measurable kac cohomology for bicrossed products we study the kac cohomology for matched pairs of locally compact groups this cohomology theory arises from the extension theory of locally compact quantum groups we prove a measurable version of the kac exact sequence and provide methods to compute the cohomology we give explicit calculations in several examples using results of moore and wigner

non semi regular quantum groups coming from number theory in this paper we study c algebraic quantum groups obtained through the bicrossed product construction examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not semi regular the crossed product of the quantum group acting on itself by translations does not contain any compact operator we describe all corepresentations of these quantum groups and the associated universal c algebras on the way we provide several remarks on c algebraic properties of quantum groups and their actions

admissibility conjecture and kazhdan s property t for quantum groups abstract we give a partial solution to a long standing open problem in the theory of quantum groups namely we prove that all finite dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups admissibility conjecture for quantum group representations we use this to study kazhdan s property t for quantum groups with non trivial scaling group strengthening and generalising some of the earlier results obtained by fima kyed and soltan chen and ng daws skalski and viselter and brannan and kerr our main results are i all finite dimensional unitary representations of locally compact quantum groups which are either unimodular or arise through a special bicrossed product construction are admissible ii a generalisation of a theorem of wang which characterises property t in terms of isolation of finite dimensional irreducible representations in the spectrum iii a very short proof of the fact that quantum groups with property t are unimodular iv a generalisation of a quantum version of a theorem of bekka valette proven earlier for quantum groups with trivial scaling group which characterises property t in terms of non existence of almost invariant vectors for weakly mixing representations v a generalisation of a quantum version of kerr pichot theorem proven earlier for quantum groups with trivial scaling group which characterises property t in terms of denseness properties of weakly mixing representations

compact lie group action with the continuous rokhlin property abstract in this paper we study the continuous rokhlin property of c dynamical systems using techniques of equivariant kk theory and quantum group theory in particular we determine the kk equivalence class and give a classification of kirchberg g algebras when the g is a compact lie group with the hodgkin condition

smooth extensions of functions on separable banach spaces let x be a banach space with a separable dual x let y subset x be a closed subspace and f y rightarrow mathbb r a c1 smooth function then we show there is a c1 extension of f to x

on locally compact quantum groups whose algebras are factors in this paper we are interested in examples of locally compact quantum groups m such that both von neumann algebras m and the dual m are factors there is a lot of known examples such that m m are respectively of type i1 i1 but there is no examples with factors of other types we construct new examples of type i1 ii1 ii1 ii1 and iii iii for each 2 0 1 also we show that there is no such example with m or m a nite factor a locally compact l c quantum group in the von neumann algebra setting see 6 7 is a pair m where m is a von neumann algebra and is a comultiplication on m with left and right invariant weight in a canonical way every l c group is a commutative l c quantum group and every von neu mann group algebra of a l c group is a cocommutative l c quantum group conversely every commutative or cocommutative l c quantum group is ob tained in this way our aim is to obtain new examples of l c quantum groups which are as far as possible from groups so we will be interested in the least commutative and cocommutative examples the formulation of the problem is the following given a pair of factors of a certain type x1 x2 is it possible to nd a l c quantum group m such that m is a type x1 factor and the dual algebra m is a type x2 factor in the sense of murray von neumann s and connes classication of factors there exists a lot of examples for the

singular stochastic control linear diffusions and optimal stopping a class of solvable problems we consider a class of singular stochastic control problems arising frequently in applications of stochastic control we state a set of conditions under which the optimal policy and its value can be derived in terms of the minimal r excessive functions of the controlled diffusion and demonstrate that the optimal policy is of the standard local time type we then state a set of weak smoothness conditions under which the value function is increasing and concave and demonstrate that given these conditions increased stochastic fluctuations decrease the value and increase the optimal threshold thus postponing the exercise of the irreversible policy in line with previous studies of singular stochastic control we also establish a connection between singular control and optimal stopping and show that the marginal value of the singular control problem coincides with the value of the associated stopping problem whenever 0 is not a regular boundary for the controlled diffusion

higher order tangents and fefferman s paper on whitney s extension problem whitney w2 proved that a function defined on a closed subset of r is the restriction of a cm function if the limiting values of all mth divided differences form a continuous function we show that fefferman s solution of whitney s problem for rn f th 1 is equivalent to a variant of our conjecture in bmp2 giving a criterion for cm extension in terms of iterated limits of finite differences

