m n coherent pairs of linear functionals and jacobi matrices abstract a pair of regular linear functionals u v in the linear space of polynomials with complex coefficients is said to be an m n coherent pair of order m if their corresponding sequences of monic orthogonal polynomials p n x n 0 and q n x n 0 satisfy a structure relation i 0 m a i n p n m i m x i 0 n b i n q n i x n 0 where m n and m are non negative integers a i n n 0 0 i m and b i n n 0 0 i n are sequences of complex numbers such that a m n 0 if n m b n n 0 if n n and a i n b i n 0 if i n when m 1 u v is called an m n coherent pair in this work we give a matrix interpretation of m n coherent pairs of linear functionals indeed an algebraic relation between the corresponding monic tridiagonal jacobi matrices associated with such linear functionals is stated as a particular situation we analyze the case when one of the linear functionals is classical finally the relation between the jacobi matrices associated with m n coherent pairs of linear functionals of order m and the hessenberg matrix associated with the multiplication operator in terms of the basis of monic polynomials orthogonal with respect to the sobolev inner product defined by the pair u v is deduced

orthogonal polynomials generated by a linear structure relation inverse problem abstract let p n n and q n n be two sequences of monic polynomials linked by a type structure relation such as q n x r n q n 1 x p n x s n p n 1 x t n p n 2 x where r n n s n n and t n n are sequences of complex numbers first we state necessary and sufficient conditions on the parameters such that the above relation becomes non degenerate when both sequences p n n and q n n are orthogonal with respect to regular moment linear functionals u and v respectively second assuming that the above relation is non degenerate and p n n is an orthogonal sequence we obtain a characterization for the orthogonality of the sequence q n n in terms of the coefficients of the polynomials and which appear in the rational transformation in the distributional sense u v some illustrative examples of the developed theory are presented

systemes dynamiques non commutatifs et moyennabilite on introduit la notion de fonction de type positif relativement a un systeme dynamique c dont les proprietes sont analogues a celles des fonctions de type positif classiques on montre que dans le cas d un groupe discret g agissant sur une algebre de von neumann m la moyennabilite de l action est equivalente a l existence d une suite bornee de fonctions de type positif sur g a support fini a valeurs dans le centre z m de m qui converge vers 1 pour la topologie de la convergence simple ultrafaible

structure relations for orthogonal polynomials on the unit circle structure relations for orthogonal polynomials with respect to hermitian linear functionals are studied firstly we prove that semi classical orthogonal polynomials satisfy structure relations of the following type k 0s1 n kpn s1 k k 0s2 n kzkpn 1 k k 0r1 n kpn s1 k 1 k 0r2 n kpn r2 k where s1 s2 r1 r2 are integers specified in the text pn is the reversed polynomial of pn pn 1 pn 1 n 1 and n k n k n k n k are complex numbers then we study the semi classical character of sequences of orthogonal polynomials rn pn connected through a structure relation of the following type k 0s1 n krn s1 k k 0s2 n krn s2 k k 0r1 n kpn r1 k 1 k 0r2 n kpn r2 k where the integers s1 s2 r1 r2 satisfy some natural conditions specified in the text

on rational transformations of linear functionals direct problem abstract let u be a quasi definite linear functional we find necessary and sufficient conditions in order to the linear functional v satisfying x a u x a v be a quasi definite one also we analyze some linear relations linking the polynomials orthogonal with respect to u and v

an equivariant brauer semigroup and the symmetric imprimitivity theorem suppose that x g is a second countable locally compact transformation group we let sg x denote the set of morita equivalence classes of separable dynamical systems a g where a is a c0 x algebra and is compatible with the given g action on x we prove that sg x is a commutative semigroup with identity with respect to the binary operation a g b g a x b g x for an appropriately defined balanced tensor product on c0 x algebras if g and h act freely and properly on the left and right of a space x then we prove that sg x h and sh g x are isomorphic as semigroups if the isomorphism maps the class of a g to the class of b h then a o g is morita equivalent to b o h

