unitary orbits in 1 dimensional nccw complexes abstract let a be a 1 dimensional nccw complex in this paper it is shown that the usual distance d u defined on the approximate unitary equivalence classes unitary orbits of self adjoints in a is equal to a distance d w defined on morphisms from the cuntz semigroup of c 0 0 1 to cu a in addition another distance d p on approximately unitary equivalence classes related to spectral information of positive elements in the classes is proved to be equal to d u and d w on simple unital inductive limits of 1 dimensional nccw complexes

classification of c homomorphisms from c0 0 1 to a c algebra abstract a class of c algebras is described for which the c homomorphisms from c 0 0 1 to the algebra may be classified by means of the cuntz semigroup functor examples are given of algebras simple and non simple for which this classification fails it is shown that a suitable suspension of the cuntz semigroup functor deals successfully with some of these counterexamples

large deviation probabilities and dominating points for open convex sets nonlogarithmic behavior the existence of a dominating point for an open convex set and a corresponding representation formula for large deviation probabilities are established in the infinite dimensional setting under conditions which are both necessary and sufficient and follow from those used previously in mathbb r d a precise nonlogarithmic estimate of large deviation probabilities applicable to gaussian measures is also included

the cuntz semigroup the elliott conjecture and dimension functions on c algebras we prove that the cuntz semigroup is recovered functorially from the elliott invariant for a large class of c algebras in particular our re sults apply to the largest class of simple c algebras for which k theoretic classification can be hoped for this work has three significant consequences first it provides new conceptual insight into elliott s classification program proving that the usual form of the elliott conjecture is equivalent among z stable algebras to a conjecture which is in general substantially weaker and for which there are no known counterexamples second and third it resolves for the class of algebras above two conjectures of blackadar and handelman concerning the basic structure of dimension functions on c algebras we also prove in passing that the cuntz pedersen semigroup is recovered functorially from the elliott invariant for a large class of simple unital c algebras

the dixmier property and tracial states for c algebras a t was partially supported by an nserc postdoctoral fellowship and through the epsrc grant ep n00874x 1 acknowledgements we are grateful to luis santiago for helpful discussions at an early stage of this investigation we would also like to thank the referee for providing helpful comments which have led to a number of improvements

nuclear dimension and mathcal z stability simple separable unital monotracial and nuclear mathrm c algebras are shown to have finite nuclear dimension whenever they absorb the jiang su algebra mathcal z tensorially this completes the proof of the toms winter conjecture in the unique trace case

unitary orbits of self adjoint operators in simple z stable c algebras abstract we prove that in a simple unital exact z stable c algebra the distance between the unitary orbits of self adjoint elements with connected spectrum is completely determined by spectral data this fails without the assumption of z stability

extremal graphs with respect to variable sum exdeg index via majorization the variable sum exdeg index of a graph g is defined as seia g uve g adegg u adegg v where a 1 is a positive real number the aim of this paper is applying the majorization technique to obtain the maximum and minimum values of variable sum exdeg index of trees unicyclic bicyclic and tricyclic graphs

on a simple unital projectionless c algebra we construct a unital separable c algebra z as an analog of the hyperfinite type ii 1 factor besides being nuclear simple projectionless and infinite dimensional z has a unique tracial state and is kk equivalent to inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the algebra of complex numbers it is shown that unital endomorphisms on z are approximately inner and that z is isomorphic to the infinite tensor product of its replicas it is also shown that a z a for certain interesting classes of unital simple nuclear c algebras a of real rank zero

nuclear dimension mathcal z stability and algebraic simplicity for stably projectionless c algebras the main result here is that a simple separable c algebra is mathcal z stable where mathcal z denotes the jiang su algebra if i it has finite nuclear dimension or ii it is approximately subhomogeneous with slow dimension growth this generalizes the main results of toms invent math 183 2 225 244 2011 and winter invent math 187 2 259 342 2012 to the nonunital setting as a consequence finite nuclear dimension implies mathcal z stability even in the case of a separable c algebra with finitely many ideals algebraic simplicity is established as a fruitful weakening of being simple and unital and the proof of the main result makes heavy use of this concept

closed unitary and similarity orbits of normal operators in purely infinite c algebras abstract we will investigate the norm closure of the unitary and similarity orbits of normal operators in unital simple purely infinite c algebras a simple proof to the classification of when two normal operators are approximately unitarily equivalent in said algebras with trivial k 1 group will be given using previously known techniques using these techniques some upper and lower bounds for the distance between unitary orbits of normal operators will be obtained in addition a complete characterization of when one normal operator is in the closed similarity orbit of another normal operator will be given for unital simple purely infinite c algebras and type iii factors with separable predual

degree based topological indices optimal trees with given number of pendents abstract a dynamic programming method is elaborated enabling the characterization of trees with a given number of pendent vertices for which a vertex degree based invariant topological index achieves its extremal value the method is applied to the chemically interesting and earlier much studied such invariants the first and second zagreb index and the atom bond connectivity index

large deviation lower bounds for arbitrary additive functionals of a markov chain a universal large deviation lower bound is proved for sums of banach space valued functions of an irreducible general state space markov chain there are no restrictions on the functions other than measurability

spectral variation of normal matrices abstract it has been a durable conjecture that the distance appropriately defined between the spectra of two normal matrices is bounded by the operator norm of their difference we report on some numerical studies that show that this conjecture is in general false in fact even in the 3 x 3 case the spectral distance may exceed the norm distance by between 1 and 2 percent we also describe the background of the conjecture and offer an analysis of extremal counterexamples