semidirect products of c quantum groups multiplicative unitaries approach c quantum groups with projection are the noncommutative analogues of semidirect products of groups radford s theorem about hopf algebras with projection suggests that any c quantum group with projection decomposes uniquely into an ordinary c quantum group and a braided c quantum group we establish this on the level of manageable multiplicative unitaries

property t discrete quantum groups and subfactors with triangle presentations abstract kazhdan s property t has been studied for several discrete group like structures including standard invariants of jones subfactors and discrete quantum groups we prove a uk type spectral gap criterion for property t in this setting this allows us to construct a family of property t discrete quantum groups as well as subfactor standard invariants given by generators and a triangle presentation these are the first examples of property t discrete quantum groups that are not directly related to a discrete group with property t

minimizing infinite time horizon discounted cost with mean variance and bounded variation controls we consider an infinite time horizon discounted cost minimization problem for a class of ito processes the available controls are drift and diffusion coefficients and the added bounded variation process we show that the value function is twice continuously differentiable and derive an optimal policy which has feedback type drift and diffusion coefficients when the absolute value of the optimal drift grows faster than the running cost function the optimal bounded variation process is identically zero when it grows weaker than the running cost function optimal bounded variation process is a local time type process in this case we relate the control problem with an optimal stopping problem we also establish the abelian limit relations between the value functions of the discounted cost problem and the stationary problem

2 cocycles and twisting of kac algebras we describe the twisting construction with the help of 2 cocycles on hopf von neumann and george kac algebras we show that twisted kac algebras are again kac algebras using this construction we give a wide class of new quantizations of the heisenberg group and describe several series of non trivial finite dimensional hopf algebras kac algebras of dimensions 4n and as twisting of finite groups

actions and coactions of finite quantum groupoids on von neumann algebras extensions of the matched pair procedure abstract in this work actions and coactions of finite c quantum groupoids are studied in an operator algebras context in particular we prove a double crossed product theorem and the existence of an universal von neumann algebra on which any finite groupoid acts outerly we give two actually different extensions of the matched pairs procedure in previous works n andruskiewitsch and s natale define for any matched pair of groupoids two c quantum groupoids in duality we give here an interpretation of them in terms of crossed products of groupoids using a single multiplicative partial isometry which gives a complete description of these structures the second extension deals only with groups to define an other type of finite c quantum groupoids

a bounded variation control problem for diffusion processes we consider an infinite horizon discounted cost minimization problem for a one dimensional stochastic differential equation model the available control is an added bounded variation process the cost structure involves a running cost function and a proportional cost for the use of the control process the running cost function is not necessarily convex we obtain sufficient conditions to guarantee the optimality of the zero control also for unbounded cost functions we provide sufficient conditions which make our optimal state process a reflecting diffusion on a compact interval in both cases the value function is a c2 function for bounded cost functions under additional assumptions we obtain a complex optimal strategy which turned out to be a mixture of jumps and local time type processes in this case we show that the value function is only a c1 function and that it fails to be a c2 function we also discuss a related variance control problem

bicrossproduct structure of the quantum weyl group abstract the quantum weyl group formula associated to a complex simple lie algebra g consists of the quantum group uq g with certain quantum simple reflections w i adjoined let kw be the group algebra of the standard covering w of the weyl group of g here k c we show that formula has the structure of a cocycle bicrossproduct formula kw uq g it consists as an algebra of a cocycle semidirect product by a cocycle action of kw on uq g defined with respect to a certain non abelian cocycle it consists as a coalgebra of an extension by a non abelian dual cocycle the dual of formula is also a bicrossproduct and consists as an algebra of an extension of the dual of uq g by the commutative algebra of functions on w via a cocycle

classifying complements for hopf algebras and lie algebras abstract let a e be a given extension of hopf respectively lie algebras we answer the classifying complements problem ccp which consists of describing and classifying all complements of a in e if h is a given complement then all the other complements are obtained from h by a certain type of deformation we establish a bijective correspondence between the isomorphism classes of all complements of a in e and a cohomological type object ha 2 h a where is the matched pair associated to h the factorization index e a f is introduced as a numerical measure of the ccp for two n th roots of unity we construct a 4 n 2 dimensional hopf algebra whose factorization index over the group algebra is arbitrary large

