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

extensions of automorphisms and gauge symmetries we characterize the automorphisms of ac algebra open image in new window which extend to automorphisms of the crossed product open image in new window by a compact group dual the case where the inclusion open image in new window is equipped with a group of automorphisms commuting with the dual action is also treated these results are applied to the analysis of broken gauge symmetries in quantum field theory to draw conclusions on the structure of the degenerate vacua on the field algebra

superselection in the presence of constraints superselection and constraints occur together in many gauge theories and here we begin a study of such systems our main focus will be to analyze compatibility questions between constraining and superselection and we will develop an example modelled on qed in which our framework is realized we proceed from a generalization of doplicher roberts superselection theory to the case of the nontrivial center and a set of dirac quantum constraints and find conditions under which the superselection structures will survive constraining in some form this involves an analysis of the restriction and factorization of superselection structures

oscillations in a class of difference equations a class of linear difference equations is considered and a necessary and sufficient condition for the oscillation of solutions is established this condition is referred to the roots of the associated characteristic equation

on the assouad dimension of self similar sets with overlaps it is known that unlike the hausdor dimension the assouad dimension of a self similar set can exceed the similarity dimension if there are overlaps in the construction our main result is the following precise dichotomy for self similar sets in the line either the weak separation property is satised in which case

necessary and sufficient conditions for oscillations of higher order delay differential equations we obtain necessary and sufficient conditions under which ail solutions of certain n th order delay differential equations oscillate when n is odd and necessary and sufficient conditions under which all bounded solutions oscillate when n is even the conditions derived involve the characteristic equation of the delay differential equation

nica toeplitz algebras associated with product systems over right lcm semigroups abstract we prove uniqueness of representations of nica toeplitz algebras associated to product systems of c correspondences over right lcm semigroups by applying our previous abstract uniqueness results developed for c precategories our results provide an interpretation of conditions identified in work of fowler and fowler raeburn and apply also to their crossed product twisted by a product system in the new context of right lcm semigroups as well as to a new doplicher roberts type c algebra associated to the nica toeplitz algebra as a derived construction we develop nica toeplitz crossed products by actions with completely positive maps this provides a unified framework for nica toeplitz semigroup crossed products by endomorphisms and by transfer operators we illustrate these two classes of examples with semigroup c algebras of right and left semidirect products

the c algebra of a vector bundle and fields of cuntz algebras abstract we study the cuntz pimsner algebra associated with the module of continuous sections of a hilbert bundle and prove that it is a continuous bundle of cuntz algebras furthermore we assign to bundles of cuntz algebras carrying a global circle action a class in the representable kk group of the zero grade bundle we explicitly compute such class for the cuntz pimsner algebra of a vector bundle

on unitary representation theories of compact quantum groups categorical structure of unitary representation of compact quantum groups is studied with relation to a metrical structure encountered in the monoidal category of bimodules of finite jones index

finite type in measure sense for self similar sets with overlaps for self similar sets with overlaps we introduce a notion named the finite type in measure sense and reveal its intrinsic relationships with the weak separation condition and the generalized finite type

oscillations of first order linear retarded differential equations the oscillatory behavior of differential equations with deviating arguments has been studied by many authors in particular several authors have studied the oscillations caused by the deviating arguments the study of such oscillations is a relatively new field and is very interesting in applications for some contributions in this area see the papers by arino gyori and jawhari 11 bowcock and yu 2 gyiiri 4 5 j hunt and yorke 6 kulenovic ladas and meimaridou 7 ladas s ladas sficas and stavroulakis 9 lo ladas and stavroulakis 111 sugie 141 tramov 15 and yan 16 we also refer to the recent book by ladde lakshmikantham and zhang 123 the purpose of this paper is to investigate the oscillatory behavior of the solutions of first order linear retarded differential equations with variable coefftcients and nonconstants delays consider the first order retarded differential equation