large deviation lower bounds for additive functionals of markov processes let x 1 x 2 ldots be a markov process with state space e a polish space let l n omega a n 1 sum n 1 j 0 1 a x j omega denote the normalized occupation time measure if mu is a probability measure on e g is a weak neighborhood of mu and if v subset e then we obtain asymptotic lower bounds for probabilities p x lbrack l n omega cdot in g x j omega in v 0 leq j leq n 1 rbrack in terms of i mu the rate function of donsker and varadhan our assumptions are weaker than those imposed by donsker and varadhan and the proof works without any essential change in the continuous time case as well in fact the same proofs apply to certain bounded additive functionals let r geq 0 and let f omega rightarrow mathbf b be bounded mathscr f 0 r measurable where omega is the sample space with the product topology skorohod topology in the continuous time case and mathbf b is a separable banach space let theta k omega rightarrow omega be the shift operator i e theta k omega j omega k j then we get lower bounds for probabilities involving n 1 f omega f theta 1 omega cdots f theta n 1 omega in place of l n omega cdot in this latter situation the rate function has to be the entropy function h q of donsker and varadhan

large deviations for vector valued functionals of a markov chain lower bounds on obtient des bornes inferieures que l on exprime en termes des parametres de convergence de certains noyaux on donne une application aux mesures empiriques de chaines de markov

perturbation of spectral subspaces and solution of linear operator equations abstract let a and b be normal operators on a hilbert space let k a and k b be subsets of the complex plane at distance at least from each other let e be the spectral projector for a belonging to k a and let f be the spectral projector for b belonging to k b our main results are estimates of the form ef c e a b f in some special situations the constant c is as low as 1 as an application we prove for an absolute constant d that if the space is finite dimensional and if a and b are normal with a b d then the spectrum of b can be obtained from that of a multiplicities counted by moving each eigenvalue by at most our main results have equivalent formulations as statements about the operator equation aq qb s let a and b be normal operators on perhaps different hilbert spaces assume a k a and b k b where k a k b and are as before then we give estimates of the forms q c aq qb

identifying a large deviation rate function assume a sequence of probabilities pn has a large deviation rate function i it is proved that i takes a form analogous to a convex conjugate if i is also assumed convex then i is a convex conjugate of an explicitly defined function ft the results are applied to the empirical law of a markov chain yielding universal bounds on i examples are given of markov chains in which the empirical law has a large deviation rate strictly between the given bounds

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

ordering chemical graphs by randic and sum connectivity numbers abstract let g be a graph with edge set e g the randic and sum connectivity indices of g are defined as r g u v e g 1 d e g g u d e g g v and s c i g u v e g 1 d e g g u d e g g v respectively where degg u denotes the vertex degree of u in g in this paper the extremal randic and sum connectivity index among all n vertex chemical trees n 13 connected chemical unicyclic graphs n 7 connected chemical bicyclic graphs n 6 and connected chemical tricyclic graphs n 8 were presented

nuclear dimension and mathcal z stability of pure c algebras in this article i study a number of topological and algebraic dimension type properties of simple c algebras and their interplay in particular a simple c algebra is defined to be tracially m bar m pure if it has strong tracial m comparison and is tracially bar m almost divisible these notions are related to each other and to nuclear dimension

classification of inductive limits of 1 dimensional nccw complexes abstract a classification result is obtained for the c algebras that are stably isomorphic to inductive limits of 1 dimensional noncommutative cw complexes with trivial k 1 group the classifying functor cu is defined in terms of the cuntz semigroup of the unitization of the algebra for the simple c algebras covered by the classification cu reduces to the ordered k 0 group the cone of traces and the pairing between them as an application of the classification it is shown that the crossed products by a quasi free action o 2 r are all isomorphic for a dense set of positive irrational numbers

an infinite family of non isomorphic c algebras with identical k theory we exhibit a countably infinite family of simple separable nuclear and mutually non isomorphic c algebras which agree on k theory and traces the algebras do not absorb the jiang su algebra z tensorially answering a question of n c phillips they are also pairwise shape and morita equivalent confirming a conjecture from our earlier work the distinguishing invariant is the radius of comparison a non stable invariant of the cuntz semigroup

the cone of lower semicontinuous traces on a c algebra the cone of lower semicontinuous traces is studied with a view to its use as an invariant its properties include compactness hausdorffness and continuity with respect to inductive limits a suitable notion of dual cone is given the cone of lower semicontinuous 2 quasitraces on a non exact c algebra is considered as well these results are applied to the study of the cuntz semigroup it is shown that if a c algebra absorbs the jiang su algebra then the subsemigroup of its cuntz semigroup consisting of the purely non compact elements is isomorphic to the dual cone of the cone of lower semicontinuous 2 quasitraces this yields a computation of the cuntz semigroup for the following two classes of c algebras c algebras that absorb the jiang su algebra and have no non zero simple subquotients and simple c algebras that absorb the jiang su algebra 1 introduction the most standard invariants in the classification of nu clear c algebras are their k groups and their traces the traces at least for simple c algebras are assumed to be bounded in the unital case and lower semicontinuous and densely finite in the non unital case if one has in mind the classification of non simple c algebras it is clear that these two kinds of traces will not suffice and a broader class should be considered in this paper we study the properties of the cone of all lower semicontinuous traces on a c algebra with the purpose of applying our results to questions in the classification of non simple c algebras one sign that this is natural is that this cone in particular determines the lattice of closed two sided ideals an important invariant in its own right see 3 4 we shall also consider lower semicontinuous 2 quasitraces since these appear naturally as functionals on the cuntz semigroup of the algebra if the