hopf von neumann algebra bicrossproducts kac algebra bicrossproducts and the classical yang baxter equations two groups g1 g2 are said to be a matched pair if each acts on the space of the other and these actions say obey a certain compatibility condition for every matched pair of locally compact groups g1 g2 we construct an associated coinvolutive hopf von neumann algebra m g1 l g2 by simultaneous cross product and cross coproduct for non trivial these bicrossproduct hopf von neumann algebras are non commutative and non cocommutative if the modules for the actions are also matched then these bicrossproducts are kac algebras in this case we show that the dual kac algebra is of the same form with the roles of g1 g2 and of interchanged examples exist with g1 a simply connected lie group and choices of g2 determined by suitable solutions of the classical yang baxter equations on the complexification of the lie algebra of g1

connections between optimal stopping and singular stochastic control ii reflected follower problems the stochastic control problem of following a brownian path by a process of bounded variation and subject to a reflecting barrier at the origin reflected follower is reduced to a question of optimal stopping with absorption direct probabilistic arguments are used to establish the equivalence of the two problems under suitable conditions on the cost functions

bicrossproducts of multiplier hopf algebras abstract in this paper we generalize majid s bicrossproduct construction we start with a pair a b of two regular multiplier hopf algebras we assume that b is a right a module algebra and that a is a left b comodule coalgebra the right action of a on b gives rise to the smash product a b the left coaction of b on a gives a possible coproduct on a b we discuss in detail the necessary compatibility conditions between the action and the coaction for to be a proper coproduct on a b the result is again a regular multiplier hopf algebra majid s construction is obtained when we have hopf algebras we also look at the dual case constructed from a pair c d of regular multiplier hopf algebras where now c is a left d module algebra while d is a right c comodule coalgebra we show that indeed these two constructions are dual to each other in the sense that a natural pairing of a with c and of b with d yields a duality between a b and the smash product c d we show that the bicrossproduct of an algebraic quantum group is again an algebraic quantum group i e a regular multiplier hopf algebra with integrals the algebra case is also considered some special cases are treated and they are related with other constructions available in the literature the basic example coming from a not necessarily finite group g with two subgroups h and k such that g k h and h k e where e is the identity of g is used to illustrate our theory more examples will be considered in forthcoming papers on the subject

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

equivariant poincare duality for quantum group actions we extend the notion of poincare duality in kk theory to the setting of quantum group actions an important ingredient in our approach is the replacement of ordinary tensor products by braided tensor products along the way we discuss general properties of equivariant kk theory for locally compact quantum groups including the construction of exterior products as an example we prove that the standard podle sphere is equivariantly poincare dual to itself

an abelian limit approach to a singular ergodic control problem we consider an ergodic stochastic control problem for a class of one dimensional ito processes where the available control is an added bounded variation process the corresponding infinite horizon discounted control problem was solved in a weerasinghe siam j control optim 44 2005 pp 389 417 here we show that as the discount factor approaches zero the optimal strategies derived in a weerasinghe siam j control optim 44 2005 pp 389 417 converge to an optimal strategy for the ergodic control problem under different assumptions two types of optimal strategies were derived also the abelian limit relationships among the ergodic control problem the infinite horizon discounted control problem and the finite time horizon control problem are established here a solution to a constrained optimization problem is obtained as an application

some results on the bellman equation of ergodic control we study the multidimensional bellman equation of ergodic control for diffusion processes without invariant measures and establish the existence of a unique solution in a certain class

the free boundary of the monotone follower this paper identifies the free boundary arising in the two dimensional monotone follower cheap control problem it proves that if a region of inaction cal a is of locally finite perimeter lfp then cal a can be replaced by a new region of inaction tilde cal a whose boundary is locally c 1 up to sets of lower dimension it then gives conditions under which the hypothesis lfp holds furthermore under these conditions even higher regularity of the free boundary is obtained namely c 2 alpha except perhaps at a single corner point

differentiable functions defined in closed sets a problem of whitney in 1934 whitney raised the question of how to recognize whether a function f defined on a closed subset x of n is the restriction of a function of class p a necessary and sufficient criterion was given in the case n 1 by whitney using limits of finite differences and in the case p 1 by glaeser 1958 using limits of secants we introduce a necessary geometric criterion for general n and p involving limits of finite differences that we conjecture is sufficient at least if x has a tame topology we prove that if x is a compact subanalytic set then there exists q q x p such that the criterion of order q implies that f is p the result gives a new approach to higher order tangent bundles or bundles of differential operators on singular spaces

