direct limit o f a system of the form

morphisms of extensions of c algebras pushing forward the busby invariant abstract we study completions of diagrams of extensions of c algebras in which all three c algebras in one of the rows and either the ideal or the quotient in the other are given along with the three morphisms between them we find universal solutions to all four of these problems under restrictions of varying severity on the given vertical maps and describe the solutions in terms of push outs and pull backs of certain diagrams our characterization of the universal solution to one of the diagrams yields a concrete description of various amalgamated free products this leads to new results about the k theory of amalgamated free products verifying the cuntz conjecture in certain cases we also obtain new results about extensions of matricial field c algebras verifying partially a conjecture of blackadar and kirchberg finally we show that almost commuting unitary matrices can be uniformly approximated by commuting unitaries when an index obstruction vanishes

the connes higson construction is an isomorphism let a be a separable c algebra and b a stable c algebra containing a strictly positive element we show that the group ext 1 2 sa b of unitary equivalence classes of extensions of sa by b modulo the extensions which are asymptotically split coincides with the group of homotopy classes of such extensions this is done by proving that the connes higson construction gives rise to an isomorphism between ext 1 2 sa b and the e theory group e a b of homotopy classes of asymptotic homomorphisms from s2a to b

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

the range of approximate unitary equivalence classes of homomorphisms from ah algebras let c be a unital ah algebra and a be a unital simple c algebras with tracial rank zero it has been shown that two unital monomorphisms phi psi c to a are approximately unitarily equivalent if and only if phi psi quad rm in quad kl c a quad rm and quad tau circ phi tau circ psi quad rm for all tau in t a where t a is the tracial state space of a in this paper we prove the following given kappa in kl c a with kappa k 0 c setminus 0 subset k 0 a setminus 0 and with 1c 1a and a continuous affine map lambda t a to t mathfrak f c which is compatible with where t mathfrak f c is the convex set of all faithful tracial states there exists a unital monomorphism phi c to a such that phi kappa quad rm and quad tau circ phi c lambda tau c for all c in c s a and tau in t a denote by rm mon au e c a the set of approximate unitary equivalence classes of unital monomorphisms we provide a bijective map lambda rm mon au e c a to klt c a where klt c a is the set of compatible pairs of elements in kl c a and continuous affine maps from t a to t mathfrak f c moreover we found that there are compact metric spaces x unital simple af algebras a and kappa in kl c x a with kappa k 0 c x setminus 0 subset k 0 a setminus 0 for which there is no homomorphism h c x a so that h

classification of simple c algebras and higher dimensional noncommutative tori we show that unital simple c algebras with tracial topological rank zero which are locally approximated by subhomogeneous c algebras can be classified by their ordered k theory we apply this classification result to show that certain simple crossed products are isomorphic if they have the same ordered k theory in particular irrational higher dimensional noncommutative tori of the form c k are in fact inductive limits of circle algebras

type ii ancient compact solutions to the yamabe flow nsf 0604657 1266172 0905749 1056387 fondo basal cmm millenium nucleus capde nc130017 fondecyt 1150066

positive solutions of elliptic equations involving supercritical growth positive radial solutions of elliptic equation involving supercritical growth are analysed as their supremum norm tends to infinity it is shown that they converge uniformly away from the origin as well as in h 1 to the unique singular solution

continuous fields of kirchberg c algebras abstract in this paper we study the c algebras associated to continuous fields over locally compact metrizable zero dimensional spaces whose fibers are kirchberg c algebras satisfying the uct we show that these algebras are inductive limits of finite direct sums of kirchberg algebras and they are classified up to isomorphism by topological invariants

existence of sign changing solutions for an equation with a weighted p laplace operator we consider radial solutions of a general elliptic equation involving a weighted p laplace operator with a subcritical nonlinearity by a shooting method we prove the existence of solutions with any prescribed number of nodes the method is based on a change of variables in the phase plane a very general computation of an angular velocity and new estimates for the decay of an energy associated with an asymptotic hamiltonian problem estimating the rate of decay for the energy requires a sub criticality condition the method covers the case of solutions which are not compactly supported or which have compact support in the last case we show that the size of the support increases with the number of nodes

