crossed products of nuclear c algebras and their traces abstract this work addresses the blackadar kirchberg question bkq as it pertains to group and reduced crossed product c algebras the bkq asks if every stably finite c algebra admits the matricial field mf property we characterize the mf property for a class of reduced crossed product c algebras and for their traces we prove that every ordered abelian group is in a sense mf and show that there is no k theoretic obstruction to an affirmative answer to the bkq using classification techniques and induced k theoretic dynamics we give an affirmative answer to the bkq for reduced crossed products of ah algebras of real rank zero by free groups combining our results with recent progress in elliott s classification program shows that the reduced crossed product of a separable simple unital nuclear uct and monotracial c algebra by the free group is always mf we also examine traces on these crossed products and show they always admit certain finite dimensional approximation properties by appealing to a result of ozawa rordam and sato we show that semidirect products of discrete amenable groups by free groups admit mf reduced group c algebras

lie bialgebroids and poisson groupoids lie bialgebras arise as infinitesimal invariants of poisson lie groups a lie bialgebra is a lie algebra g with a lie algebra structure on the dual g which is compatible with the lie algebra g in a certain sense for a poisson group g the multiplicative poisson structure induces a lie algebra structure on the lie algebra dual g which makes g g into a lie bialgebra in fact there is a one one correspondence between poisson lie groups and lie bialgebras if the lie groups are assumed to be simply connected 7 16 19 the importance of poisson lie groups themselves arises in part from their role as classical limits of quantum groups 8 and in part because they provide a class of poisson structures for which the realization problem is tractable 15 poisson groupoids were introduced by weinstein 24 as a generalization of both poisson lie groups and the symplectic groupoids which arise in the integration of arbitrary poisson manifolds 4 11 he noted that the lie algebroid dual a g of a poisson groupoid g itself has a lie algebroid structure but did not develop the infinitesimal structure further in this paper we introduce and study a natural infinitesimal invariant for poisson groupoids the lie bialgebroids of the title our ultimate purpose is to develop a lie theory for poisson groupoids parallel to that for poisson groups in this paper we are primarily concerned with the first half of this process that is with the construction of the lie bialgebroid of a poisson groupoid after the preliminary 2 in which we describe the generalization to arbitrary lie algebroids of the exterior calculus and schouten calculus in 3 we define a lie bialgebroid to be a lie algebroid a whose dual a is also equipped with a lie algebroid structure such that the coboundary operator d a a associated to a satisfies a cocycle equation with respect to a the lie algebra of sections of a this is clearly a straightforward extension of the concept of a lie bialgebra 16 but cannot be formalized in lie algebroid cohomological terms since there is no satisfactory adjoint representation for a general lie algebroid most of 3 is devoted to proving that this definition is self dual if a a is a lie bialgebroid then a a is also in 4 we briefly consider the special case of lie bialgebroids satisfying a triangularity condition which include some important examples such as the usual triangular lie bialgebras and lie bialgebroids associated to poisson manifolds the techniques used in 2 4 are similar to those known for lie bialgebras it would be possible by suitably generalizing the proof for poisson groups to prove

representations versus numberings on the relationship of two computability notions abstract this paper gives an answer to weihrauch s computability springer berlin 1987 question whether and if not always when an effective map between the computable elements of two represented sets can be extended to a partial computable map between the represented sets examples are known showing that this is not possible in general a condition is introduced and for countably based topological t 0 spaces it is shown that exactly the partial effective maps meeting the requirement are extendable for total effective maps the extra condition is satisfied in the standard cases of effectively given separable metric spaces and continuous directed complete partial orders in which the extendability is already known in the first case a similar result holds also for partial effective maps but not in the second

independence properties in subalgebras of ultraproduct ii1 factors let mn be a sequence of finite factors with dim mn and denote by m mn their ultraproduct over a free ultrafilter we prove that if q m is either an ultraproduct q qn of subalgebras qn mn with qn mnqn mn n or the centralizer q b m of a separable amenable subalgebra b m then for any separable subspace x m q m there exists a diffuse abelian von neumann subalgebra in q which is free independent to x relative to q m some related independence properties for subalgebras in ultraproduct ii1 factors are also discussed

parametrised second order complexity theory with applications to the study of interval computation abstract we extend the framework for complexity of operators in analysis devised by kawamura and cook 2012 to allow for the treatment of a wider class of representations the main novelty is to endow represented spaces of interest with an additional function on names called a parameter which measures the complexity of a given name this parameter generalises the size function which is usually used in second order complexity theory and therefore also central to the framework of kawamura and cook the complexity of an algorithm is measured in terms of its running time as a second order function in the parameter as well as in terms of how much it increases the complexity of a given name as measured by the parameters on the input and output side as an application we develop a rigorous computational complexity theory for interval computation in the framework of kawamura and cook the representation of real numbers based on nested interval enclosures does not yield a reasonable complexity theory in our new framework this representation is polytime equivalent to the usual cauchy representation based on dyadic rational approximation by contrast the representation of continuous real functions based on interval enclosures is strictly smaller in the polytime reducibility lattice than the usual representation which encodes a modulus of continuity furthermore the function space representation based on interval enclosures is optimal in the sense that it contains the minimal amount of information amongst those representations which render evaluation polytime computable