a new approach to induction and imprimitivity results given a closed quantum subgroup of a locally compact quantum group we study induction of unitary corepresentations of the quantum subgroup to the ambient quantum group more generally we study induction given a coaction of the quantum subgroup on a c algebra we prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups this means that we deflne quantum homogeneous spaces as c algebras and that we prove morita equivalence of crossed products and homogeneous spaces we essentially use von neumann algebraic techniques to prove these morita equivalences between c algebras the theory of induced representations of locally compact l c groups was introduced by mackey 21 who discovered the imprimitivity theorem characterizing induced representations through the presence of a covariant representation of a homogeneous space riefiel 26 provided a modern approach using the language of hilbert c modules after the work of riefiel several induction procedures and imprimitivity results have been obtained both for actions and for coactions of groups we shall brie y review them below the purpose of this paper is to develop such an induction and imprimitivity machinery in the setting of locally compact quantum groups 16 17 in this way we provide a unifled approach to several results on actions and coactions of groups on c algebras at the same time our proofs in the setting of l c quantum groups are simpler than the classical proofs dealing with coactions of groups another motivation comes from quantum group theory meyer and nest have undertaken a reformulation of the baum connes conjecture 24 in which induction and restriction play a crucial role the development of induction and imprimitivity in the current paper should play an equally important role in the formulation of a baum connes conjecture for quantum groups

embeddable quantum homogeneous spaces abstract we discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups on the von neumann algebra level we recall an interesting duality for such objects studied earlier by m izumi r longo s popa for compact kac algebras and by m enock in the general case of locally compact quantum groups a definition of a quantum homogeneous space is proposed along the lines of the pioneering work of vaes on induction and imprimitivity for locally compact quantum groups the concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces among various examples we single out the quantum analog of the quotient of the cartesian product of a quantum group with itself by the diagonal subgroup analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles the former turns out to be an interesting application of the duality mentioned above

deformation of a kac algebra by an abelian subgroup some examples of quantum groups in literature arise as deformations of a locally compact group by a dual 2 cocycle we make this construction in the framework of kac algebras we show that these deformations are still kac algebras using this construction we give new quantizations of the heisenberg group from this point of view we analyse the dimension 8 non trivial example of kac and paljutkin and give a new example of non trivial dimension 12 semi simple hopf algebras a dimension 12 kac algebra

whitney s extension problem for c m let f be a real valued function on a compact set in rn and let m be a positive integer we show how to decide whether f extends to a cm function on rn

asymptotically optimal admission control of a queue with impatient customers we consider a gi gi 1 queue with impatient customers in heavy traffic we use the solution of an approximating singular diffusion control problem to construct an admission control policy for the queue the approximating control problem does not admit a so called pathwise solution hence the resulting admission control policy depends on second moment data we prove asymptotic optimality of the constructed policy using weak convergence methods rh

cross product bialgebras i abstract the subject of this article is bialgebra factorizations or cross product bialgebras without cocycles we establish a theory characterizing cross product bialgebras universally in terms of projections and injections especially all known types of biproduct double cross product and bicross product bialgebras can be described by this theory furthermore the theory provides new families of cocycle free cross product bialgebras besides the universal characterization we find an equivalent co modular description of certain types of cross product bialgebras in terms of so called hopf data with the help of hopf data construction we recover again all known cross product bialgebras as well as new and more general types of cross product bialgebras we are working in the general setting of braided monoidal categories which allows us to apply our results in particular to the braided category of hopf bimodules over a hopf algebra majid s double biproduct is seen to be a twisting of a certain tensor product bialgebra in this category this resembles the case of the drinfel d double which can be constructed as a twist of a specific cross product

on the principle of smooth fit for a class of singular stochastic control problems for diffusions this paper considers the principle of smooth fit for a class of one dimensional singular stochastic control problems allowing the system to be of nonlinear diffusion type the existence and the uniqueness of a convex c 2 solution to the corresponding variational inequality are obtained it is proved that this solution gives the value function of the control problem and the optimal control process is constructed as an example of the degenerate case it is proved that the conclusion is also true for linear systems and the explicit formula for the smooth fit points is derived

