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

actions of symbolic dynamical systems on c algebras ii simplicity of c symbolic crossed products and some examples a c symbolic dynamical system mathcal a rho sigma consists of a unital c algebra mathcal a and a finite family rho alpha alpha in sigma of endomorphisms of mathcal a indexed by symbols of satisfying some conditions the endomorphisms rho alpha alpha in sigma yield both a subshift and a c algebra of a hilbert c bimodule the obtained c algebra is regarded as a crossed product of mathcal a by the subshift we will study simplicity condition of these c algebras some examples such as irrational rotation cuntz krieger algebras will be studied

representations of etale groupoids on l p spaces for p 1 we study representations of etale groupoids on l p spaces our main result is a generalization of renault s disintegration theorem for representations of etale groupoids on hilbert spaces we establish a correspondence between l p representations of an etale groupoid g contractive l p representations of c c g and tight regular l p representations of any countable inverse semigroup of open slices of g that is a basis for the topology of g we define analogs f p g and f red p g of the full and reduced groupoid c algebras using representations on l p spaces as a consequence of our main result we deduce that every contractive representation of f p g or f red p g is automatically completely contractive examples of our construction include the following natural families of banach algebras discrete group l p operator algebras the analogs of cuntz algebras on l p spaces and the analogs of af algebras on l p spaces our results yield new information about these objects their matricially normed structure is uniquely determined more generally groupoid l p operator algebras provide analogs of several families of classical c algebras such as cuntz krieger c algebras tiling c algebras and graph c algebras

duals of compact lie groups realized in the cuntz algebras and their actions on c algebras abstract as a first step towards a new duality theorem for compact groups we consider a representation category t g of a compact lie group g u d c whose objects are the tensor powers of the defining representation and whose arrows are the intertwiners we associate to t g in a natural way a c algebra og which can be identified with the fixed points of the cuntz algebra od under the natural action of g og carries a canonical endomorphism g and t g can be reconstructed from og g if g su d then og is simple we give conditions for an endomorphism of a unital c algebra a to determine an action of t su d on a such actions correspond to monomorphisms of osu d into a with natural intertwining properties

risk bounds for embedded variable selection in classification trees the problems of model and variable selections for classification trees are jointly considered a penalized criterion is proposed which explicitly takes into account the number of variables and a risk bound inequality is provided for the tree classifier minimizing this criterion this penalized criterion is compared to the one used during the pruning step of the cart algorithm it is shown that the two criteria are similar under some specific margin assumptions in practice the tuning parameter of the cart penalty has to be calibrated by hold out or cross validation a simulation study is performed to compare the form of the theoretical penalized criterion we propose with the form obtained after tuning the regularization parameter via cross validation

polish group actions dichotomies and generalized elementary embeddings the results in this paper involve two different topics in the descriptive theory of polish group actions the book becker kechris 6 is an introduction to that theory our two topics and two collections of theorems are rather unrelated but the proofs for both topics are essentially the same locally compact polish groups i e second countable locally compact groups are the traditional objects of study in the field known as topological groups more recently there has been interest in nonlocally compact polish groups such as s the symmetric group on n topologized as a subspace of nn much of this paper is concerned with a proper subclass of the class of polish groups namely those polish groups which admit a complete left invariant metric we call these cli groups the class of cli groups includes all locally compact groups proposition 3 c 2 d but also much more for example all solvable groups are cli hjorth solecki see proposition 3 c 2 f and a closed subgroup g of s is cli iff g is closed in nn see example 3 c 3 a we study continuous actions by polish groups on polish spaces and the associated orbit equivalence relation for locally compact groups all of our results are known or trivial or both but some of our results can be viewed as generalizations of known theorems about locally compact groups to the larger class of cli groups one of the two topics considered in this paper is dichotomy theorems for equivalence relations these are theorems which assert that either the quotient space is small or else it contains a copy of a specific large set we prove some dichotomy theorems for the orbit space of actions by cli groups there are two types of dichotomies which we call the silver vaught dichotomy and the glimm effros dichotomy the silver vaught dichotomy asserts that either there are only countably many equivalence classes or else there is a perfect set of pairwise inequivalent elements assuming the negation of the continuum hypothesis the silver vaught dichotomy is equivalent to the proposition that there are either countably many equivalence classes or 2 0 equivalence classes the topological vaught conjecture which is open is for any continuous action by a polish group g on a polish space x the orbit equivalence relation satisfies the silver vaught dichotomy this conjecture implies and was motivated by the vaught conjecture in model theory indeed

