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

cuntz krieger algebras of infinite graphs and matrices we show that the cuntz krieger algebras of infinite graphs and infinite 0 1 matrices can be approximated by those of finite graphs we then use these approximations to deduce the main uniqueness theorems for cuntz krieger algebras and to compute their k theory since the finite approximating graphs have sinks we have to calculate the k theory of cuntz krieger algebras of graphs with sinks and the direct methods we use to do this should be of independent interest

l kernel classification nonparametric kernel methods are widely used and proven to be successful in many statistical learning problems well known examples include the kernel density estimate kde for density estimation and the support vector machine svm for classification we propose a kernel classifier that optimizes the l2 or integrated squared error ise of a difference of densities we focus on the gaussian kernel although the method applies to other kernels suitable for density estimation like a support vector machine svm the classifier is sparse and results from solving a quadratic program we provide statistical performance guarantees for the proposed l2 kernel classifier in the form of a finite sample oracle inequality and strong consistency in the sense of both ise and probability of error a special case of our analysis applies to a previously introduced ise based method for kernel density estimation for dimensionality greater than 15 the basic l2 kernel classifier performs poorly in practice thus we extend the method through the introduction of a natural regularization parameter which allows it to remain competitive with the svm in high dimensions simulation results for both synthetic and real world data are presented

inductive limit automorphisms of the irrational rotation algebra it is shown that the flip automorphismu u v v of the irrational rotation algebra a is an inductive limit automorphism here the algebra a is generated by unitariesu v satisfyingvu e2 i uv where is an irrational number recently elliott and evans proved that a can be approximated by unital subalgebras isomorphic to a direct sum of two matrix algebras over c mathbb t the algebra of continuous functions on the unit circle this is the central result which they used to obtain their structure theorem on a namely that a is the inductive limit of an increasing sequence of subalgebras each isomorphic to a direct sum of two matrix algebras over c mathbb t in their proof they devised a subtle construction of two complementary towers of projections in the present paper it is shown that the two towers can be chosen so that each summand of their approximating basic building blocks is invariant under the flip automorphism and in particular that the unit projection of the first summand is unitarily equivalent to the complement of the unit of the second by a unitary which is fixed under the flip also an explicit computation of the flip on the approximating basic building blocks of a is given further combining this result along with others including a theorem of su and a spectral argument of bratteli evans and kishimoto a two tower proof is obtained of the fact established by bratteli and kishimoto that the fixed point subalgebra b under the flip is approximately finite dimensional also used here is the fact that b has the cancellation property and is gifted with four basic unbounded trace functionals the question is raised whether other finite order automorphisms of a arising from a matrix in sl 2 are inductive limit automorphisms or evenalmost inductive limit automorphisms in the sense of voiculescu

graphs groupoids and cuntz krieger algebras we associate to each locally finite directed graphgtwo locally compact groupoidsgandg the unit space ofgis the space of one sided infinite paths ing andg is the reduction ofgto the space of paths emanating from a distinguished vertex we show that under certain conditions theirc algebras are morita equivalent the groupoidc algebrac g is the cuntz krieger algebra of an infinite 0 1 matrix defined byg and that the algebrasc g contain thec algebras used by doplicher and roberts in their duality theory for compact groups we then analyse the ideal structure of these groupoidc algebras using the general theory of renault and calculate theirk theory

a cauchy problem for some fractional abstract differential equation with non local conditions abstract we discuss in this paper the existence and uniqueness of solutions to the cauchy problem for the fractional differential equation with non local conditions d q x t f t x t t 0 t x 0 g x x 0 where 0 q 1 in a banach space we use classical tools from functional analysis to obtain the results

symmetries of the car algebra a automorphism of period two of the uhf c algebra of type 2 is constructed for which the fixed point algebra is not approximately finite dimensional the construction also yields a diagonal maximal commutative c subalgebra whose spectrum is not totally disconnected variations and consequences of the construction are discussed

the c algebras of finitely aligned higher rank graphs we generalise the theory of cuntz krieger families and graph algebras to the class of finitely aligned k graphs this class contains in particular all row finite k graphs the cuntz krieger relations for non row finite k graphs look significantly different from the usual ones and this substantially complicates the analysis of the graph algebra we prove a gauge invariant uniqueness theorem and a cuntz krieger uniqueness theorem for the c algebras of finitely aligned k graphs

cuntz pimsner algebras crossed products and k theory abstract suppose a is a c algebra and h is a c correspondence over a if h is regular in the sense that the left action of a is faithful and is given by compact operators then we compute the k theory of o a h t where the action is the usual gauge action the case where a is an af algebra is carefully analyzed in particular if a is af we show o a h t is af combining this with takai duality and an af embedding theorem of n brown we show the conditions af embeddability quasidiagonality and stable finiteness are equivalent for o a h if h is also assumed to be regular these finiteness conditions can be characterized in terms of the ordered k theory of a

when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional evans d g and sims a when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional preprint arxiv 1112 4549v1