tensor products of operator spaces abstract in this paper we lay the foundations for a systematic study of tensor products of subspaces of c algebras to accomplish this various notions of duality are introduced and employed elementary proofs of the complete injectivity of the haagerup norm and of the extension theorem for completely bounded maps are given pisier s gamma norms are examined and found to be special cases of the haagerup norm we identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense indeed most of the elementary theory of banach space tensor norms generalizes to the category of operator spaces

optimal policies for n dimensional singular stochastic control problems part i the skorokhod problem an n dimensional brownian motion is controlled by adding a process of locally bounded variation to it so as to minimize an expected infinite horizon discounted cost we show by direct probabilistic techniques that the optimal control is a solution of a generalized skorokhod problem

extensions of hopf algebras and lie bialgebras let f g be finite dimensional lie algebras over a field of characteristic zero regard f and g the dual lie coalgebra of g as lie bialgebras with zero cobracket and zero bracket respectively suppose that a matched pair f g of lie bialgebras is given which has structure maps then it induces a matched pair u f ug of hopf algebras where u f is the universal envelope of f and ug is the hopf dual of ug we show that the group opext u f ug of cleft hopf algebra extensions associated with u f ug is naturally isomorphic to the group opext f g of lie bialgebra extensions associated with f g an exact sequence involving either of these groups is obtained which is a variation of the exact sequence due to g i kac if g g g there follows a bijection between the set ext u f ug of all cleft hopf algebra extensions of u f by ug and the set ext f g of all lie bialgebra extensions of f by g

the whitney extension problem and lipschitz selections of set valued mappings in jet spaces we study a variant of the whitney extension problem 1934 for the space c k r n we identify c k r n with a space of lipschitz mappings from r n into the space p k x r of polynomial fields on r n equipped with a certain metric this identification allows us to reformulate the whitney problem for c k r n as a lipschitz selection problem for set valued mappings into a certain family of subsets of p k x r we prove a helly type criterion for the existence of lipschitz selections for such set valued mappings defined on finite sets with the help of this criterion we improve estimates for finiteness numbers in finiteness theorems for c k r n due to c fefferman

on cocycle twisting of compact quantum groups abstract we provide a class of examples of compact quantum groups and unitary 2 cocycles on them such that the twisted quantum groups are non compact but still locally compact quantum groups in the sense of kustermans and vaes this also gives examples of cocycle twists where the underlying c algebra of the quantum group changes

extensions of locally compact quantum groups and the bicrossed product construction abstract in the framework of locally compact quantum groups we study cocycle actions we develop the cocycle bicrossed product construction starting from a matched pair of locally compact quantum groups we define exact sequences and establish a one to one correspondence between cocycle bicrossed products and cleft extensions in this way we obtain new examples of locally compact quantum groups

mapping ideals of quantum group multipliers abstract we study the dual relationship between quantum group convolution maps l 1 g l g and completely bounded multipliers of g for a large class of locally compact quantum groups g we completely isomorphically identify the mapping ideal of row hilbert space factorizable convolution maps with m c b l 1 g yielding a quantum gilbert representation for completely bounded multipliers we also identify the mapping ideals of completely integral and completely nuclear convolution maps the latter coinciding with l 1 b g where b g is the quantum bohr compactification of g for quantum groups whose dual has bounded degree we show that the completely compact convolution maps coincide with c b g our techniques comprise a mixture of operator space theory and abstract harmonic analysis including fubini tensor products the non commutative grothendieck inequality quantum eberlein compactifications and a suitable notion of inner co amenable quantum group that we introduce and for which we exhibit examples from the bicrossed product construction our main results are new even in the setting of group von neumann algebras v n g for quasi sin locally compact groups g

existence of markov controls and characterization of optimal markov controls given a solution of a controlled martingale problem it is shown under general conditions that there exists a solution having markov controls which has the same cost as the original solution this result is then used to show that the original stochastic control problem is equivalent to a linear program over a space of measures under a variety of optimality criteria existence and characterization of optimal markov controls then follows an extension of echeverria s theorem characterizing stationary distributions for uncontrolled markov processes is obtained as a corollary in particular this extension covers diffusion processes with discontinuous drift and diffusion coefficients