borel subgroups of polish groups abstract we study three classes of subgroups of polish groups borel subgroups polishable subgroups and maximal divisible subgroups the membership of a subgroup in each of these classes allows one to assign to it a rank that is a countable ordinal measuring in a natural way complexity of the subgroup we prove theorems comparing these three ranks and construct subgroups with prescribed ranks in particular answering a question of mauldin we establish the existence of borel subgroups which are 0 complete 3 and 0 complete 2 in each uncountable polish group also for every 1 we construct an abelian locally compact second countable group which is densely divisible and of ulm length 1 all previously known such groups had ulm length 0 or 1

the complexity of the homeomorphism relation between compact metric spaces abstract we determine the exact complexity of classifying compact metric spaces up to homeomorphism more precisely the homeomorphism relation on compact metric spaces is borel bi reducible with the complete orbit equivalence relation of polish group actions consequently the same holds for the isomorphism relation between separable commutative c algebras and the isometry relation between c k spaces

operator algebras for analytic varieties we study the isomorphism problem for the multiplier algebras of irreducible complete pick kernels these are precisely the restrictionsmv of the multiplier algebram of drury arveson space to a holomorphic subvariety v of the unit ball bd we nd that mv is completely isometrically isomorphic tomw if and only if w is the image of v under a biholomorphic auto morphism of the ball in this case the isomorphism is unitarily implemented this is then strengthend to show that when d 1 every isometric isomorphism is completely isometric the problem of characterizing when two such algebras are al gebraically isomorphic is also studied when v and w are each a nite union of irreducible varieties and a discrete variety in bd with d 1 then an isomorphism betweenmv andmw deter mines a biholomorphism with multiplier coordinates between the varieties and the isomorphism is composition with this function these maps are automatically weak continuous we present a number of examples showing that the converse fails in several ways we discuss several special cases in which the converse does hold particularly smooth curves and blaschke sequences

the mackey glimm dichotomy for foliations and other polish groupoids abstract for polish groups acting on polish spaces e effros gave a condition under which 14 properties of the orbits or the orbit space of an anti ergodic character are equivalent he also raised the question of extending these results to polish groupoids and this paper gives an affirmative answer to that question the paper also includes results about totally disconnected quotient spaces and an application to indecomposable continua

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

a going down principle for ample groupoids and the baum connes conjecture abstract we study a going down or restriction principle for ample groupoids and its applications the going down principle for locally compact groups was developed by chabert echterhoff and oyono oyono and allows to study certain functors that arise in the context of the topological k theory of a locally compact group in terms of their restrictions to compact subgroups we extend this principle to the class of ample hausdorff groupoids using le gall s groupoid equivariant version of kasparov s bivariant kk theory moreover we provide an application to the baum connes conjecture for ample groupoids which are strongly amenable at infinity this result in turn is then used to relate the baum connes conjecture for an ample groupoid group bundle which is strongly amenable at infinity to the baum connes conjecture for the fibres

equilibrium and ground states from cayley graphs abstract we present a general framework for the study of kms states of generalized gauge actions on the c algebra of a cayley graph which is pointed by considering the neutral element of the group as a distinguished vertex we use this framework to give a concrete description of a particular kind of kms states on the c algebras that we call abelian kms states if the group is nilpotent all kms states are abelian and our analysis gives the full picture in this case we then describe the ground states that are limits of abelian kms states when the temperature goes to zero

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

purely infinite partial crossed products let a g be a partial dynamical system we show that there is a bijective correspondence between g invariant ideals of a and ideals in the partial crossed product a rg provided the action is exact and residually topologically free assuming in addition a technical condition automatic when a is abelian we show that a rg is purely infinite if and only if the positive nonzero elements in a are properly infinite in a rg as an application we verify pure infiniteness of various partial crossed products including realisations of the cuntz algebras on oa on and oz as partial crossed products

the nuclear dimension of graph c algebras abstract consider a graph c algebra c e with a purely infinite ideal i possibly all of c e such that i has only finitely many ideals and c e i is approximately finite dimensional we prove that the nuclear dimension of c e is 1 if i has infinitely many ideals then the nuclear dimension of c e is either 1 or 2