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

compact lie groups associated with endomorphisms of c algebras on donne des conditions sur un endomorphisme d une c algebre 7b a avec unite et centre trivial en termes d operateurs d entrelacements pour des puissances de cet endomorphisme qui suffisent a determiner un groupe de lie compact g et une action d un g dual sur 7b a

a necessary and sufficient condition for the solutions of a functional differential equation to be oscillatory or tend to zero abstract it is shown that every solution of the nonhomogeneous functional differential equation d dt x t px t q t g x t f t wheref q c t 0 0 0 p 0 forx 0 gis nondecreasing lipschitzian and satisfy a sublinear condition 0 k dx g x and f s ds is either oscillatory or tends to zero asymptoticallyif and only if t q s ds

oscillations of advanced differential inequalities on considere le comportement oscillatoire d inegalites differentielles avancees de la forme 1 2 x n t sign x t p t i 1 m x g i t i et 1 z x n t sign x t i 1 m p i t x g i t ou n 2 z 1 2 n n 1 i i 1 m sont des nombres non negatifs avec 1 m 1 les fonctions p p i r r 0 sont integrables sur chaque segment fini et ne sont pas identiquement nulles dans tout voisinage de l infini

systems with outer constraints gupta bleuler electromagnetism as an algebraic field theory since there are some important systems which have constraints not contained in their field algebras we develop here in ac context the algebraic structures of these the constraints are defined as a groupg acting as outer automorphisms on the field algebra g aut g inn and we find that the selection ofg invariant states on is the same as the selection of states onm g m g mathop times limits alpha f by ug 1 g g whereug m g m g mathop times limits alpha f are the canonical elements implementing g these states are taken as the physical states and this specifies the resulting algebraic structure of the physics inm g m g mathop times limits alpha f and in particular the maximal constraint free physical algebra a nontriviality condition is given for to exist and we extend the notion of a crossed product to deal with a situation whereg is not locally compact this is necessary to deal with the field theoretical aspect of the constraints next thec algebra of the ccr is employed to define the abstract algebraic structure of gupta bleuler electromagnetism in the present framework the indefinite inner product representation structure is obtained and this puts gupta bleuler electromagnetism on a rigorous footing finally as a bonus we find that the algebraic structures just set up provide a blueprint for constructive quadratic algebraic field theory

oscillation of third order trinomial delay differential equations the purpose of this paper is to study oscillation and asymptotic behavior of solutions to a third order linear delay differential equation y t p t y t q t y tau t 0 new comparison th

oscillation theorems for neutral difference equations abstract oscillation criteria for first order difference equations of the form x n q n x n 0 and x n px n k q n x n f n are established here xn xn 1 xn is the forward difference operator 1 p is a real number and k are nonnegative integers qn fn are sequences of nonnegative real numbers and r r is continuous oscillation criteria for nonlinear difference equations of sublinear as well as superlinear type are also established

the c0 x algebra of a net and index theory abstract given a connected and locally compact hausdorff space x with a good base k we assign in a functorial way a c 0 x algebra to any precosheaf of c algebras a defined over k afterwards we consider the representation theory and the kasparov k homology of a and interpret them in terms respectively of the representation theory and the k homology of the associated c 0 x algebra when a is an observable net over the spacetime x in the sense of algebraic quantum field theory this yields a geometric description of the recently discovered representations affected by the topology of x

oscillations caused by several retarded and advanced arguments in this paper we study the oscillatory behavior of equations of the forms y t i 1npiy t i 0 and y t i 1npiy t i 0 where piand i i 1 2 n are positive constants we prove that each one of the following conditions 1 pi i1 e for some i i 1 2 n 2 i 1npi 1 e where min 1 2 n 3 i 1npi 1 n i 1n i 1 e or 4 1 n i 1n pi i 1 2 21 e implies that every solution of and oscillates a generalization in the case where the coefficientspi i 1 2 n are positive and continuous functions is also presented

morita homotopy theory of c categories abstract in this article we establish the foundations of the morita homotopy theory of c categories concretely we construct a cofibrantly generated simplicial symmetric monoidal quillen model structure denoted by m mor on the category c 1 cat of small unital c categories the weak equivalences are the morita equivalences and the cofibrations are the functors which are injective on objects as an application we obtain an elegant description of brown green rieffel s picard group in the associated homotopy category ho m mor we then prove that ho m mor is semi additive by group completing the induced abelian monoid structure at each hom set we obtain an additive category ho m mor 1 and a composite functor c 1 cat ho m mor ho m mor 1 which is characterized by two simple properties inversion of morita equivalences and preservation of all finite products finally we prove that the classical grothendieck group functor becomes co represented in ho m mor 1 by the tensor unit object