the toda system and clustering interfaces in the allen cahn equation we consider the allen cahn equation varepsilon 2 delta u 1 u 2 u 0 in a bounded smooth domain in mathbb r 2 under zero neumann boundary conditions where varepsilon 0 is a small parameter let 0 be a segment contained in connecting orthogonally the boundary under certain nondegeneracy and nonminimality assumptions for 0 satisfied for instance by the short axis in an ellipse we construct for any given n 1 a solution exhibiting n transition layers whose mutual distances are o varepsilon log varepsilon and which collapse onto 0 as varepsilon to 0 asymptotic location of these interfaces is governed by a toda type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary

classification of homomorphisms and dynamical systems let a be a unital simple c algebra with tracial rank zero and let x be a compact metric space suppose that h 1 h 2 c x a are two unital monomorphisms we show that h 1 and h 2 are approximately unitarily equivalent if and only if h 1 h 2 in kl c x a and o h 1 f o h 2 f for every f c x and every trace r of a inspired by a theorem of tomiyama we introduce a notion of approximate conjugacy for minimal dynamical systems let x be a compact metric space and let x x be two minimal homeomorphisms using the above mentioned result we show that two dynamical systems are approximately conjugate in that sense if and only if a k theoretical condition is satisfied in the case that x is the cantor set this notion coincides with the strong orbit equivalence of giordano putnam and skau and the k theoretical condition is equivalent to saying that the associate crossed product c algebras are isomorphic another application of the above mentioned result is given for c dynamical systems related to a problem of kishimoto let a be a unital simple ah algebra with no dimension growth and with real rank zero and let a aut a we prove that if r fixes a large subgroup of k 0 a and has the tracial rokhlin property then ax z is again a unital simple ah algebra with no dimension growth and with real rank zero

multi peak solutions for super critical elliptic problems in domains with small holes abstract this paper deals with the slightly super critical elliptic problem u u n 2 n 2 e in u 0 in u 0 on 1 where e 0 is a small parameter and r n is a bounded domain with smooth boundary assuming that the domain exhibits k sufficiently small holes multiple solutions are constructed by gluing double spike patterns located near each of the holes

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

phase plane analysis and classification of solutions of a mixed sublinear superlinear elliptic problem abstract in this paper some mixed sublinear superlinear critical problem extending the famous problem of brezis nirenberg are analysed the existence of solutions is discussed a phase plane analysis is performed in order to transform the problem into an ordinary differential equation finally a full classification of radial solutions according to their behavior at the origin is performed for subcritical critical and supercritical cases

geometry of phase space and solutions of semilinear elliptic equations in a ball we consider the problem u u p u in in b u 0 in b u 0 on b where b denotes the unit ball in r n n 3 a 0 and p 1 merle and peletier showed that for p n 2 n 2 there is a unique value a 0 such that a radial singular solution exists this value is the only one at which an unbounded sequence of classical solutions of 1 may accumulate here we prove that if additionally p 0 0 in b u 0 on b where f s s p s q 1 q p and p satisfies the same condition as above

the classification of simple separable unital z stable locally ash algebras let a be a unital simple separable locally approximately subhomogeneous c algebra locally ash algebra it is shown that a qa q can be tracially approximated by unital elliott thomsen algebras with trivial k1k1 group where q is the universal uhf algebra in particular it follows that a is classifiable by the elliott invariant if a is jiang su stable

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

stability of symmetric vortices for two component ginzburg landau systems abstract we study ginzburg landau equations for a complex vector order parameter c 2 we consider the dirichlet problem in the disk in r 2 with a symmetric degree one boundary condition and study its stability in the sense of the spectrum of the second variation of the energy we find that the stability of the degree one equivariant solution depends on the ginzburg landau parameter as well as the sign of the interaction term in the energy

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

e theory for c algebras over topological spaces abstract we define e theory for separable c algebras over second countable topological spaces and establish its basic properties this includes an approximation theorem that relates the e theory over a general space to the e theories over finite approximations to this space we obtain effective criteria for determining the invertibility of e theory elements over possibly infinite dimensional spaces furthermore we prove a universal multicoefficient theorem for c algebras over totally disconnected metrisable compact spaces

boundary towers of layers for some supercritical problems we consider the supercritical problem u u p 1uin d u 0on d where d is a bounded smooth domain in rn and p is smaller than the th critical sobolev exponent 2n n 2n 2 with 1 n 3 we show that in some suitable torus like domains d there exists an arbitrary large number of sign changing solutions with alternate positive and negative layers which concentrate at different rates along a dimensional submanifold of d as p approaches 2n from below

a short remark regarding pohozaev type results on general domains assuming finite morse index we are interested in nonnegative nontrivial solutions of u up in u 0 on 1 where 1 p and a bounded smooth domain in rn with 3 n 9 we show that given a nonnegative integer m there is some large p m such that the only nonnegative solution u of morse index at most m is u 0 2010 mathematics subject classification