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

c algebras associated to boolean dynamical systems copyright 2017 elsevier b v or its licensors or contributors this is the authors accepted and refereed manuscript to the article

turbulence orbit equivalence and the classification of nuclear c algebras we bound the borel cardinality of the isomorphism relation for nuclear simple separable c algebras it is turbulent yet borel reducible to the action of the automorphism group of the cuntz algebra o2 on its closed subsets the same bounds are obtained for affine homeomorphism of metrizable choquet simplexes as a by product we recover a result of kechris and solecki namely that homeomorphism of compacta in the hilbert cube is borel reducible to a polish group action these results depend intimately on the classification theory of nuclear simple c algebras by k theory and traces both of necessity and in order to lay the groundwork for further study on the borel complexity of c algebras we prove that many standard c algebra constructions and relations are borel and we prove borel versions of kirchberg s o2 stability and embedding theorems we also find a c algebraic witness for a k hard equivalence relation the authors dedicate this article to the memory of greg hjorth

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

rejoinder local rademacher complexities and oracle inequalities in risk minimization of the true risk function f 3 f pf the first quantity of interest is the l2 diameter of this set d f 8 and the second one is the function fa f 8 that is equal to the expected supremum of the empirical process indexed by the differences g f g e f 8 these two functions are then combined in the expression n 8 t that has its roots in talagrand s concentration inequalities for empirical processes the j transform of n t which is just a way to write solu tions of fixed point type equations is then used to define the localized complexities that provide upper bounds on the excess risk under further assumptions such as mean variance relationships discussed in detail by shen and wang bartlett and mendelson also discuss this and call the function classes satisfying these relation ships bernstein classes these complexities can be redefined in terms of the local l2 continuity modulus of empirical processes since the rademacher process can be used as a data dependent bootstrap type estimate of the empirical process this approach also leads to data dependent local rademacher complexities the use of the whole minimal set is not the only possibility one can also look at its slices f 8 82 f 82 f 8 and define the excess risk bounds in terms of the accuracy of empirical approximation on the slices one can even make the slices really thin and look at g f pf inf pg 8 this was the approach taken by peter bartlett and shahar mendelson under an additional and relatively innocent assumption that the class f is star shaped they established excess risk bounds and also ratio type bounds in terms of complexities of such thin slices

sharp relaxation rates for plane waves of general reaction diffusion models it is a well known and classical result that spectrally stable traveling waves of a general reaction diffusion system in one spatial dimension are asymptotically stable with exponential relaxation rates in a series of works in the 1990s goodman trans am math soc 311 683 1989 kapitula trans am math soc 349 1901 1997 and xin commun partial differ equations 17 1889 1992 considered plane traveling waves for such systems and they have succeeded in showing asymptotic stability for such objects interestingly the estimates for the relaxation rates that they have exhibited are all algebraic and dimension dependent it was heuristically argued that as the spectral gap closes in dimensions n 2 algebraic rates are the best possible in this paper we revisit this issue we rigorously calculate the sharp relaxation rates in l based spaces both for the asymptotic phase and the radiation terms they are indeed algebraic but about twice better than the best ones obtained in these early works although this can be mostly attributed to the inefficiencies of using sobolev embeddings to control l norms by high order l2 based sobolev spaces finally we explicitly construct the leading order profiles both for the phase and for the radiation terms our approach relies on the method of scaling variables as introduced in the work of gallay and wayne arch ration mech anal 163 209 2002 and gallay and wayne philos trans r soc a 360 2155 2002 and in fact provides sharp relaxation rates in a class of weighted l2 spaces as well