presheaves of superselection structures in curved spacetimes we show that superselection structures on curved spacetimes that are expected to describe quantum charges affected by the underlying geometry are categories of sections of presheaves of symmetric tensor categories when an embedding functor is given the superselection structure is a tannaka type dual of a locally constant group bundle which hence becomes a natural candidate for the role of the gauge group indeed we show that any locally constant group bundle with suitable structure group acts on a net of c algebras fulfilling normal commutation relations on an arbitrary spacetime we also give examples of gerbes of c algebras defined by wightman fields and constructed using projective representations of the fundamental group of the spacetime which we propose as solutions for the problem that existence and uniqueness of the embedding functor are not guaranteed

from path representations to global morphisms for a class of minimal models we construct global observable algebras and global dhr morphisms for the virasoro minimal models with central charge c 2 q q odd to this end we pass from the irreducible highest weight modules to path representations which involve fusion graphs of the c 2 q models the paths have an interpretation in terms of quasi particles which capture some structure of non conformal perturbations of the c 2 q models the path algebras associated to the path spaces serve as algebras of bounded observables global morphisms which implement the superselection sectors are constructed using quantum symmetries we argue that there is a canonical semi simple quantum symmetry algebra for each quasi rational cft in particular for the c 2 q models these symmetry algebras act naturally on the path spaces which allows to define a global field algebra and covariant multiplets therein

multifractal measures and a weak separation condition abstract we define a new separation property on the family of contractive similitudes that allows certain overlappings this property is weaker than the open set condition of hutchinson it includes the well known class of infinite bernoulli convolutions associated with the p v numbers and the solutions of the two scale dilation equations our main purpose in this paper is to prove the multifractal formalism under such condition

oscillation criteria of a certain class of third order nonlinear delay differential equations with damping abstract in this paper we are concerned with the oscillation of third order nonlinear delay differential equations of the form r 2 t r 1 t y p t y q t f y g t 0 by using a generalized riccati transformation and integral averaging technique we establish some new sufficient conditions which insure that any solution of this equation oscillates or converges to zero in particular several examples are given to illustrate the importance of our results

galois theory for braided tensor categories and the modular closure abstract given a braided tensor category c with conjugate dual objects and irreducible unit together with a full symmetric subcategory s we define a crossed product c s this construction yields a tensor category with conjugates and an irreducible unit a category is a category enriched over vect c with positive operation a galois correspondence is established between intermediate categories sitting between c and c s and closed subgroups of the galois group gal c s c aut c c s of c the latter being isomorphic to the compact group associated with s by the duality theorem of doplicher and roberts denoting by d c the full subcategory of degenerate objects i e objects which have trivial monodromy with all objects of c the braiding of c extends to a braiding of c s iff s d under this condition c s has no non trivial degenerate objects iff s d if the original category c is rational i e has only finitely many isomorphism classes of irreducible objects then the same holds for the new one the category c c d is called the modular closure of c since in the rational case it is modular i e gives rise to a unitary representation of the modular group sl 2 z if all simple objects of s have dimension one the structure of the category c s can be clarified quite explicitly in terms of group cohomology

on self similar sets with overlaps and inverse theorems for entropy we study the dimension of self similar sets and measures on the line we show that if the dimension is less than the generic bound of minf1 sg where s is the similarity dimension then there are superexponentially close cylinders at all small enough scales this is a step towards the conjecture that such a dimension drop implies exact overlaps and conrms it when the generating similarities have algebraic coecients as applications we prove furstenberg s conjecture on projections of the one dimensional sierpinski gasket and achieve some progress on the bernoulli convolutions problem and more generally on problems about parametric families of self similar measures the key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale 2 n has only a small amount of growth under convolution

higher dimensional algebra ii 2 hilbert spaces abstract a 2 hilbert space is a category with structures and properties analogous to those of a hilbert space more precisely we define a 2 hilbert space to be an abelian category enriched over hilb with a structure conjugate linear on the hom sets satisfying fg h g f h f hg we also define monoidal braided monoidal and symmetric monoidal versions of 2 hilbert spaces which we call 2 h algbebras braided 2 h algebras and symmetric 2 h algbebras and we describe the relation between these and tangles in two three and four dimensions respectively we prove a generalized doplicher roberts theorem starting that every symmetric 2 h algebra is equivalent to the category rep g of continuous unitary finite dimensional representations of some compact supergroupoid g the equivalence is given by a categorified version of the gelfand transform we also construct a categorified version of the fourier transform when g is a compact abelian group finally we characterize rep g by its universal properties when g is a compact classical group for example rep u n is the free connected symmetric 2 h algebra on one even object of dimension n