reduction of courant algebroids and generalized complex structures we present a theory of reduction for courant algebroids as well as dirac structures generalized complex and generalized kahler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction the enhanced symmetry group of a courant algebroid leads us to define extended actions and a generalized notion of moment map key examples of generalized kahler reduced spaces include new explicit bi hermitian metrics on cp2

effectiveness of the global modulus of continuity on metric spaces abstract in the e definition of continuity of a function between metric spaces the value of depends on e and function as well this kind of dependence can be described by a function so called global modulus of continuity which maps the triple e to corresponding by a recent result of repovs and semenov proc int conf topol trieste 1993 g gentili ed rent ist mat univ trieste vol 25 1993 pp 441 446 there is a continuous global modulus of continuity for the function space c x y of all continuous functions from a locally compact metric space x to an arbitrary metric space y based on weihrauch s framework on computable metric spaces k weihrauch theoret comput sci 113 1993 191 210 we show that there is a computable global modulus of continuity for c x y if x is an effectively locally compact metric space and y is a computable metric space the proof is a direct construction not depending on the proof of repovs and semenov

l 2 homology for von neumann algebras the aim of this paper is to introduce a notion of l homology in the context of von neumann algebras finding a suitable co homology theory for von neumann algebras has been a dream for several generations see kr71a kr71b jkr72 ss95 and references therein one s hope is to have a powerful invariant to distinguish von neumann algebras unfortunately little positive is known about the kadison ringrose cohomology h b m m except that it vanishes in many cases furthermore there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von neumann algebra our interest in developing an l cohomology theory was revived by the introduction of lcohomology invariants in the field of ergodic equivalence relations in the paper of gaboriau gab02 his results in particular imply that l betti numbers 2 i of a discrete group are the same for measure equivalent groups i e for groups that can generate isomorphic ergodic measure preserving equivalence relations parallels between the worlds of von neumann algebras and measurable equivalence relations have been noted for a long time starting with the parallel between the work of murray and von neumann mvn and that of h dye dy thus there is hope that an invariant of a group that survives measure equivalence will survive also von neumann algebra equivalence i e will be an invariant of the von neumann algebra of the group the original motivation for our construction comes from the well understood analogy between the theory of ii1 factors and that of discrete groups based on the theory of correspondences con con94 this analogy has been remarkably efficient to transpose analytic properties such as amenability or property t from the group context to the factor context con80 cj and more recently in the breakthrough work of popa popa con03 we use the theory of correspondences together with the algebraic description of l betti numbers given by luck his definition involves the computation of the algebraic group homology with coefficients in the group von neumann algebra following the guiding idea that the category of bimodules over a von neumann algebra is the analogue of the category of modules over a group we are led to the following algebraic definition of l homology of a von neumann algebra m

group cocycles and the ring of affiliated operators in this article we study cocycles of discrete countable groups with values in 2 g and the ring of affiliated operators ug we clarify properties of the first cohomology of a group g with coefficients in 2 g and answer several questions from de cornulier et al transform groups 13 1 125 147 2008 moreover we obtain strong results about the existence of free subgroups and the subgroup structure provided the group has a positive first 2 betti num ber we give numerous applications and examples of groups which satisfy our assumptions

computability on the probability measures on the borel sets of the unit interval while computability theory on many countable sets is well established and for computability on the real numbers several mutually non equivalent definitions are applied for most other uncountable sets in particular for measures no generally accepted computability concepts at all ha ve been available until now in this contribution we introduce computability on the set m of probability measures on the borel subsets of the unit interval 0 1 its main purpose is to demonstrate that this concept of computability is not merely an ad hoc definition but has very natural properties although the definitions and many results can of course be transferred to more general spaces of measures we restrict our attention to m in order to keep the technical details simple and concentrate on the central ideas in particular we show that simple obvious reqirements exclude a number of similar definitions that the definition leads to the expected computability results that there are other natural definitions inducing the same computability theory and that the theory is embedded smoothly into classical measure theory as background we consider tte type 2 theory of effectivity kw84 kw85 which provides a frame for very realistic computability definitions in this approach computability is defined on finite and infinite sequences of symbols explicitly by turing machines and on other sets by means of notations and representations canonical representations are derived from information structures wei97 we introduce a standard representation delta m subseteq sum omega to m via some natural information structure defined by a subbase the atomic properties of some topology on m and a standard notation of while several modifications of m suggesting themselves at first glance violate simple and obvious requirements m has several very natural properties and hence should induce an important computability theory many interesting functions on measures turn out to be computable in particular linear combination integration of continuous functions and any transformation defined by a computable iterated function system with probabilities some other natural representations of m are introduced among them a cauchy representation associated with the hutchinson metric and proved to be equivalent to m as a corollary the final topology of m is the well known weak topology on m