countable sections for free actions of groups let the group g act on the set x we will call the set y a countable section if each gx n y is countable p forrest showed in 2 that if a locally compact separable metrizable group g acts freely on a standard measurable space x with quasi invariant probability measure u then there is a countable section y which is in and such that the set gy which is the image of g x y by a measurable function and therefore p measurable has pmeasure 1 in the terminology of 11 y is a complete measurable countable section we prove the following converse to forrest s result

the noncommutative geometry of graph c algebras i the index theorem we investigate conditions on a graph c algebra for the existence of a faithful semifinite trace using such a trace and the natural gauge action of the circle on the graph algebra we construct a smooth 1 summable semi finite spectral triple the local index theorem allows us to compute the pairing with k theory this produces invariants in the k theory of the fixed point algebra and these are invariants for a finer structure than the isomorphism class of c e

essential countability of treeable equivalence relations we establish a dichotomy theorem characterizing the circumstances under which a treeable borel equivalence relation e is essentially countable under additional topological assumptions on the treeing we in fact show that e is essentially countable if and only if there is no continuous embedding of e1 into e our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations and allow us to show that up to continuous kakutani embeddability there is a minimum borel function which is not essentially countable to one

stability of viscous scalar shock fronts in several dimensions we prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions the proof uses the integrated equation and an effective equation for the motion of the front itself we derive energy estimates that balance terms from the integrated equation with terms from the front motion equation in this paper we prove that viscous shock profiles for scalar conservation laws are stable in two or more space dimensions these multidimensional stability questions are separate from their one dimensional analogues because of the possibility of transverse instabilities such as those that occur in combustion fronts lu and in shock waves with phase changes the proof here is a rigorous version of arguments that are used to derive effective equations such as the kuramoto sivashinsky equation to describe the behavior of fronts the one dimensional stability for scalar conservation laws was proven by ii in and oleinik io using the integrated equation see below and a max imum principle another proof based on weighted norms and spectral theory for the linearized problem was given by sattinger s the multidimensional stability proof below has more in common with the stability proofs for systems of conservation laws in one space dimension begun by kawashima and mat sumura km and goodman go and completed by liu li these proofs use l2 energy estimates for the integrated equation we consider equations of the form ut f u x g u y uxx uyy where f u is a strictly convex function of u

endomorphisms of c algebras cross products and duality for compact groups on definit une nouvelle sorte de produit croise d une c algebre par des semigroupes d endomorphismes avec des proprietes reliees a la symetrie de permutation et a l existence de conjugues un groupe compact est alors defini intrinsequement par son action duale sur le produit croise cette construction donne une caracterisation des duaux de groupes compacts abstraits qui est hors d atteinte de la theorie de dualite de taunaka kein et est independante de cette theorie ceci resout le probleme de demontrer l existence d un groupe de jauge global compact en physique des particules etant seulement donnees les observables locales

semigroupoid c algebras abstract a semigroupoid is a set equipped with a partially defined associative operation given a semigroupoid we construct a c algebra o from it we then present two main examples of semigroupoids namely the markov semigroupoid associated to an infinite 0 1 matrix and the semigroupoid associated to a row finite higher rank graph without sources in both cases the semigroupoid c algebra is shown to be isomorphic to the algebras usually attached to the corresponding combinatorial object namely the cuntz krieger algebras and the higher rank graph c algebras respectively in the case of a higher rank graph d it follows that the dimension function d is superfluous for defining the corresponding c algebra

the brauer group of a locally compact groupoid we define the brauer group br g of a locally compact groupoid g to be the set of morita equivalence classes of pairs a consisting of an elementary c bundle a over g 0 satisfying fell s condition and an action of g on a by isomorphisms when g is the transformation groupoid x h then br g is the equivariant brauer group br h x in addition to proving that br g is a group we prove three isomorphism results first we show that if g and h are equivalent groupoids then br g and br h are isomorphic this generalizes the result that if g and h are groups acting freely and properly on a space x say g on the left and h on the right then br g x h and br h g x are isomorphic secondly we show that the subgroup br0 g of br g consisting of classes a with a having trivial dixmier douady invariant is isomorphic to a quotient e g of the collection tw g of twists over g finally we prove that br g is isomorphic to the inductive limit ext g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of the groups e g x where x varies over all principal g spaces x and g x is the imprimitivity groupoid associated to x