diagram projection rules for recoupling diagrams in the racah wigner category this paper gives three important techniques for recoupling coefficients of an arbitrary number or irreducible representations factorization as a product of smaller degree recoupling coefficients diagram projection projecting a recoupling diagram to obtain constraints on recoupling coefficients and primitive expansion expressing a recoupling coefficient in terms of primitive recoupling coefficients the first technique reduces a recoupling coefficient to its essential and simplest form the second is many faceted allowing one to work at the recoupling level importantly recoupling diagrams can be used to construct a complete set of constraints on the recoupling coefficients the third allows large degree recoupling coefficients to be expressed in terms of primitive recoupling coefficients an example of this is the fact that the 9j symbol is defined by the 3j and 6j symbols

liberation of orthogonal lie groups abstract we show that under suitable assumptions we have a one to one correspondence between classical groups and free quantum groups in the compact orthogonal case we classify the groups under correspondence with the result that there are exactly 6 of them o n s n h n b n s n b n we investigate the representation theory aspects of the correspondence with the result that for o n s n h n b n this is compatible with the bercovici pata bijection finally we discuss some more general classification problems in the compact orthogonal case notably with the construction of a new quantum group

oscillation of third order differential equations with noncanonical operators abstract new oscillation criteria for third order delay differential equations with noncanonical operators are presented contrary to existing results oscillation of the studied equation is attained via only two conditions our criteria not only improve extend and significantly simplify existing ones but the newly proposed approach could hopefully serve as a reference in the less developed theory of noncanonical equations of higher order the importance of the results obtained is illustrated via euler type equations

the fusion rules of some free wreath product quantum groups and applications abstract in this article we find the fusion rules of the free wreath product quantum groups s n for any discrete group to do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify the irreducible corepresentations we then apply the knowledge of the fusion rules to prove in most cases several operator algebraic properties of the associated reduced c algebras such as simplicity and uniqueness of the trace we also prove that the associated von neumann algebra is a full type ii 1 factor and that the dual of s n has the haagerup approximation property for all finite groups

necessary and sufficient conditions for oscillations of neutral differential equations abstract consider the neutral delay differential equation x t px t qx t 0 t t 0 where q and are positive constants while p is a real parameter then a necessary and sufficient condition for all solutions of to oscillate is that the characteristic equation p e qe 0 of has no real roots

oscillation of third order nonlinear damped delay differential equations this paper is concerned with the oscillation of certain third order nonlinear delay differential equations with damping we give new characterizations of oscillation of the third order equation in terms of oscillation of a related well studied second order linear differential equation without damping we also establish new oscillation results for the third order equation by using the integral averaging technique due to philos numerous examples are given throughout

topological aperiodicity for product systems over semigroups of ore type we prove a version of uniqueness theorem for cuntz pimsner algebras of discrete product systems over semigroups of ore type to this end we introduce doplicher roberts picture of cuntz pimsner algebras and the semigroup dual to a product system of regular c correspondences under a certain aperiodicity condition on the latter we obtain the uniqueness theorem and a simplicity criterion for the algebras in question these results generalize the corresponding ones for crossed products by discrete groups due to archbold and spielberg and for exel s crossed products due to exel and vershik they also give interesting conditions for topological higher rank graphs and p graphs and apply to the new cuntz c algebra qn arising from the ax b semigroup over n

a new duality theory for compact groups the tannaka krein duality theory characterizes the category g of finite dimensional continuous unitary representations of a compact group as a subcategory of the category of hilbert spaces we prove a more powerful result characterizing g as an abstract category every strict symmetric monoidalc category with conjugates which has subobjects and direct sums and for which thec algebra of endomorphisms of the monoidal unit reduces to the complex numbers is isomorphic to a category g for a compact groupg unique up to isomorphism

continuous fields of c algebras arising from extensions of tensor c categories abstract the notion of extension of a given c category c by a c algebra a is introduced in the commutative case a c the objects of the extension category are interpreted as fiber bundles over of objects belonging to the initial category it is shown that the doplicher roberts algebra dr algebra in the following associated to an object in the extension of a strict tensor c category is a continuous field of dr algebras coming from the initial one in the case of the category of the hermitian vector bundles over the general result implies that the dr algebra of a vector bundle is a continuous field of cuntz algebras some applications to pimsner c algebras are given

completely positive maps of the cuntz algebras we construct a covariant functor from the category whose objects are the complex infinite dimensional separable hilbert spaces and whose morphisms are the contractions into the category whose objects are the unitalc algebras and whose morphisms are the completely positive identity preserving maps thec algebras involved are isomorphic to the cuntz algebra o and our functor extends the canonical action of the group of unitaries on the generating hilbert space our result cannot apply to on n