the unitary implementation of a locally compact quantum group action abstract in this paper we study actions of locally compact quantum groups on von neumann algebras and prove that every action has a canonical unitary implementation paralleling haagerup s classical result on the unitary implementation of a locally compact group action this result is an important tool in the study of quantum groups in action we will use it in this paper to study subfactors and inclusions of von neumann algebras when is an action of the locally compact quantum group m on the von neumann algebra n we can give necessary and sufficient conditions under which the inclusion n n m n is a basic construction here n denotes the fixed point algebra and m n is the crossed product when is an outer and integrable action on a factor n we prove that the inclusion n n is irreducible of depth 2 and regular giving a converse to the results of m enock and r nest 1996 j funct anal 137 466 543 1998 j funct anal 154 67 109 finally we prove the equivalence of minimal and outer actions and we generalize the main theorem of yamanouchi 1999 math scand 84 297 319 every integrable outer action with infinite fixed point algebra is a dual action

cohomology and coquasi bialgebra extensions associated to a matched pair of bialgebras abstract by using braid diagrams we explicitly reconstruct the cohomology associated to a matched pair of cocommutative bialgebras in order to give a method of constructing coquasi bialgebras which generalize bialgebras and classifying them up to monoidal equivalence of their comodule categories an alternative homological proof is given for schauenburg s generalized kac sequence involving the abelian group opext h k of bialgebra extensions we define an abelian group opext h k of coquasi bialgebra extensions associated to a singer pair h k of bialgebras and prove a variant of schauenburg s sequence which involves the group it is also proved that there is a natural isomorphism opext h 1 k opext h 2 k that preserves monoidal equivalence classes if h 1 k and h 2 k arise from such matched pairs that are shifts of each other

calculations of some groups of hopf algebra extensions w let us recall the theory of group extensions with abelian kernels mc x chap iv let g be a group and m a g module the equivalence classes of group extensions m a e a g giving rise to the g module action on m form a group with respect to the baer sum furthermore this group is 2 isomorphic to the cohomology group h g m the theory is extended to the theory of hopf algebra extensions which w x are abelian in some sense h s we restrict our interests to hopf algebra extensions over a field k which are of the form

deformation of c algebras by cocycles on locally compact quantum groups given a c algebra a with a left action of a locally compact quantum group g on it and a unitary 2 cocycle on g we define a deformation a of a the construction behaves well under certain additional technical assumptions on the most important of which is regularity meaning that c0 g g is isomorphic to the algebra of compact operators on some hilbert space in particular then a is stably isomorphic to the iterated twisted crossed product g op g a also in good situations the c algebra a carries a left action of the deformed quantum group g and we have an isomorphism g a g a when g is a genuine locally compact group we show that the action of g on c0 g cr g is always integrable stronger assumptions of properness and saturation of the action imply regularity as an example we make a preliminary analysis of the cocycles on the duals of some solvable lie groups recently constructed by bieliavsky et al and discuss the relation of our construction to that of bieliavsky and gayral

closed quantum subgroups of locally compact quantum groups abstract we investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group two definitions one due to s vaes and one due to s l woronowicz are analyzed and relations between them discussed among many reformulations we prove that the former definition can be phrased in terms of quasi equivalence of representations of quantum groups while the latter can be related to an old definition of podle from the theory of compact quantum groups the cases of classical groups duals of classical groups compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context a deep relationship with the quantum group generalization of the herz restriction theorem from classical harmonic analysis is also established in particular in the course of our analysis we give a new proof of the herz restriction theorem