product systems over right angled artin semigroups we build upon mac lane s definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid g we show that the existing notions of product systems fit into our categorical framework as do the k graphs of kumjian and pask we then specialize to product systems over right angled artin semigroups these are semigroups that interpolate between free semigroups and free abelian semigroups for such a semigroup we characterize all product systems which take values in a given tensor groupoid g in particular we obtain necessary and sufficient conditions under which a collection of k 1 graphs form the coordinate graphs of a k graph

on the k theory of twisted higher rank graph c algebras abstract we investigate the k theory of twisted higher rank graph algebras by adapting parts of elliott s computation of the k theory of the rotation algebras we show that each 2 cocycle on a higher rank graph taking values in an abelian group determines a continuous bundle of twisted higher rank graph algebras over the dual group we use this to show that for a circle valued 2 cocycle on a higher rank graph obtained by exponentiating a real valued cocycle the k theory of the twisted higher rank graph algebra coincides with that of the untwisted one

affine buildings tiling systems and higher rank cuntz krieger algebras to an r dimensional subshift of finite type satisfying certain special properties we associate a c algebra a this algebra is a higher rank version of a cuntz krieger algebra in particular it is simple purely infinite and nuclear we study an example if is a group acting freely on the vertices of an a2 building with finitely many orbits and if is the boundary of that building then c o is the algebra associated to a certain two dimensional subshift

nonparametric estimation of regression functions with both categorical and continuous data abstract in this paper we propose a method for nonparametric regression which admits continuous and categorical data in a natural manner using the method of kernels a data driven method of bandwidth selection is proposed and we establish the asymptotic normality of the estimator we also establish the rate of convergence of the cross validated smoothing parameters to their benchmark optimal smoothing parameters simulations suggest that the new estimator performs much better than the conventional nonparametric estimator in the presence of mixed data an empirical application to a widely used and publicly available dynamic panel of patent data demonstrates that the out of sample squared prediction error of our proposed estimator is only 14 20 of that obtained by some popular parametric approaches which have been used to model this data set

endomorphisms of o n which preserve the canonical uhf subalgebra abstract unital endomorphisms of the cuntz algebra o n which preserve the canonical uhf subalgebra f n o n are investigated we give examples of such endomorphisms u for which the associated unitary element u in o n which satisfies s j u s j for all j does not belong to f n one such example in the case where n 2 arises from a construction of a unital endomorphism on o 2 which preserves the canonical uhf subalgebra and where the relative commutant of its image in o 2 contains a copy of o 2

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

an alternative method of cross validation for the smoothing of density estimates cross validation with kullback leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation a framework for this problem is constructed and used to derive an alternative method of cross validation based on integrated squared error recently also proposed by rudemo 1982 hall 1983 has established the consistency and asymptotic optimality of the new method for small and moderate sized samples the performances of the two methods of cross validation are compared on simulated data and specific examples

non commutative shifts and crossed products abstract when a is a unital simple af c algebra and has a unique tracial state it is shown that the crossed product of the two sided infinite tensor product z a by the shift is a tracially af c algebra a similar result is given to the crossed product of a certain non unital two sided infinite tensor product by the shift applying a far reaching classification result of such c algebras by h lin we obtain an example of a one parameter automorphism group on some af c algebra which is not approximately inner a counter example to the af version of the so called powers sakai conjecture 23

non commutative spheres iii irrational rotations leta be the irrational rotation algebra i e thec algebra generated by two unitariesu v satisfyingvu e 2 i uv with irrational and consider the fixed point subalgebrab under the flip automorphismu u 1 v v 1 we prove thatb is an af algebra

on nonparametric multivariate binary discrimination aitchison aitken 1976 introduced a novel and ingenious nonparametric method for estimating probabilities in a multidimensional binary space the technique is designed for use in multivariate binary discrimination their estimator depends crucially on an unknown smoothing parameter a and aitchison aitken proposed a maximum likelihood method for determining a from the sample unfortunately this leads to an adaptive estimator which can behave very erratically when there are a number of empty or near empty cells present we demonstrate this both theoretically and by example to overcome these difficulties we introduce another method of estimating a which is designed to minimize a global function of the mean squared error

on nonparametric discrimination using density differences summary we propose a technique for nonparametric discrimination in which smoothing parameters are chosen jointly according to a criterion based on the difference between two densities the approach is suitable for categorical continuous and mixed data and uses information from both populations to determine the smoothing parameter for any one population in the case of categorical data optimal performance is sometimes achieved using negative smoothing parameters a property which does not emerge if the smoothing parameters are chosen individually

discrimination with polychotomous predictor variables using orthogonal functions abstract three discriminant functions proposed by bahadur 1961 a b martin and bradley 1972 and ott and kronmal 1976 are generalized to accommodate polychotomous predictor variables the discrete fourier orthogonal functions used in the generalization provide computational advantages with the use of the fast fourier transform data dependent inclusion rules for each expansion s coefficients are developed to smooth observed frequencies and improve discrimination the performances of these methods are compared in a simulation study and populations in which these rules are expected to perform well are identified