on the stability of traveling waves in weighted l spaces abstract i present new techniques and results concerning the stability of travelling waves to semilinear parabolic equations i am concerned with those problems in which the essential spectrum of the linearization about the wave contains 0 in these circumstances the stability results of d henry in lecture notes in mathematics vol 840 springer verlag new york 1981 and p bates and c jones dynamics rep 2 1989 1 38 are not applicable as the spectral information does not yield an exponential time decay estimate on the semigroup i show that under certain relatively easily computable conditions the wave is stable in polynomially weighted l spaces furthermore the rate at which the perturbation decays to the wave as t depends on the growth rate of the polynomial near infinity the method of proof uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates for the semigroup are derived through a new technique for estimating the resolvent

turbulence representations and trace preserving actions we establish criteria for turbulence in certain spaces of c algebra repre sentations and apply this to the problem of nonclassiability by countable structures for group actions on a standard atomless probability space x and on the hypernite ii1 factor r we also prove that the conjugacy action on the space of free actions of a countably innite amenable group on r is turbulent and that the conjugacy action on the space of ergodic measure preserving ows on x is generically turbulent

graph inverse semigroups their characterization and completion abstract graph inverse semigroups generalize the polycyclic inverse monoids and play an important role in the theory of c algebras in this paper we characterize such semigroups and show how they may be completed under suitable conditions to form what we call the cuntz krieger semigroup of the graph this semigroup is proved to be the ample semigroup of a topological groupoid associated with the graph and the semigroup analogue of the leavitt path algebra of the graph

hopf bifurcation and exchange of stability in diffusive media we consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses a continuous spectrum up to the imaginary axis for all values of the bifurcation parameter and that a pair of imaginary eigenvalues crosses the imaginary axis for a reaction diffusion convection system we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations prove the occurrence of a hopf bifurcation and the nonlinear stability of the bifurcating time periodic solutions again with respect to spatially localized perturbations

the classification problem for operator algebraic varieties and their multiplier algebras we study from the perspective of borel complexity theory the classification problem for multiplier algebras associated with operator algebraic varieties these algebras are precisely the multiplier algebras of irreducible complete nevanlinna pick spaces we prove that these algebras are not classifiable up to algebraic isomorphism using countable structures as invariants in order to prove such a result we develop the theory of turbulence for polish groupoids which generalizes hjorth s turbulence theory for polish group actions we also prove that the classification problem for multiplier algebras associated with varieties in a finite dimensional ball up to isometric isomorphism has maximum complexity among the essentially countable classification problems in particular this shows that blaschke sequences are not smoothly classifiable up to conformal equivalence via automorphisms of the disc

the classification of hypersmooth borel equivalence relations this paper is a contribution to the study of borel equivalence relations in standard borel spaces i e polish spaces equipped with their borel structure a class of such equivalence relations which has received particular attention is the class of hyperfinite borel equivalence relations these can be defined as the increasing unions of sequences of borel equivalence relations all of whose equivalence classes are finite or as it turns out equivalently those induced by the orbits of a single borel auto morphism hyperfinite equivalence relations have been classified in djk under two notions of equivalence borel bi reducibility and borel isomorphism

the classification problem for von neumann factors abstract we prove that it is not possible to classify separable von neumann factors of types ii 1 ii or iii 0 1 up to isomorphism by a borel measurable assignment of countable structures as invariants in particular the isomorphism relation of type ii 1 factors is not smooth we also prove that the isomorphism relation for von neumann ii 1 factors is analytic but is not borel

non classification of cartan subalgebras for a class of von neumann algebras abstract we study the complexity of the classification problem for cartan subalgebras in von neumann algebras we construct a large family of ii1 factors whose cartan subalgebras up to unitary conjugacy are not classifiable by countable structures providing the first such examples additionally we construct examples of ii1 factors whose cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures finally we show directly that the cartan subalgebras of the hyperfinite ii1 factor up to unitary conjugacy are not classifiable by countable structures and deduce that the same holds for any mcduff ii1 factor with at least one cartan subalgebra