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

sign changing bubble towers for asymptotically critical elliptic equations on riemannian manifolds given a smooth compact riemannian n manifold m g we consider the equation gu hu u 2 2 eu where h is a c1 function on m the exponent 2 2n n 2 is the critical sobolev exponent and e is a small positive real parameter such that e 0 we prove the existence of blowing up families of sign changing solutions which develop bubble towers at some point where the function h is greater than the yamabe potential n 24 n 1 scalg

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

asymptotically unitary equivalence and asymptotically inner automorphisms let c be a unital ah algebra and let a be a unital separable simple c algebra with tracial rank zero suppose that phi 1 phi 2 colon c to a are two unital monomorphisms we show that there is a continuous path of unitaries u t colon t in 0 infty of a such that lim t to infty u t phi 1 a u t phi 2 a quad hbox rm for all quad a in c if and only if phi 1 phi 2 in kk c a tau circ phi 1 tau circ phi 2 for all tau in t a and the rotation map tilde eta phi 1 phi 2 associated with phi 1 and phi 2 is zero in particular an automorphism alpha on a unital separable simple c algebra a in cal n with tracial rank zero is asymptotically inner if and only if alpha rm id a quad rm in quad kk a a and the rotation map tilde eta phi 1 phi 2 is zero let a be a unital ah algebra not necessarily simple and let alpha in aut a be an automorphism as an application we show that the associated crossed product a rtimes alpha bbb z can be embedded into a unital simple af algebra if and only if a admits a strictly positive alpha invariant tracial state

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

equivariant homomorphisms rokhlin constraints and equivariant uhf absorption abstract we classify equivariant homomorphisms between c dynamical systems associated to actions of finite groups with the rokhlin property in addition the given actions are classified an obstruction is obtained for the cuntz semigroup of a c algebra allowing such an action we also obtain an equivariant uhf absorption result

semilinear elliptic equations and supercritical growth where p p the aim of the present paper is to understand for the case p pc the behaviour of solutions to 1 i with large supremum norm to this end 0 is restricted to be the unit ball f3c r3 as in this case the behaviour of solutions to 1 1 can be described in some detail according to the results proved in gidas ni and nirenberg s all positive solutions of 1 1 must be radial when q b with the scaling iid ilip in the case n 3 problem 1 1 reduces to the following ordinary differential equation problem

bubble towers for supercritical semilinear elliptic equations we construct positive solutions of the semilinear elliptic problem u u up 0 with dirichet boundary conditions in a bounded smooth domain rn n 4 when the exponent p is supercritical and close enough to n 2n 2 and the parameter r is small enough as p n 2n 2 the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves green s function our result extends the result of del pino et al j differential equations 193 2 2003 280 when is a ball and the solutions are radially symmetric

the classification of certain non simple c algebras of tracial rank zero abstract it is proved that the z 2 graded ordered k theory group with order unit k a k a 1 a is a complete invariant for the class of unital c algebras which are inductive limits of finite direct sums of unital amenable separable simple c algebras with tracial rank zero which satisfy the uct

the automorphism group of a simple tracially ai algebra the structure of the automorphism group of a simple tai algebra is studied in particular we show that frac overline rm inn a overline rm inn 0 a is isomorphic as a topological group to an inverse limit of discrete abelian groups for a unital simple ah algebra with bounded dimension growth consequently frac overline rm inn a overline rm inn 0 a is totally disconnected

boundary bubbling solutions for a supercritical neumann problem with mixed nonlinearities abstract in this paper we investigate the existence of boundary bubbling solutions for a supercritical neumann problem with mixed nonlinearities under a suitable assumption on the mean curvature of the boundary the behavior of the solutions is as a tower of k bubbles when the exponent goes to sobolev critical exponent and the blow up point is a critical point of the mean curvature

nodal solutions for an elliptic problem involving large nonlinearities abstract let us consider the problem 0 1 u a x u u p 1 u in b 1 u 0 on b 1 where b 1 is the unit ball in r n n 3 and a x 0 is a smooth radial function under some suitable assumptions on the regular part of the green function of the operator u n 1 r u a r u we prove the existence of a changing sign solution to 0 1 for p large enough