bundles of c categories ii c dynamical systems and dixmier douady invariants we introduce a cohomological invariant arising from a class in nonabelian cohomology this invariant generalises the dixmier douady class and encodes the obstruction to a c algebra bundle being the fixed point algebra of a gauge action as an application the duality breaking for group bundles vs tensor c categories with nonsimple unit is discussed in the setting of nistor troitsky gauge equivariant k theory there is a map assigning a nonabelian gerbe to a tensor category and triviality of the gerbe is equivalent to the existence of a dual group bundle at the c algebraic level this corresponds to studying c algebra bundles with fibre a fixed point algebra of the cuntz algebra and in this case our invariant describes the obstruction to finding an embedding into the cuntz pimsner algebra of a vector bundle

algebraic quantization of systems with a gauge degeneracy systems with a gauge degeneracy are characterized either by supplementary conditions or by a set of generators of gauge transformations or by a set of constraints deriving from dirac s canonical constraint method these constraints can be expressed either as conditions on the field algebra or on the states on in ac algebra framework we show that the state conditions give rise to a factor algebra of a subalgebra of the field algebra this factor algebra is free of state conditions in this formulation we show also that the algebraic conditions can be treated in the same way as the state conditions the connection between states on and states on is investigated further within this framework as is also the set of transformations which are compatible with the set of constraints it is also shown that not every set of constraints can give rise to a nontrivial system finally as an example the abstract theory is applied to the electromagnetic field and this treatment can be generalized to all systems of bosons with linear constraints the question of dynamics is not discussed

a generalized finite type condition for iterated function systems abstract we study iterated function systems ifss of contractive similitudes on r d with overlaps we introduce a generalized finite type condition which extends a more restrictive condition in s m ngai y wang hausdorff dimension of self similar sets with overlaps j london math soc 2 63 3 2001 655 672 and allows us to include some ifss of contractive similitudes whose contraction ratios are not exponentially commensurable we show that the generalized finite type condition implies the weak separation property under this condition we can identify the attractor of the ifs with that of a graph directed ifs and by modifying a setup of mauldin and williams r d mauldin s c williams hausdorff dimension in graph directed constructions trans amer math soc 309 1988 811 829 we can compute the hausdorff dimension of the attractor in terms of the spectral radius of certain weighted incidence matrix

conjugation in representation categories of multiplicative unitaries and their actions on c algebras abstract a conjugation functor f on a full subcategory of r v the representation category of a multiplicative unitaryv is defined ifvhas a conjugate it is also regular and the domain of f is all r v examples of selfconjugate multiplicative unitaries are discussed a coaction of the hopfc algebra associated withvon the cuntz algebra o dis canonically defined by a unitary objectwof r v acting on ad dimensional hilbert space as in the group action case ifd andwbelongs to the domain of f ergodic coactions are often characterized by the absence of finite dimensional subrepresentations ofw furthermore model actions of compact quantum group duals onc algebras are defined

quantum charges and spacetime topology the emergence of new superselection sectors a new form of superselection sectors of topological origin is developed by that it is meant a new investigation that includes several extensions of the traditional framework of doplicher haag and roberts in local quantum theories at first we generalize the notion of representations of nets of c algebras then we provide a brand new view on selection criteria by adopting one with a strong topological flavour we prove that it is coherent with the older point of view hence a clue to a genuine extension in this light we extend roberts cohomological analysis to the case where 1 cocycles bear non trivial unitary representations of the fundamental group of the spacetime equivalently of its cauchy surface in the case of global hyperbolicity a crucial tool is a notion of group von neumann algebras generated by the 1 cocycles evaluated on loops over fixed regions one proves that these group von neumann algebras are localized at the bounded region where loops start and end and to be factorial of finite type i all that amounts to a new invariant in a topological sense which can be defined as the dimension of the factor we prove that any 1 cocycle can be factorized into a part that contains only the charge content and another where only the topological information is stored this second part much resembles what in literature is known as geometric phases indeed by the very geometrical origin of the 1 cocycles that we discuss in the paper they are essential tools in the theory of net bundles and the topological part is related to their holonomy content at the end we prove the existence of net representations

oscillatory behavior in linear retarded functional differential equations abstract this paper is devoted to the investigation of when all solutions of autonomous and nonautonomous linear retarded functional differential equations are oscillatory