the coset equivalence relation and topologies on subgroups the paper studies the structure of the homogeneous space g h for g a polish group and h g a borel not necessarily closed subgroup of g from the point of view of the theory of definable equivalence relations it makes a connection between the complexity of the natural it coset equivalence relation associated with g h and it polishability of h that is the possibility of introducing a polish group topology on h respecting its borel structure in particular it is proved that if h is an abelian borel subgroup of a polish group g then either h is polishable or bbb e 1 continuously embeds into the coset equivalence relation induced by h on g the same conclusion is shown to hold if h is an increasing union of a sequence of polishable subgroups of g

multidimensional stability of planar travelling waves the multidimensional stability of planar travelling waves for systems of reaction diffusion equations is considered in the case that the diffusion matrix is the identity it is shown that if the wave is exponentially orbitally stable in one space dimension then it is stable for x rn n 2 furthermore it is shown that the perturbation of the wave decays like t n 1 4 as t the result is proved via an application of linear semigroup theory

congruences on graph inverse semigroups abstract graph inverse semigroups were defined by ash and hall in 1975 they found necessary and sufficient conditions for the semigroups to be congruence free in this paper we give a classification of congruences on a graph inverse semigroup in terms of the underlying graph as a consequence we show that the inverse graph semigroup of a finite graph is congruence noetherian

the space of composants of an indecomposable continuum abstract the family of all composants of an indecomposable continuum is studied we investigate the equivalence relation induced on an indecomposable continuum by its partition into composants we show that up to borel bireducibility such an equivalence relation can be of only two types e 0 and e 1 the e 0 type being simple and the e 1 type being complicated as a consequence of this we show that each indecomposable continuum carries a borel probability measure which assigns 0 to each composant and 0 or 1 to each borel set which is the union of a family of composants in particular it follows that there is no borel transversal for the family of all composants this solves an old problem in the theory of continua we prove that all hereditarily indecomposable continua are of the complicated type that is they fall into the e 1 type we analyze the properties of being of type e 1 or of type e 0 we show using effective descriptive set theory that the first of these properties is analytic and so the second one is coanalytic we construct examples of continua of both types in fact we produce a family of indecomposable continua and use it to prove that these properties are complete analytic and complete coanalytic respectively hence non borel so they do not admit simple topological characterizations we also use continua from this family to show that an indecomposable continuum may be of type e 1 only because of the behavior of composants on a small subset of the continuum this in particular shows that certain natural approaches to solving kuratowski s problem on generic ergodicity of the composant equivalence relation will not work we finish with some open problems

borel complexity and automorphisms of c algebras abstract we show that if a is z o 2 o a uhf algebra of infinite type or the tensor product of a uhf algebra of infinite type and o then the conjugation action aut a aut a is generically turbulent for the point norm topology we moreover prove that if a is either i a separable c algebra which is stable under tensoring with z or k or ii a separable ii 1 factor which is mcduff or a free product of ii 1 factors then the automorphisms of a are not classifiable up to conjugacy by countable structures

on the classification of positions and complex structures in banach spaces abstract a topological setting is defined to study the complexities of the relation of equivalence of embeddings or position of a banach space into another and of the relation of isomorphism of complex structures on a real banach space the following results are obtained a if x is not uniformly finitely extensible then there exists a space y for which the relation of position of y inside x reduces the relation e 0 and therefore is not smooth b the relation of position of l p inside l p or inside l p p 2 reduces the relation e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group c the relation of position of a space inside another can attain the maximum complexity e max d there exists a subspace of l p 1 p 2 on which isomorphism between complex structures reduces e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group

polish groupoids and functorial complexity we introduce and study the notion of functorial borel complexity for polish groupoids such a notion aims to measure the complexity of classifying the objects of a category in a constructive and functorial way in the particular case of principal groupoids such a notion coincides with the usual borel complexity of equivalence relations our main result is that on one hand for polish groupoids with an essentially treeable orbit equivalence relation functorial borel complexity coincides with the borel complexity of the associated orbit equivalence relation on the other hand for every countable equivalence relation e that is not treeable there are polish groupoids with different functorial borel complexity both having e as orbit equivalence relation in order to obtain such a conclusion we generalize some fundamental results about the descriptive set theory of polish group actions to actions of polish groupoids answering a question of arlan ramsay these include the becker kechris results on polishability of borel g spaces existence of universal borel g spaces and characterization of borel g spaces with borel orbit equivalence relations