computability theory of generalized functions the theory of generalized functions is the foundation of the modern theory of partial differential equations pde as computers are playing an ever larger role in solving pdes it is important to know those operations involving generalized functions in analysis and pde that can be computed on digital computers in this article we introduce natural concepts of computability on test functions and generalized functions as well as computability on schwartz test functions and tempered distributions type 2 turing machines are used as the machine model weihrauch 2000 it is shown here that differentiation and integration on distributions are computable operators and various types of fourier transforms and convolutions are also computable operators as an application it is shown that the solution operator of the distributional inhomogeneous three dimensional wave equation is computable

recursion and topology on 2 ω for possibly infinite computations in the context of possibly infinite computations yielding finite or infinite binary outputs the space 2 2 u 2 appears to be one of the most fundamental spaces in computer science though underconsidered next to 2 this space can be viewed section 3 5 2 as the simplest compact space native to computer science in this paper we study some of its properties involving topology and computability though 2 can be considered as a computable metric space in the sense of computable analysis a direct and self contained study based on its peculiar properties related to words is much illuminating it is well known that computability for maps 2 2 reduces to continuity with recursive modulus of continuity with 2 things get less simple maps 2 2 or 2 2 induced by input output behaviours of turing machines on finite or infinite words which we call semicomputable maps are not necessarily continuous but merely lower semicontinuous with respect to the prefix partial ordering on 2 continuity asks for a stronger notion of computability we prove for semi continuous and semi computable maps f f o with f o 2 2 a detailed representation theorem theorem 81 via functions f 2 2 following two approaches bottom up from f to f and top down from f to f

computability on subsets of euclidean space i closed and compact subsets in this paper we introduce and compare computability concepts on the set of closed subsets of euclidean space we use the language and framework of type 2 theory of effectivity tte which supplies a concise language for distinguishing a variety of effectivity properties and which admits highly effective versions of classical theorems in particular type 2 theory of effectivity allows to separate topological from computational aspects of effectivity we consider three different computability concepts on the set of closed subsets each of which is characterized by several representations which are proved to be equivalent the three induced types of computable closed sets have already been considered by many authors however under different and partly inconsistent names our characterizations show that they can be regarded as straightforward generalizations of the r e co r e and recursive subsets of natural numbers therefore we suggest to call them the recursively enumerable the co recursively enumerable and the recursive closed subsets of euclidean space open subsets obtain the dual names we extend the investigation by introducing several natural representations of the compact subsets of euclidean space and proving equivalences the paper extends and generalizes earlier definitions adds new ones and compares them in a single framework the resultant canonical computability concepts induce computability of objects as well as computability of operators on the space of closed and compact subsets

free entropy dimension in amalgamated free products we calculate the microstates free entropy dimension of natural generators in an amalgamated free product of certain von neumann algebras with amalgamation over a hyperfinite subalgebra in particular some exotic popa algebra generators of free group factors are shown to have the expected free entropy dimension we also show that microstates and non microstates free entropy dimension agree for generating sets of many groups in the appendix the first l 2 betti number for certain amalgamated free products of groups is calculated

constraint metric approximations and equations in groups abstract we introduce notions of a constraint metric approximation and of a constraint stability of a metric approximation this is done in the language of group equations with coefficients we give an example of a group which is not constraintly sofic in building it we find a sofic representation of free group with trivial commutant among extreme points of the convex structure on the space of sofic representations we consider the centralizer equation in permutations as an instance of this new general setting we characterize permutations p s k whose centralizer is stable in permutations with respect to the normalized hamming distance that is a permutation which almost centralizes p is near a centralizing permutation this answers a question of gorenstein sandler mills 1962

type 2 computational complexity of functions on cantor s space abstract continuity and computability on cantor s space c has turned out to be a very natural basis for a type 2 theory of effectivity tte in particular the investigation of type 2 computational complexity e g complexity in analysis requires the study of oracle machines which w l g operate on cantor s space however no general type 2 complexity theory has been developed so far in this paper we lay a foundation for this theory by investigating computational complexity of functions c 0 1 and c c and the related concepts of dependence and input lookahead in the former case results of gordon and shamir are extended and generalized for continuous functions c c the relation between the three concepts is studied in detail compact sets are proved to be natural domains of resource bounded functions on c finally we demonstrate that an optimization of the input lookahead used by a machine may result in a rapid increase of computation time