on the structure of fractional degree vortices in a spinor ginzburg landau model abstract we consider a ginzburg landau functional for a complex vector order parameter whose minimizers exhibit vortices with half integer degree by studying the associated system of equations in r 2 which describes the local structure of these vortices we show some new and unconventional properties of these vortices in particular one component of the solution vanishes but the other does not we also prove the existence and uniqueness of equivariant entire solutions and provide a second proof of uniqueness valid for a large class of systems with variational structure

uniqueness of symmetric vortex solutions in the ginzburg landau model of superconductivity abstract symmetric vortices are finite energy solutions a to the ginzburg landau equations of superconductivity with the form f r e id a s r r 2 y x the existence regularity and asymptotic form of the solutions f r s r for any d z 0 have been established by plohr and by burger and chen in this paper we prove the uniqueness of these solutions when the ginzburg landau parameter satisfies 2 2 d 2 for any fixed d z 0 to do this we show that any such solution is a non degenerate relative minimizer of the free energy functional constrained to a convex set then use a version of the mountain pass theorem to derive a contradiction should there be more than one solution

a new proof of kirchberg s 2 stable classification i present a new proof of kirchbergs q2 stable classification theorem two separable nuclear stable unital q2 stable c algebras are isomorphic if and only if their ideal lattices are order isomorphic or equivalently their primitive ideal spaces are homeomorphic many intermediate results do not depend on pure infiniteness of any sort

a complete invariant foradalgebras with real rank zero and bounded torsion ink1 abstract for each integerp 1 we consider an algebraic invariant forc algebras the invariant consists ofk0 k1 thek0 group with z pcoefficients the order structures these groups possess and the natural maps between the three groups we prove that this invariant is complete for the class ofadalgebras of real rank zero ifpannihilates every torsion element ofk1 theadalgebras arec algebras which are inductive limits of finite direct sums ofc algebras of continuous matrix valued functions over the circle or over the interval where in the interval case we require that the value over the endpoints is a scalar multiple of the unit matrix examples show that the condition on the torsion ink1is necessary when only one modp k0 group is considered applying a theorem of d d rlat the result also applies to a subclass of theahalgebras

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

the kk lifting problem for dimension drop interval algebras abstract in this paper we investigate the kk theory of generalized dimension drop interval algebras with possibly different dimension drops at the endpoints with special emphasis on the problem which kk class is representable by a homomorphism between two such c algebras allowing the tensor product with a matrix algebra for the codomain algebra this lifting problem makes sense in its own right in kk theory and also has application to the classification of c algebras which are inductive limits of these building blocks it turns out that when the dimension drops at the two endpoints are different there exist kk elements which preserve the order structure defined by m dadarlat and t a loring in 4 on the mod p k theory but cannot be lifted to a homomorphism this is different from the equal dimension drop case as shown by s eilers in 6

when almost multiplicative morphisms are close to homomorphisms absrract it is shown that approximately multiplicative contractive positive morphisms from c x with dim x 2 into a simple c algebra a of real rank zero and of stable rank one are close to homomorphisms provided that certain k theoretical obstacles vanish as a corollary we show that a homomorphism h c x a is approximated by homomorphisms with finite dimensional range if h gives no k theoretical obstacle

asymptotic behavior of solutions for nonlinear elliptic problems with the fractional laplacian abstract in this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional laplacians and the critical exponents this work can be seen as a nonlocal analog of the results of han 1991 24 and rey 1990 35

finite group actions on c algebras with the rohlin property ii abstract we classify finite group actions on some classes of c algebras with the rohlin property in terms of the k groups a group cohomological obstruction is obtained for the k groups of a c algebra allowing such an action for example the tate cohomology groups with the k groups as the coefficient modules must vanish model actions with the rohlin property are constructed for given k theoretical invariants quasi free actions of finite cyclic groups on the cuntz algebras and their dual actions are investigated from the view point of the rohlin property

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

vortex structure in p wave superconductors we study vortices in p wave superconductors in a ginzburg landau setting the state of the superconductor is described by a pair of complex wave functions and the p wave symmetric energy functional couples these in both the kinetic gradient and potential energy terms giving rise to systems of partial differential equations which are nonlinear and coupled in their second derivative terms we prove the existence of energy minimizing solutions in bounded domains 2 and consider the existence and qualitative properties such as the asymptotic behavior of equivariant solutions defined in all of 2 the coupling of the equations at highest order changes the nature of the solutions and many of the usual properties of classical ginzburg landau vortices either do not hold for the p wave solutions or are not immediately evident