a class of c algebras generalizing both graph algebras and homeomorphism c algebras i fundamental results we introduce a new class of c algebras which is a generalization of both graph algebras and homeomorphism c algebras this class is very large and also very tractable we prove the so called gauge invariant unique ness theorem and the cuntz krieger uniqueness theorem and compute the k groups of our algebras

the structure of gauge invariant ideals of labelled graph c algebras abstract in this paper we consider the gauge invariant ideal structure of a c algebra c e l b associated to a set finite receiver set finite and weakly left resolving labelled space e l b where l is a labelling map assigning an alphabet to each edge of the directed graph e with no sinks it is obtained that if an accommodating set b is closed under relative complements there is a one to one correspondence between the set of all hereditary saturated subsets of b and the gauge invariant ideals of c e l b for this we introduce a quotient labelled space e l b r arising from an equivalence relation r on b and show the existence of the c algebra c e l b r generated by a universal representation of e l b r finally we give necessary and sufficient conditions for simplicity of certain labelled graph c algebras

measure reducibility of countable borel equivalence relations we show that every basis for the countable borel equivalence relations strictly above mathbb e 0 under measure reducibility is uncountable thereby ruling out natural generalizations of the glimm effros dichotomy we also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base using arguments substantially simpler than those previously employed

on the complexity of the uniform homeomorphism relation between separable banach spaces we investigate the uniform homeomorphism relation between separable banach spaces and the related relation of local equivalence we completely characterize the descriptive complexity of local equivalence in the borel reducibility hierarchy this also provides a lower bound for the complexity of the uniform homeomorphism

completeness of the isomorphism problem for separable c algebras this paper studies the descriptive set theoretical complexity of the isomorphism problem for separable c algebras we prove that the isomorphism problem for separable simple ai c algebras is complete in the class of orbit equivalence relations this means that any isomorphism problem arising from a continuous action of a separable completely metrizable group can be reduced to the isomorphism of simple separable ai c algebras

risk bounds for cart classifiers under a margin condition non asymptotic risk bounds for classification and regression trees cart classifiers are obtained in the binary supervised classification framework under a margin assumption on the joint distribution of the covariates and the labels these risk bounds are derived conditionally on the construction of the maximal binary tree and allow to prove that the linear penalty used in the cart pruning algorithm is valid under the margin condition it is also shown that conditionally on the construction of the maximal tree the final selection by test sample does not alter dramatically the estimation accuracy of the bayes classifier

unitary equivalence of automorphisms of separable c algebras abstract we prove that the automorphisms of any separable c algebra that does not have continuous trace are not classifiable by countable structures up to unitary equivalence this implies a dichotomy for the borel complexity of the relation of unitary equivalence of automorphisms of a separable unital c algebra such relation is either smooth or not even classifiable by countable structures

leavitt r algebras over countable graphs embed into l2 r abstract for a commutative ring r with unit we show that the leavitt path algebra l r e of a graph e embeds into l 2 r precisely when e is countable before proving this result we prove a generalised cuntz krieger uniqueness theorem for leavitt path algebras over r

strength of convergence in the orbit space of a groupoid abstract let g be a second countable locally compact hausdorff groupoid with a haar system and let x n be a sequence in the unit space g 0 of g we show that the notions of strength of convergence of x n in the orbit space g 0 g and measure theoretic accumulation along the orbits are equivalent ways of realising multiplicity numbers associated to a sequence of induced representation of the groupoid c algebra

the primitive ideals of the cuntz krieger algebra of a row finite higher rank graph with no sources abstract we catalogue the primitive ideals of the cuntz krieger algebra of a row finite higher rank graph with no sources each maximal tail in the vertex set has an abelian periodicity group of finite rank at most that of the graph the primitive ideals in the cuntz krieger algebra are indexed by pairs consisting of a maximal tail and a character of its periodicity group the cuntz krieger algebra is primitive if and only if the whole vertex set is a maximal tail and the graph is aperiodic