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

admissible representations of effective cpo s abstract the elements of a cpo complete partial order d are abstract objects in general a concrete machine cannot operate with abstract objects but only with names of objects in this paper the set f of total functions from n to n is suggested as the set of names admissible representations f d for effective cpo s are defined by two axioms which generalize the axioms for godel numberings of the partial recursive functions topological and recursion theoretic considerations show that the definition is very natural it is also proved that p the set of subsets of n is not as suitable as a set of names finally it is proved that for admissible representations the computable functions over cpo s are represented by the computable extensional functions over the names

on the computability of walsh functions the haar and the walsh functions are proved to be computable with respect to the fine metric df which is induced from the infinite product 0 1 1 2 with the weighted product metric dc of the discrete metric on 0 1 although they are discontinuous functions on 0 1 with respect to the euclidean metric they are continuous functions on dc and on 0 1 df on dc computable real valued cylinder functions which include the walsh functions become computable and every computable function can be approximated effectively by a computable sequence of cylinder functions the metric space 0 1 df is separable but not complete nor effectively complete we say that a function on 0 1 is uniformly fine computable if it is sequentially computable and effectively uniformly continuous with respect to the metric df it is proved that a uniformly fine computable function is essentially a computable function on it is also proved that walsh fourier coefficients of a uniformly fine computable function f form a computable sequence of reals and there exists a subsequence of the walsh fourier series which fine converges effectively uniformly to f

equality is a jump abstract we define a notion of degree of unsolvability for subsets of r n where r is a real closed archimedean field and prove that in contrast to type 2 computability the presence of exact equality in the bss model forces exactly one jump of the unsolvability degree of decidable sets

effective symbolic dynamics random points statistical behavior complexity and entropy we consider the dynamical behavior of martin lof random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure we use computable partitions to define a sort of effective symbolic model for the dynamics through this construction we prove that such points have typical statistical behavior the behavior which is typical in the birkhoff ergodic theorem and are recurrent we introduce and compare some notions of complexity for orbits in dynamical systems and prove i that the complexity of the orbits of random points equals the kolmogorov sinai entropy of the system ii that the supremum of the complexity of orbits equals the topological entropy

computability of probability measures and martin lof randomness over metric spaces in this paper we investigate algorithmic randomness on more general spaces than the cantor space namely computable metric spaces to do this we first develop a unified framework allowing computations with probability measures we show that any computable metric space with a computable probability measure is isomorphic to the cantor space in a computable and measure theoretic sense we show that any computable metric space admits a universal uniform randomness test without further assumption

towards a descriptive set theory for domain like structures this is a survey of results in descriptive set theory for domains and similar spaces with the emphasis on the algebraic domains we try to demonstrate that the subject is interesting in its own right and is closely related to some areas of theoretical computer science since the subject is still in its beginning we discuss in detail several open questions and possible future development we also mention some relevant facts of effective descriptive set theory

computation by while programs on topological partial algebras the language of while programs is a fundamental model for imperative programming on any data type it leads to a generalisation of the theory of computable functions on the natural numbers to the theory of computable functions on any many sorted algebra the language is used to express many algorithms in scientific computing where while programs are applied to continuous data in the theory of data continuous data types are modelled by topological many sorted algebras we study both exact and approximate computations by while programs and while programs with arrays over topological many sorted algebras with partial operations first we establish that partial operations are necessary in order to compute a wide range of continuous functions we prove basic continuity properties of our abstract computability any partial function computable over a partial topological algebra by a while array program is continuous any set semicomputable or computable over a partial topological algebra by a while array program is open or clopen respectively secondly we contrast exact and approximate computations the class of functions that can be computed exactly can be quite limited we show that on connected total algebras the while and while array computable functions are precisely those that are explicitly definable by terms we show that for certain general classes of topological algebras the total functions that can be approximated by while programs are precisely those that can be effectively approximated by terms this property we call generalised weierstrass approximation an application of this result is that a function on the set r of reals is computable in the sense of computable analysis if and only if it is while program approximable on a simple algebra based on r

concrete models of computation for topological algebras a concrete model of computation for a topological algebra is based on a representation of the algebra made from functions on the natural numbers the functions computable in a concrete model are computable in the representation in the classical sense of the chruch turing thesis moreover the functions turn out to be continuous in the topology of the algebra in this paper we consider different concrete models for computing in topological algebras and prove their mutual equivalence in certain commonly occurring circumstances for topological algebras the concrete models we use are effective representation by algebraic domains stoltenberg hansen and tucker effective representation by continuous domains edelat effective representation by type two recursion on baire space weihrauch and for metric and normed algebras we use effective metric spaces moschovakis and computability structures pour el and richards the result are evidence that computability theory for topological algebras is a stable theory independent of the specific models of computation just as classical computability theory for discrete algebras is stable