m n coherent pairs of order m k and sobolev orthogonal polynomials a pair of regular linear functionals u v is said to be a m n coherent pair of order m k if their corresponding sequences of monic orthogonal polynomials p n x n 0 and q n x n 0 satisfy a structure relation such as i 0ma i np n m i m x i 0nb i nq n k i k x n 0 where a i n and b i n are complex numbers such that a m n 0 if n m b n n 0 if n n and a i n b i n 0 when i n in the first part of this work we focus our attention on the algebraic properties of an m n coherent pair of order m k to be more precise let us assume that m k if m k then u and v are related by a rational factor in the distributional sense if m k then u and v are semiclassical and they are again related by a rational factor in the second part of this work we deal with a sobolev type inner product defined in the linear space of polynomials with real coefficients p as l rp x q x d m 0 x l rp m x q m x d m 1 x p q p where l is a positive real number m is a positive integer number and m 0 m 1 is a m n coherent pair of order m of positive borel measures supported on an infinite subset of the real line meaning that the sequences of monic orthogonal polynomials p n x n 0 and q n x n 0 with respect to m 0 and m 1 respectively satisfy a structure relation as above with k 0 a i n and b i n being real numbers fulfilling the above mentioned conditions we generalize several recent results known in the literature in the framework of sobolev orthogonal polynomials and their connections with coherent pairs introduced in a iserles p e koch s p norsett j m sanz serna on polynomials orthogonal with respect to certain sobolev inner products j approx theory 65 2 1991 151 175 and their extensions in particular we show how to compute the coefficients of the fourier expansion of functions on an appropriate sobolev space defined by the above inner product in terms of the sequence of sobolev orthogonal polynomials s n x l n 0

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

symmetry of quantum torus with crossed product algebra in this paper we study the symmetry of quantum torus with the concept of crossed product algebra as a classical counterpart we consider the orbifold of classical torus with complex structure and investigate the transformation property of classical theta function an invariant function under the group action is constructed as a variant of the classical theta function then our main issue the crossed product algebra representation of a quantum torus with complex structure under the symplectic group is analyzed as a quantum version of orbifolding we perform this analysis with manin s so called model ii quantum theta function approach the symplectic group sp 2n z satisfies the consistency condition of a crossed product algebra representation of quantum torus times the algebra of functions on the siegel space however only a subgroup of sp 2n z satisfies the consistency condition for orbifolding of the quantum torus

variations around classical orthogonal polynomials connected problems abstract we survey and prove the principal characterizations of the classical orthogonal polynomials with the method of the dual sequence two new characterizations are given connected problems are discussed

a matrix approach for the semiclassical and coherent orthogonal polynomials we obtain a matrix characterization of semiclassical orthogonal polynomials in terms of the jacobi matrix associated with the multiplication operator in the basis of orthogonal polynomials and the lower triangular matrix that represents the orthogonal polynomials in terms of the monomial basis of polynomials we also provide a matrix characterization for coherent pairs of linear functionals

induction in stages for crossed products of c algebras by maximal coactions let be a maximal coaction of a locally compact group g on a c algebra b and let n and h be closed normal subgroups of g with n h we show that the process indg hg which uses mansfield s bimodule to induce representations of b g from those of b g h is equivalent to the two stage induction process indg ng indg hg n the proof involves a calculus of symmetric imprimitivity bimodules which relates the bimodule tensor product to the fibred product of the underlying spaces

a new look at crossed product correspondences and associated c algebras abstract when a locally compact group acts on a c correspondence it also acts on the associated cuntz pimsner algebra in a natural way hao and ng have shown that when the group is amenable the cuntz pimsner algebra of the crossed product correspondence is isomorphic to the crossed product of the cuntz pimsner algebra in this paper we have a closer look at this isomorphism in the case where the group is not necessarily amenable we also consider what happens at the level of toeplitz algebras

k theory of equivariant quantization abstract using an equivariant version of connes thom isomorphism we prove that equivariant k theory is invariant under strict deformation quantization for a compact lie group action

effect of non gaussian noises on the stochastic resonance like phenomenon in gated traps we exploit a simple one dimensional trapping model introduced before prompted by the problem of ion current across a biological membrane the voltage sensitive channels are open or closed depending on the value taken by an external potential that has two contributions a deterministic periodic and a stochastic one here we assume that the noise source is colored and non gaussian with a q dependent probability distribution where q is a parameter indicating the departure from gaussianity we analyze the behavior of the oscillation amplitude as a function of both q and the noise correlation time the main result is that in addition to the resonant like maximum as a function of the noise intensity there is a new resonant maximum as a function of the parameter q