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

kernel estimates for categorical data a new method of estimation of the smoothing parameters of the aitchison and aitken kernel method for categorical data is proposed this is based on minimizing an unbiased estimator of the mean squared error and it provides a natural approach to multivariate estimation thus allowing for the interaction structure of data in a simple way the kernel smoothing technique is shown theoretically to offer a potential improvement over the direct use of cell frequencies by means of an existence theorem akin to that used by hoerl and kennard for ridge regression the proposed smoothing parameter estimates are compared with competing methods by means of a simulation in the simple binomial case and are illustrated on a set of multivariate data used in titterington 1980 a property of invariance to the reversal of estimate formation and marginalization is proved for any multivariate kernel and estimation procedure

classification using kernel density estimates multiscale analysis and visualization the use of kernel density estimates in discriminant analysis is quite well known among scientists and engineers interested in statistical pattern recognition using a kernel density estimate involves properly selecting the scale of smoothing namely the bandwidth parameter the bandwidth that is optimum for the mean integrated square error of a class density estimator may not always be good for discriminant analysis where the main emphasis is on the minimization of misclassification rates on the other hand cross validation based methods for bandwidth selection which try to minimize estimated misclassification rates may require huge computation when there are several competing populations besides such methods usually allow only one bandwidth for each population density estimate whereas in a classification problem the optimum bandwidth for a class density estimate may vary significantly depending on its competing class densities and their prior probabilities therefore in a multiclass problem it

an alternative choice of smoothing for kernel based density estimates in discrete discriminant analysis summary the kernel method of estimating the cell probabilities of a multivariate categorical distribution due to aitchison aitken 1976 depends crucially on an unknown smoothing parameter a a method of estimating a is introduced which is explicitly connected to multivariate discrimination the method based on maximization of the leaving one out estimator of the nonerror rate is shown to be bayes risk strongly consistent an example is given to illustrate the application

quantum spheres and projective spaces as graph algebras the c algebras of continuous functions on quantum spheres quantum real projective spaces and quantum complex projective spaces are realized as cuntz krieger algebras corresponding to suitable directed graphs structural results about these quantum spaces especially about their ideals and k theory are then derived from the general theory of graph algebras it is shown that the quantum even and odd dimensional spheres are produced by repeated application of a quantum double suspension to two points and the circle respectively

a class of smooth estimators for discrete distributions summary this paper presents a class of smooth weight function estimators for discrete distributions any estimator in the class depends on choosing a parameterized set of weights the resulting estimators are strongly consistent and asymptotically normal under mild regularity conditions a general procedure for choosing the weight function smoothing parameter is given along with specific solutions in some cases mean squared error comparisons with the maximum likelihood estimator based on large sample theory and small sample simulations are obtained typically the weight function estimates yield significantly smaller mean squared error in these comparisons

compact group actions on uhf algebras obtained by folding the interval we show that for any integer p greater than one there exists a c algebra which is not af such that mp a mp where mp mp is the uhf algebra of type p using this construction of uhf algebras as inductive limits of subhomogeneous algebras on an interval embedded in each other via a folding operation on the interval we exhibit actions of compact groups on uhf algebras whose fixed point algebras are not af in particular for any compact group g e there exists an action of g on mp whose fixed point algebra is not af

a comparison of two methods of discriminant analysis applied to binary data the results of applying classical linear discriminant analysis and kernel discriminant analysis to several real sets of multivariate binary data are presented classical discriminant analysis is intrinsically parametric and is usually presented as being well suited to continuous variables it is also well known to be optimal when the two classes have normal distributions with identical covariance matrices the kernel method on the other hand is nonparametric and in the form used here is ideally suited to binary data the apparent error rates of the kernel method are found to be consistently less than those of the classical method however when the true error rates are estimated either by applying the classifiers to independent test sets or by the leaving one out method from the design sets no significant difference is discernible between the two types of classifier

af embeddability of crossed products of cuntz algebras abstract we investigate crossed products of cuntz algebras by quasi free actions of abelian groups we prove that our algebras are af embeddable when actions satisfy a certain condition we also give a necessary and sufficient condition that our algebras become simple and purely infinite and consequently our algebras are either purely infinite or af embeddable when they are simple

cross validation in nonparametric estimation of probabilities and probability densities summary we examine large sample properties of cross validation for estimating cell probabilities starting from a completely general measure of loss necessary and sufficient conditions on the loss function are derived for the resulting estimator to be consistent or to minimize expected loss these results reveal that cross validation is extremely sensitive to the shape of the loss function nevertheless when the loss function is chosen correctly cross validation can be relied on to perform well for large samples we provide a simple method of generating loss functions with optimal properties extension to the estimation of univariate probability density functions is discussed at a heuristic level

a consistent model specification test with mixed discrete and continuous data in this paper we propose a nonparametric kernel based model specification test that can be used when the regression model contains both discrete and continuous regressors we employ discrete variable kernel functions and we smooth both the discrete and continuous regressors using least squares cross validation cv methods the test statistic is shown to have an asymptotic normal null distribution we also prove the validity of using the wild bootstrap method to approximate the null distribution of the test statistic the bootstrap being our preferred method for obtaining the null distribution in practice simulations show that the proposed test has significant power advantages over conventional kernel tests which rely upon frequency based nonparametric estimators that require sample splitting to handle the presence of discrete regressors