a semi finite algebra associated to a subfactor planar algebra we canonically associate to any planar algebra two type ii factors m the subfactors constructed previously by the authors in guionnet et al 2010 6 are isomorphic to compressions of m to finite projections we show that each m is isomorphic to an amalgamated free product of type i von neumann algebras with amalgamation over a fixed discrete type i von neumann subalgebra in the finite depth case existing results in the literature imply that m m is the amplification a free group factor on a finite number of generators as an application we show that the factors mj constructed in guionnet et al in press 6 are isomorphic to interpolated free group factors l f rj rj 1 2 2j 1 i where 2 is the index of the planar algebra and i is its global index other applications include computations of laws of jones wenzl projections

algebraic constructions in the category of lie algebroids the aim of this paper is to give the first general and abstract treatment of the algebraic properties of lie algebroids the concept of lie algebroid was introduced in 1967 by pradines ls as the basic infinitesimal invariant of a differentiable groupoid the construction of the lie algebroid of a differentiable groupoid is a direct generalization of the construction of the lie algebra of a lie group and 183 described a full lie theory for differentiable groupoids and lie algebroids encompassing many phenomena in the foundations of differential geometry a more detailed account and further references are given in 13 however at this stage the algebraic properties of lie algebroids were not pursued in 13 iii section 2 iv section 11 one of us gave a fairly detailed account of the abstract algebra of lie algebroids over a fixed base lie algebroids are vector bundles with several additional structures and a category of lie algebroids on a given base and morphisms which preserve that base has properties similar to those of the category of lie algebras using this algebra it was demonstrated in 13 that the lie theory of a category of locally trivial differentiable lie groupoids over a fixed base and the corresponding category of transitive lie algebroids over that base is coextensive with the standard theory of connections in principal bundles we now propose to show that the lie functor from the category of all differentiable groupoids over arbitrary bases and arbitrary smooth morphisms to the category of all lie algebroids preserves the basic algebraic constructions known to be possible in the category of differentiable groupoids the main work is to give abstract formulations of the corresponding lie algebroid concepts the groupoid results with which we

embedding metric spaces into cpo s abstract we show that metric spaces with polish topology mp can be represented very well by bounded complete algebraic cpo s with continuous weight and distance cwd an operator cwd mp is defined such that 1 and 2 hold 1 1 d can be embedded topologically into d such that a partial function on d is metrically continuous if it can be extended to a total continuous function on d 2 2 every m mp is isometric to some d therefore the continuous cpo functions are exactly adequate for describing the continuous functions on the corresponding metric spaces the approach could be refined in order to study computability on metric spaces with polish topology examples of application are the euclidean space the space of total functions on the natural numbers baire space and spaces of infinite trees

from axiomatics to intrinsic characterization some open problems in computable analysis abstract five open problems are presented the framework for these problems is the notion of a computability structure on a banach space which is discussed below the study of banach spaces in particular the study of hilbert space is an important topic in mathematics and its applications problems a and b are concerned with transfer principles for obtaining effective versions of classical theorems of mathematics problems c and d deal with separability and effective separability problem e is concerned with computability for classes of mathematical structures other than banach spaces

random elements in effective topological spaces with measure following a suggestion of zvonkin and levin we generalize martin lof s definition of infinite random sequences over a finite alphabet via randomness tests to effective topological spaces with a measure we show that under weak computability conditions there is a universal randomness test we prove a theorem on randomness preserving functions which corrects and extends a result by schnorr and apply it to a number of examples in particular we show that a real number is random if and only if it has a random b ary representation for any b 2 we show that many computable continuously differentiable real functions preserve randomness especially all computable analytic functions which are not constant on any open subset of their domain preserve randomness finally we introduce a new randomness concept for subsets of natural numbers which we characterize in terms of random sequences surprisingly it turns out that there are infinite co r e random sets

the turing closure of an archimedean field a bss machine is uniform if it does not use exact tests such machines are equivalent modulo parameters to type 2 turing machines we define a notion of closure related to turing machines for archimedean fields and show that such fields admit nontrivial uniformly decidable sets iff they are not turing closed then the partially ordered set of turing closed fields is proved isomorphic to the ideal completion of unsolvability degrees

the free entropy dimension of hyperfinite von neumann algebras suppose m is a hyperfinite von neumann algebra with a normal tracial state and 1 n is a set of selfadjoint generators for m we calculate 0 1 n the modified free entropy dimension of 1 n moreover we show that 0 1 n depends only on m and consequently 0 1 n is independent of the choice of generators for m in the course of the argument we show that if b 1 b n is a set of selfadjoint generators for a von neumann algebra r with a normal tracial state and b 1 b n has finite dimensional approximants then 0 n 0 b 1 b n for any hyperfinite von neumann subalgebra n of r combined with a result by voiculescu this implies that if r has a regular diffuse hyperfinite von neumann subalgebra then 0 b 1 b n 1