continuous orbit equivalence flow equivalence of markov shifts and circle actions on cuntz krieger algebras we will study circle actions on cuntz krieger algebras trivially acting on its canonical maximal abelian c subalgebras from the view points of continuous orbit equivalence of one sided topological markov shifts and flow equivalence of two sided topological markov shifts

sobolev orthogonal polynomials and m n coherent pairs of measures we introduce the notion of m n coherent pair of measures as a generalization of the concept of coherent pair of measures introduced by iserles et al a iserles p e koch s p norsett j m sanz serna on polynomials orthogonal with respect to certain sobolev inner products j approx theory 65 2 1991 151 175 and subsequently generalized by several authors a pair of measures email protected 0 email protected 1 is called m n coherent if the corresponding orthogonal polynomial sequences p n n and q n n resp satisfy a non zero structure relation such as i 0nr i np n i x email protected i 0ms i nq n i 1 x for all n 0 1 2 where m and n are fixed non negative integer numbers and r i n and s i n are given real parameters satisfying some natural conditions we prove that the regular moment linear functionals associated to an m n coherent pair are semiclassical and they are related by a rational modification in the usual sense of distribution theory we also discuss the converse statement under the assumption that email protected 0 email protected 1 form an m n coherent pair we study the sequence s n l n of the monic orthogonal polynomials with respect to the sobolev inner product l email protected email protected email protected f g email protected 1 defined in the space of all polynomials with real coefficients where l 0 an efficient algorithm is stated to compute the coefficients in the fourier sobolev type series f x email protected n 0 c n ls n l x with respect to l for suitable smooth functions f such that email protected l m 0 2 r and f l m 1 2 r finally some illustrative computational examples are presented

on linearly related sequences of difference derivatives of discrete orthogonal polynomials let be either c 0 or q c 0 1 and let d be the corresponding difference operator defined in the usual way either by d p x p x p x or d q p x p q x p x q 1 x let u and v be two moment regular linear functionals and let p n x n 0 and q n x n 0 be their corresponding orthogonal polynomial sequences ops we discuss an inverse problem in the theory of discrete orthogonal polynomials involving the two ops p n x n 0 and q n x n 0 assuming that their difference derivatives d of higher orders m and k resp are connected by a linear algebraic structure relation such as i 0 m a i n d m p n m i x i 0 n b i n d k q n k i x n 0 where m n m k n 0 a m n 0 for n m b n n 0 for n n and a i n b i n 0 for i n under certain conditions we prove that u and v are related by a rational factor in the distributional sense moreover when m k then both u and v are d semiclassical functionals this leads us to the concept of m n d coherent pair of order m k extending to the discrete case several previous works as an application we consider the ops with respect to the following sobolev type inner product p x r x u p x r x v d m p x d m r x 0 assuming that u and v which eventually may be represented by discrete measures supported either on a uniform lattice if or on a q lattice if q constitute a m n d coherent pair of order m that is an m n d coherent pair of order m 0 m n being fixed

proper actions on imprimitivity bimodules and decompositions of morita equivalences we consider a class of proper actions of locally compact groups on imprimitivity bimodules over c algebras which behave like the proper actions on c algebras introduced by rieffel in 1988 we prove that every such action gives rise to a morita equivalence between a crossed product and a generalized fixed point algebra and in doing so make several innovations which improve the applicability of rieffel s theory we then show how our construction can be used to obtain canonical tensor product decompositions of important morita equivalences our results show for example that the different proofs of the symmetric imprimitivity theorem for actions on graph algebras yield isomorphic equivalences and this gives new information about the amenability of actions on graph algebras

orthogonal polynomials associated with an inverse quadratic spectral transform let p n n 0 be a sequence of monic orthogonal polynomials with respect to a quasi definite linear functional u and q n n 0 a sequence of polynomials defined by q n x p n x s np n 1 x t np n 2 x n 1 with t n 0 for n 2 we obtain a new characterization of the orthogonality of the sequence q n n 0 with respect to a linear functional v in terms of the coefficients of a quadratic polynomial h such that h x v u we also study some cases in which the parameters s n and t n can be computed more easily and give several examples finally the interpretation of such a perturbation in terms of the jacobi matrices associated with p n n 0 and q n n 0 is presented