the structure of hyperfinite borel equivalence relations we study the structure of the equivalence relations induced by the orbits of a single borel automorphism on a standard borel space we show that any two such equivalence relations which are not smooth i e do not admit borel selectors are borel embeddable into each other this utilizes among other things work of effros and weiss using this and also results of dye varadarajan and recent work of nadkarni we show that the cardinality of the set of ergodic invariant measures is a complete invariant for borel isomorphism of aperiodic nonsmooth such equivalence relations in particular since the only possible such cardinalities are the finite ones countable infinity and the cardinality of the continuum there are exactly countably infinitely many isomorphism types canonical examples of each type are also discussed

dichotomy theorems for families of non cofinal essential complexity we prove that for every borel equivalence relation e either e is borel reducible to mathbb e 0 or the family of borel equivalence relations incompatible with e has cofinal essential complexity it follows that if f is a borel equivalence relation and cal f is a family of borel equivalence relations of non cofinal essential complexity which together satisfy the dichotomy that for every borel equivalence relation e either e in cal f or f is borel reducible to e then cal f consists solely of smooth equivalence relations thus the dichotomy is equivalent to a known theorem

on c algebras associated with c correspondences abstract we study c algebras arising from c correspondences which were introduced by the author we prove the gauge invariant uniqueness theorem and obtain conditions for our c algebras to be nuclear exact or satisfy the universal coefficient theorem we also obtain a 6 term exact sequence of k groups involving the k groups of our c algebras

the suspension of a graph and associated c algebras abstract given a directed graph e we construct for each real number l a quiver whose vertex space is the topological realisation of e and whose edges are directed paths of length l in the vertex space these quivers are not topological graphs in the sense of katsura nor topological quivers in the sense of muhly and tomforde we prove that when l 1 and e is finite the infinite path space of the associated quiver is homeomorphic to the suspension of the one sided shift of e we call this quiver the suspension of e we associate both a toeplitz algebra and a cuntz krieger algebra to each of the quivers we have constructed and show that when l 1 the cuntz krieger algebra admits a natural faithful representation on the l 2 space of the suspension of the one sided shift of e for graphs e in which sufficiently many vertices both emit and receive at least two edges and for rational values of l we show that the toeplitz algebra and the cuntz krieger algebra of the associated quiver are homotopy equivalent to the toeplitz algebra and cuntz krieger algebra respectively of a graph that can be regarded as encoding the lth higher shift associated to the one sided shift space of e

finite digraphs and kms states abstract the paper contains a description of the kms states of a generalized gauge action on the c algebra of a finite graph

turbulence and araki woods factors abstract using baire category techniques we prove that araki woods factors are not classifiable by countable structures as a result we obtain a far reaching strengthening as well as a new proof of the well known theorem of woods that the isomorphism problem for itpfi factors is not smooth we derive as a consequence that the odometer actions of z that preserve the measure class of a finite non atomic product measure are not classifiable up to orbit equivalence by countable structures

semigroups of non unital endomorphisms of c algebras and compact group duality we show that the fixed point algebra og of the cuntz algebra on n 2 3 under the canonical action of a strongly compact group g of unitaries on the generating hubert space h is simple if the defining representation of g is a direct sum of finite dimensional representations whose determinant takes values in finite subgroups of the circle we use this result to define the doplicher roberts cross product of a unital c algebra a with centre cl by a semigroup of not necessarily unital endomorphisms generated by a countable orthogonal collection p i i n fulfilling natural conditions as an application we give necessary and sufficient conditions in order that a p is isomorphic to o g with the regular representation of a compact and metrizable group g and the set of endomorphisms induced by an irreducible decomposition of

polish group actions and the vaught conjecture we consider the topological vaught conjecture if a polish group g acts continuously on a polish space s then s has either countably many or perfectly many orbits we show 1 the conjecture is true for abelian groups 2 the conjecture is true whenever g s are recursively presented the action of g is recursive and for x e s the orbit of x is of borel multiplicative rank wx assertion 1 holds also for analytic s specializing g to a closed subgroup of co we prove that nonempty invariant borel sets not having perfectly many orbits have orbits of about the same borel rank an upper bound is derived for the borel rank of orbits when the set of orbits is finite