manin triples for lie bialgebroids in his study of dirac structures a notion which includes both poisson structures and closed 2 forms t courant introduced a bracket on the direct sum of vector fields and 1 forms this bracket does not satisfy the jacobi identity except on certain subspaces in this paper we systematize the properties of this bracket in the definition of a courant algebroid this structure on a vector bundle e rightarrow m consists of an antisymmetric bracket on the sections of e whose jacobi anomaly has an explicit expression in terms of a bundle map e rightarrow tm and a field of symmetric bilinear forms on e when m is a point the definition reduces to that of a lie algebra carrying an invariant nondegenerate symmetric bilinear form for any lie bialgebroid a a over m a notion defined by mackenzie and xu there is a natural courant algebroid structure on a oplus a which is the drinfel d double of a lie bialgebra when m is a point conversely if a and a are complementary isotropic subbundles of a courant algebroid e closed under the bracket such a bundle with dimension half that of e is called a dirac structure there is a natural lie bialgebroid structure on a a whose double is isomorphic to e the theory of manin triples is thereby extended from lie algebras to lie algebroids our work gives a new approach to bihamiltonian structures and a new way of combining two poisson structures to obtain a third one we also take some tentative steps toward generalizing drinfel d s theory of poisson homogeneous spaces from groups to groupoids

lagrangian submanifolds and dynamics on lie algebroids in some previous papers a geometric description of lagrangian mechanics on lie algebroids has been developed in this topical review we give a hamiltonian description of mechanics on lie algebroids in addition we introduce the notion of a lagrangian submanifold of a symplectic lie algebroid and we prove that the lagrangian hamiltonian dynamics on lie algebroids may be described in terms of lagrangian submanifolds of symplectic lie algebroids the lagrangian hamiltonian formalism on lie algebroids permits us to deal with lagrangian hamiltonian functions not defined necessarily on tangent cotangent bundles thus we may apply our results to the projection of lagrangian hamiltonian functions which are invariant under the action of a symmetry lie group as a consequence we obtain that lagrange poincare hamilton poincare equations are the euler lagrange hamilton equations associated with the corresponding atiyah algebroid moreover we prove that lagrange poincare hamilton poincare equations are the local equations defining certain lagrangian submanifolds of symplectic atiyah algebroids

computational complexity of real functions in this chapter we introduce the formal definitions of computational complexity of real functions we first review the definitions of computable real numbers and computable real functions and their basic properties then the oracle turing machine is introduced as the formal model for computing real functions this model allows us to define the complexity measures for computing real functions in a natural way the class of polynomial time computable real functions is then defined and several characterizations of this class will be given other complexity classes of real numbers and real functions such as np real functions and log space computable real functions will be defined in later chapters

theory of representations abstract an approach for a simple general and unified theory of effectivity on sets with cardinality not greater than that of the continuum is presented a standard theory of effectivity on f f n n has been developed in a previous paper by representations f m this theory is extended to other sets m topological and recursion theoretical properties of representations are studied where the final topology of a representation plays an essential role it is shown that for any separable t0 space an up to equivalence unique admissible representation can be defined which reflects the topological properties correctly

recursive quasi metric spaces in computable analysis recursive metric spaces play an important role since these are roughly speaking spaces with computable metric and limit operation unfortunately the concept of a metric space is not powerful enough to capture all interesting phenomena which occur in computable analysis some computable objects are naturally considered as elements of asymmetric spaces which are not metrizable nevertheless most of these spaces are t0 spaces with countable bases and thus at least quasi metrizable we introduce a definition of recursive quasi metric spaces in analogy to recursive metric spaces we show that this concept leads to similar results as in the metric case and we prove that the most important spaces of computable analysis can be naturally considered as recursive quasi metric spaces especially we discuss some hyper and function spaces

computability on continuous lower semi continuous and upper semi continuous real functions in this paper we extend computability theory to the spaces of continuous upper and lower semi continuous real functions we apply the framework of tte type 2 theory of effectivity where not only computable objects but also computable functions on the spaces can be considered first some basic facts about tte are summarized for each of the function spaces we introduce several natural representations based on different intiuitive concepts of effectivity and prove their equivalence computability of several operations on the function spaces is investigated among others limits mappings to open sets images of compact sets and preimages of open sets maximum and minimum values the positive results usually show computability in all arguments negative results usually express non continuity several of the problems have computable but not extensional solutions since computable functions map computable elements to computable elements many previously known results on computability are obtained as simple corollaries

graphs of quantum groups and k amenability building on a construction of j p serre we associate to any graph of c algebras a maximal and a reduced fundamental c algebra and use this theory to construct the fundamental quantum group of a graph of discrete quantum groups this construction naturally gives rise to a quantum bass serre tree which can be used to study the k theory of the fundamental quantum group to illustrate the properties of this construction we prove that if all the vertex quantum groups are amenable then the fundamental quantum group is k amenable this generalizes previous results of p julg a valette r vergnioux and the first author our proof even for classical groups is quite different from the original proof of julg and valette which does not seem to extend straightforwardly to the quantum setting