an algebraic framework for group duality abstract a hopf algebra is a pair a whereais an associative algebra with identity and a homomorphism formatoa asatisfying certain conditions if we drop the assumption thatahas an identity and if we allow to have values in the so called multiplier algebram a a we get a natural extension of the notion of a hopf algebra we call this a multiplier hopf algebra the motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group also for these multiplier hopf algebras there is a natural notion of left and right invariance for linear functionals called integrals in hopf algebra theory we show that if such invariant functionals exist they are unique up to a scalar and faithful for a regular multiplier hopf algebra a i e with invertible antipode with invariant functionals we construct in a canonical way the dual it is again a regular multiplier hopf algebra with invariant functionals it is also shown that the dual of is canonically isomorphic with the original multiplier hopf algebra a it is possible to generalize many aspects of abstract harmonic analysis here one can define the fourier transform one can prove plancherel s formula because any finite dimensional hopf algebra is a regular multiplier hopf algebra and has invariant functionals our duality theorem applies to all finite dimensional hopf algebras then it coincides with the usual duality for such hopf algebras but our category of multiplier hopf algebras also includes in a certain way the discrete quantum groups and the compact quantum groups our duality includes the duality between discrete quantum groups and compact quantum groups in particular it includes the duality between compact abelian groups and discrete abelian groups one of the nice features of our theory is that we have an extension of this duality to the non abelian case but within one category this is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type we prove that also these are dual to each other we treat an example that is sufficiently general to illustrate most of the different features of our theory it is also possible to construct the quantum double of drinfel d within this category this provides a still wider class of examples so we obtain many more than just the compact and discrete quantum within this setting

degenerate variance control of a one dimensional diffusion consider an ito equation for a scalar valued process that is controlled through a dynamic and adaptive choice of its diffusion coefficient such a control is called a variance control and is said to degenerate when it becomes zero we consider the problem of choosing a control to minimize a discounted infinite horizon cost that penalizes state values close to an equilibrium point of the drift and also imposes a control cost admissible controls are required to take values in the closed bounded interval 0 sigma 0 where sigma 0 0 in particular the control can be degenerate in general there will be a bang bang optimal control that takes the value sigma 0 in some open set and is zero otherwise we discuss the existence and properties of solutions to stochastic differential equations with such controls and characterize the value function and optimal control in more detail in the case of both linear and nonlinear drift employing the hamilton jacobi bellman equation and results of n v krylov theory probab appl 17 1973 pp 114 131 and p l lions comm pure appl math 34 1981 pp 121 147 we derive sufficient conditions for the existence of single region optimal controls construct examples of multiple region controls and provide bounds on the number and size of the regions in which the optimal control is positive

whitney s extension problems and interpolation of data given a function f e r with e rn we explain how to decide whether f extends to a cm function f on rn if e is finite then one can efficiently compute an f as above whose cm norm has the least possible order of magnitude joint work with b klartag

relative matched pairs of finite groups from depth two inclusions of von neumann algebras to quantum groupoids in this work we give a generalization of matched pairs of finite groups to describe a general class of depth two inclusions of factor von neumann algebras and the c quantum groupoids associated with using double groupoids

quantum groups with projection on von neumann algebra level abstract we introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study their properties our approach is slightly different from the one introduced in the thesis of s roy 16 and unlike the investigations of roy we work within the von neumann algebraic picture this allows the use of powerful techniques related to crossed products by actions of locally compact quantum groups in particular we show existence of a braided comultiplication on the algebra spanned by slices of braided multiplicative unitary

connections between optimal stopping and singular stochastic control i monotone follower problems the stochastic control problem of tracking a brownian motion by a nondecreasing process monotone follower is related to a question of optimal stopping direct probabilistic arguments are employed to show that the two problems are equivalent and that both admit optimal solutions

regularity of the value function for a two dimensional singular stochastic control problem it is desired to control a two dimensional brownian motion by adding a possibly singularly continuous process to it so as to minimize an expected infinite horizon discounted running cost the hamilton jacobi bellman characterization of the value function v is a variational inequality which has a unique twice continuously differentiable solution the optimal control process is constructed by solving the skorokhod problem of reflecting the two dimensional brownian motion along a free boundary in the nabla v direction

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h

cross product bialgebras part ii abstract this is the central article of a series of three papers on cross product bialgebras we present a universal theory of bialgebra factorizations or cross product bialgebras with cocycles and dual cocycles we also provide an equivalent co modular co cyclic formulation all known examples as for instance bi or smash doublecross and bicross product bialgebras as well as double biproduct bialgebras and bicrossed or cocycle bicross product bialgebras are now united within a single theory furthermore our construction yields various novel types of cross product bialgebras

classifying complements for associative algebras abstract for a given extension a e of associative algebras we describe and classify up to an isomorphism all a complements of e i e all subalgebras x of e such that e a x and a x 0 let x be a given complement and a x the canonical matched pair associated with the factorization e a x we introduce a new type of deformation of the algebra x by means of the given matched pair and prove that all a complements of e are isomorphic to such a deformation of x several explicit examples involving the matrix algebra are provided