cattaneo type subdiffusion reaction equation subdiffusion in a system in which mobile particles a can chemically react with static particles b according to the rule a b b is considered within a persistent random walk model this model which assumes a correlation between successive steps of particles provides hyperbolic cattaneo normal diffusion or fractional subdiffusion equations starting with the difference equation which describes a persistent random walk in a system with chemical reactions using the generating function method and the continuous time random walk formalism we will derive the cattaneo type subdiffusion differential equation with fractional time derivatives in which the chemical reactions mentioned above are taken into account we will also find its solution over a long time limit based on the obtained results we will find the cattaneo type subdiffusion reaction equation in the case in which mobile particles of species a and b can chemically react according to a more complicated rule

isomorphism and morita equivalence classes for crossed products of irrational rotation algebras by cyclic subgroups of sl2 z abstract let be irrational numbers and a b be matrices in s l 2 z of infinite order we compute the k theory of the crossed product a a z and show that a a z and a b z are isomorphic if and only if mod z and i a 1 is matrix equivalent to i b 1 combining this result and an explicit construction of equivariant bimodules we show that a a z and a b z are morita equivalent if and only if and are in the same g l 2 z orbit and i a 1 is matrix equivalent to i b 1 finally we determine the morita equivalence class of a f for any finite subgroup f of s l 2 z

enveloping actions and takai duality for partial actions we show that any partial action on a topological space x is the restriction of a suitable global action called enveloping action that is essentially unique in the case of c algebras we prove that any partial action has a unique enveloping action up to morita equivalence and that the corresponding reduced crossed products are morita equivalent the study of the enveloping action up to morita equivalence reveals the form that takai duality takes for partial actions by applying our constructions we prove that the reduced crossed product of the reduced cross sectional algebra of a fell bundle by the dual coaction is liminal postliminal or nuclear if and only if so is the unit fiber of the bundle we also give a non commutative generalization of the well known fact that the integral curves of a vector field on a compact manifold are defined on all of r

on the morita equivalence of tensor algebras our objective is two fold first we want to develop a notion of morita equivalence for c correspondences that guarantees that if two c correspondences e and f are morita equivalent then the tensor algebras of e and f taoeu and taofu are strongly morita equivalent in the sense of 8 the toeplitz algebras toeu and tofu are strongly morita equivalent in the sense of rieffel 32 see 33 also and the cuntz pimsner algebras 28 ooeu and oofu are strongly morita equivalent in the same sense we were inspired by earlier work of curto williams and the first author in 15 and combes 14 these papers investigate conditions implying that the crossed product of two c dynamical systems are strongly morita equivalent given that the coefficient algebras are strongly morita equivalent abadie eilers and exel 1 were similarly inspired and developed a theory of morita equivalence for special c correspondences our analysis extends theirs but our arguments are different see example 2 5 and remarks 3 3 and 3 6 for a comparison of their results and ours second we want to investigate the converse implication that is we study the following question if taoeu and taofu are strongly morita equivalent in the sense of 8 when are e and f morita equivalent in the sense we define for this converse direction we focus on the tensor algebras rather than the toeplitz and cuntz pimsner algebras because very little can be said in the self adjoint setting here we were inspired by arveson s paper 2 in which it is proved that two ergodic measure preserving transformations are conjugate if and only if certain non self adjoint crossed products associated with the transformations are algebraically isomorphic see 3 also his analysis was further extended by k saito and the first author to cover non self adjoint crossed products built from automorphisms of general von neumann algebras they showed that non self adjoint crossed products built from properly outer automorphisms of von neumann algebras are isomorphic if and only if the automorphisms are conjugate see 22 and 23 our principal result in the converse direction theorem 7 2 asserts that if e and f are two aperiodic correspondences and if taoeu is strongly morita equivalent to taofu in the sense of 8 then e and f are morita equivalent as we shall see in 5 the notion of aperiodicity is an exact analogue for correspondences of the concept of free action in ergodic theory free action was a central hypothesis in arveson s

an equivariant brauer group and actions of groups onc algebras suppose that g t is a second countable locally compact trans formation group given by a homomorphism l g homeo t and that a is a separable continuous trace c algebra with spectrum t an action g aut a is said to cover l if the induced action of g on t coincides with the original one we prove that the set brg t of morita equivalence classes of such systems forms a group with multiplication given by the balanced tensor product a b a c0 t b and we refer to brg t as the equivariant brauer group we give a detailed analysis of the structure of brg t in terms of the moore cohomology of the group g and the integral cohomology of the space t using this we can characterize the stable continuous trace c algebras with spectrum t which admit actions covering l in particular we prove that if g r then every stable continuous trace c algebra admits an essentially unique action covering l thereby substantially improving results of raeburn and rosenberg