computable banach spaces via domain theory this paper extends the order theoretic approach to computable analysis via continuous domains to complete metric spaces and banach spaces we employ the domain of formal balls to define a computability theory for complete metric spaces for banach spaces the domain specialises to the domain of closed balls ordered by reversed inclusion we characterise computable linear operators as those which map computable sequences to computable sequences and are effectively bounded we show that the domain theoretic computability theory is equivalent to the well established approach by pour el and richards

topological properties of real number representations we prove three results about representations of real numbers or elements of other topological spaces by infinite strings such representations are useful for the description of real number computations performed by digital computers or by turing machines first we show that the so called admissible representations a topologically natural class of representations introduced by kreitz and weihrauch are essentially the continuous extensions with a well behaved domain of continuous and open representations second we show that there is no admissible representation of the real numbers such that every real number has only finitely many names third we show that a rather interesting property of admissible real number representations holds true also for a certain non admissible representation namely for the naive cauchy representation the property that continuity is equivalent to relative continuity with respect to the representation

extended admissibility we give a new definition of admissible representations which allows to handle also non countably based topological spaces in the framework of type 2 theory of effectivity we prove that admissible representations x y of topological spaces x y have the desirable property that every partial function f x y is continuously realizable with respect to x y if and only if f is sequentially continuous furthermore we characterize the class of the spaces having an admissible representation many interesting operators creating new topological spaces from old ones are shown to preserve the property of having an admissible representation in particular the class of sequential spaces with admissible representations turns out to be cartesian closed thus a reasonable computability theory is possible on important non countably based spaces

the momentum map in poisson geometry every action on a poisson manifold by poisson diffeomorphisms lifts to a hamiltonian action on its symplectic groupoid which has a canonically defined momentum map we study various properties of this momentum map as well as its use in reduction

large deviation bounds for matrix brownian motion we prove large deviation bounds for the convergence of hermitian matrix valued brownian motion towards free brownian motion as a consequence we obtain upper and lower bounds on the microstates entropy introduced by voiculescu 24

reduction of a symplectic like lie algebroid with momentum map and its application to fiberwise linear poisson structures this paper addresses the problem of developing an extension of the marsden weinstein reduction process to symplectic like lie algebroids and in particular to the case of the canonical cover of a fiberwise linear poisson structure whose reduction process is the analog to cotangent bundle reduction in the context of lie algebroids dedicated to the memory of jerrold e marsden

lie algebroids holonomy and characteristic classes abstract we extend the notion of connection in order to study singular geometric structures namely we consider a notion of connection on a lie algebroid which is a natural extension of the usual concept of a covariant connection it allows us to define holonomy of the orbit foliation of a lie algebroid and prove a stability theorem we also introduce secondary or exotic characteristic classes thus providing invariants which generalize the modular class of a lie algebroid

topological free entropy dimension in unital c algebras abstract the notion of topological free entropy dimension of n tuple of elements in a unital c algebra was introduced by voiculescu in the paper we compute topological free entropy dimension of one self adjoint element and topological free orbit dimension of one self adjoint element in a unital c algebra we also calculate the values of topological free entropy dimensions of any families of self adjoint generators of some unital c algebras including irrational rotation c algebra uhf algebra and minimal tensor product of two reduced c algebras of free groups

asymptotic infinitesimal freeness with amalgamation for haar quantum unitary random matrices we consider the limiting distribution of u na nu n and bn and more general expressions where an and bn are n n matrices with entries in a unital c algebra mathcal b which have limiting mathcal b valued distributions as n and un is a n n haar distributed quantum unitary random matrix with entries independent from mathcal b under a boundedness assumption we show that u na nu n and bn are asymptotically free with amalgamation over mathcal b moreover this also holds in the stronger infinitesimal sense of belinschi shlyakhtenko

recursive characterization of computable real valued functions and relations abstract corresponding to the definition of recursive functions we introduce a class of recursive relations in metric spaces such that each relation is generated from a class of basic relations by a finite number of applications of some specified operators we prove that our class of recursive relations essentially coincides with our class of densely computable relations defined via turing machines in the special case of the real numbers our subclass of recursive functions coincides with the classical class of computable real valued functions defined via turing machines by grzegorczyk lacombe and others

computability on computable metric spaces in previous papers e g weihrauch 1987 type 2 theory of effectivity tte has been shown to be a very general and powerful computer oriented theory for studying effectivity in mathematics in this contribution computability on certain computable separable metric spaces is studied in detail by applying the framework of tte computationally admissible representations of metric spaces are introduced after showing that four different effective representations are computationally equivalent for extending computability to the set of continuous functions several effective namings which are related to definitions of continuity are introduced and compared the definitions are used to prove effective g extension theorems for continuous functions and an effective g characterization of the domains of strongly continous functions which generalize the known properties of real functions kreitz 1984