generalized coherent pairs abstract a pair of quasi definite moment functionals u0 u1 is a generalized coherent pair if monic orthogonal polynomials pn x n 0 and rn x n 0 relative to u0 and u1 respectively satisfy a relation r n x 1 n 1 p n 1 x n n p n x n 1 n 1 p n 1 x n 2 where n and n are arbitrary constants which may be zero if u0 u1 is a generalized coherent pair then u0 and u1 must be semiclassical we find conditions under which either u0 or u1 is classical in such a case we also determine the types of the companion moment functionals also some illustrating examples and two ways of generating generalized coherent pairs are given we also discuss the corresponding sobolev orthogonal polynomials

on the structure of twisted group c algebras we first give general structural results for the twisted group algebras c g of a locally compact group g with large abelian subgroups in particular we use a theorem of williams to realise c g as the sections of a c bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras next we use a device of rosenberg to show that when is a discrete subgroup of a solvable lie group g the k groups k c are isomorphic to certain twisted k groups k g of the homogeneous space g and we discuss how the twisting class h 3 g z depends on the cocycle

on linearly related sequences of derivatives of orthogonal polynomials abstract we discuss an inverse problem in the theory of standard orthogonal polynomials involving two orthogonal polynomial families p n n and q n n whose derivatives of higher orders m and k resp are connected by a linear algebraic structure relation such as i 0 n r i n p n i m m x i 0 m s i n q n i k k x for all n 0 1 2 where m and n are fixed nonnegative integer numbers and r i n and s i n are given complex parameters satisfying some natural conditions let u and v be the moment regular functionals associated with p n n and q n n resp assuming 0 m k we prove the existence of four polynomials m m i and n k i of degrees m m i and n k i resp such that d k m m m i u n k i v i 0 1 the k m th derivative as well as the left product of a functional by a polynomial being defined in the usual sense of the theory of distributions if k m then u and v are connected by a rational modification if k m 1 then both u and v are semiclassical linear functionals which are also connected by a rational modification when k m the stieltjes transform associated with u satisfies a non homogeneous linear ordinary differential equation of order k m with polynomial coefficients

on the linear functionals associated to linearly related sequences of orthogonal polynomials abstract an inverse problem is solved by stating that the regular linear functionals u and v associated to linearly related sequences of monic orthogonal polynomials p n n and q n n respectively in the sense p n x i 1 n r i n p n i x q n x i 1 m s i n q n i x for all n 0 1 2 where r i n and s i n are complex numbers satisfying some natural conditions are connected by a rational modification i e there exist polynomials and with degrees m and n respectively such that u v we also make some remarks concerning the corresponding direct problem stating a characterization theorem in the case n 1 and m 2 as an example we give a linear relation of the above type involving jacobi polynomials with distinct parameters

on linearly related orthogonal polynomials and their functionals let pn be a sequence of polynomials orthogonal with respect a linear functional u and qn a sequence of polynomials defined by pn x snpn 1 x qn x tnqn 1 x we find necessary and sufficient conditions in order to qn be a sequence of polynomials orthogonal with respect to a linear functional v furthermore we prove that the relation between these linear functionals is x a u x a v even more if u and v are linked in this way we get that pn and qn satisfy a formula as above

exotic crossed products and the baum connes conjecture we study general properties of exotic crossed product functors and characterise those which extend to functors on equivariant c algebra categories based on correspondences we show that every such functor allows the construction of a descent in kk theory and we use this to show that all crossed products by correspondence functors of k amenable groups are kk equivalent we also show that for second countable groups the minimal exact morita compatible crossed product functor used in the new formulation of the baum connes conjecture by baum guentner and willett 4 extends to correspondences when restricted to separable g c algebras it therefore allows a descent in kk theory for separable systems

duality of restriction and induction for c coactions consider a coaction of a locally compact group g on a c algebra a and a closed normal subgroup n of g we prove following results of echterhoff for abelian g that mansfield s imprimitivity between a g n and a g r n implements equivalences between mansfield induction of representations from a g n to a g and restriction of representations froma g rn to a g and between restriction of representations from a g to a g n and green induction of representations from a g to a g r n this allows us to deduce properties of mansfield induction from the known theory of ordinary crossed products