computability on subsets of metric spaces the notions recursively enumerable and recursive are the basic notions of effectivity in classical recursion theory in computable analysis these notions are generalized to closed subsets of euclidean space using their metric distance functions we study a further generalization of these concepts to subsets of computable metric spaces it appears that different characterizations which coincide in case of euclidean space lead to different notions in the general case however under certain additional conditions such as completeness and effective local compactness the situation is similar to the euclidean case we present all results in the framework of type 2 theory of effectivity which allows to express effectivity properties in a very uniform way instead of comparing properties of single subsets we compare corresponding representations of the hyperspace of closed subsets such representations do not only induce a concept of computability for single subsets but they even yield a concept of computability for operations on hyperspaces such as union intersection etc we complete our investigation by studying the special situation of compact subsets

real number computation through gray code embedding we propose an embedding g of the unit open interval to the set 0 1 1 of infinite sequences of 0 1 with at most one undefined element this embedding is based on gray code and it is a topological embedding with a natural topology on 0 1 1 we also define a machine called an indeterministic multihead type 2 machine which input output sequences in 0 1 1 and show that the computability notion induced on real functions through the embedding g is equivalent to the one induced by the signed digit representation and type 2 machines we also show that basic algorithms can be expressed naturally with respect to this embedding

an effective riemann mapping theorem the main results of the paper are two effective versions of the riemann mapping theorem the first uniform version is based on the constructive proof of the riemann mapping theorem by cheng and by bishop and bridges and formulated in the computability framework developed by kreitz and weihrauch it states which topological information precisely one needs about a nonempty proper open connected and simply connected subset of the complex plane in order to compute a description of a holomorphic bijection from the unit disk onto this set and vice versa which topological information about the set can be obtained from a description of a holomorphic bijection the second version which is derived from the first by considering the sets and the functions with computable descriptions characterizes the subsets of the complex plane for which there exists a computable holomorphic bijection from the unit disk this gives a partial answer to a problem posed by pour el and richards we also show that this class of sets is strictly larger than a class of sets considered by zhou which solves an open problem posed by him in preparation recursively enumerable open subsets and closed subsets of euclidean spaces are considered and several effective results in complex analysis are proved

computability on di domains in this paper we investigate a subset of the class of scott computable stable functions called berry computable furthermore we introduce a corresponding notion of an effectively given di domain we obtain the result that the category with these domains as objects and berry computable functions as morphisms has all the important attributes of recursive domains and scott computable continuous functions e g cartesian closedness moreover we investigate the relationship between stable functions and sequential algorithms we show that any berry computable function can be computed in a sequential way by a relative algorithm additionally if the codomain of the considered stable function is an atomic di domain we obtain the result that the reverse holds as well

type 2 recursion theory abstract an attempt is made to lay a basis for a general unified concise and simple theory of computable and continuous functions from f to f or n where f f n n the theoy is formally very similar to ordinary recursion theory it splits into a purely topological version and more special theory of computability the basic definitions are given and fundamental properties are proved as an example it is shown how the theory of recursively enumerable subsets of n can be transferred to a theory of open and a theory of computably open subsets of f

effective metric spaces and representations of the reals based on standard notions of classical recursion theory a natural model of approximate computability for partial functions between effective metric spaces is presented it generalizes the ko friedman approach to computability of real functions by means of oracle turing machines follows the main ideas of weihrauch s type 2 theory of effectivity but it avoids the explicit use of representations the topological arithmetical hierarchy is introduced and shown to be strict if the underlying space contains an effectively discrete sequence the domains of computable functions are exactly the 2 sets of this hierarchy if the space admits a finitary stratification finally this framework is used to investigate and characterize the standard representations of the real numbers they are just those functions from the name space onto the reals which have both computable extensions and inversions that are computable as relations

three concepts of decidability for general subsets of uncountable spaces there is no uniquely standard concept of an effectively decidable set of real numbers or real n tuples here we consider three notions decidability up to measure zero m w parker undecidability in rn riddled basins the kam tori and the stability of the solar system phil sci 70 2 2003 359 382 which we abbreviate d m z recursive approximability or r a k i ko complexity theory of real functions birkhauser boston 1991 and decidability ignoring boundaries d i b w c myrvold the decision problem for entanglement in r s cohen et al eds potentiality entanglement and passion at a distance quantum mechanical studies fo abner shimony vol 2 kluwer academic publishers great britain 1997 pp 177 190 unlike some others in the literature these notions apply not only to certain nice sets but to general sets in rn and other appropriate spaces we consider some motivations for these concepts and the logical relations between them it has been argued that d m z is especially appropriate for physical applications and on rn with the standard measure it is strictly stronger than r a m w parker undecidability in rn riddled basins the kam toil and the stability of the solar system phil sci 70 2 2003 359 382 here we show that this is the only implication that holds among our three decidabilities in that setting under arbitrary measures even this implication fails yet for intervals of non zero length and more generally convex sets of non zero measure the three concepts are equivalent