twisted dual group algebras equivariant deformations of c0 g abstract we consider a family of twisted fourier algebras a g of a locally compact group g which in the case of a abelian group g are the fourier transforms of the usual twisted group algebras of over cap g o the corresponding c algebras c g o are deformations of c 0 g which are equivariant in the sense that g still acts by left translation the main examples come from cocycles on the dual of an abelian subgroup h of g we prove that for such cocycles the twisted dual group algebras c g o are induced from the twisted group algebras c h o and we give detailed formulas for the multiplication on a g which extend to larger dense subalgebras of c 0 g and c b g we anticipate that these larger subalgebras will be useful for constructing deformations of homogeneous spaces c 0 g

linear quadratic differential games closed loop saddle points the object of this paper is to revisit the results of bernhard j optim theory appl 27 1979 pp 51 69 on two person zero sum linear quadratic differential games and generalize them to utility functions without positivity assumptions on the matrices acting on the state variable and to linear dynamics with bounded measurable data matrices our paper specializes to state feedback via lebesgue measurable affine closed loop strategies with possible non l 2 integrable singularities after sharpening the recent results of delfour siam j control optim 46 2007 pp 750 774 on the characterization of the open loop lower and upper values of the game it first deals with l 2 integrable closed loop strategies and then with the larger family of strategies that may have non l 2 integrable singularities a new conceptually meaningful and mathematically precise definition of a closed loop saddle point is introduced to simultaneously handle state feedbacks of the l 2 type and smooth locally bounded ones except at most in the neighborhood of finitely many instants of time a necessary and sufficient condition is that the free end problem be normalizable almost everywhere this relaxation of the classical notion allows singularities in the feedback law at an infinite number of instants including accumulation points that are not isolated a complete classification of closed loop saddle points is given in terms of the convexity concavity properties of the utility function and connections are given with the open loop lower value upper value and value of the game

rieffel deformation via crossed products we start from rieffel data a where a is a c algebra is an action of an abelian group on a and is a 2 cocycle on the dual group using landstad theory of crossed product we get a deformed c algebra a in the case of rn we obtain a very simple proof of invariance of k groups under the deformation in the general case we also get a very simple proof that nuclearity is preserved under the deformation we show how our approach leads to quantum groups and investigate their duality the general theory is illustrated by an example of the deformation of sl 2 c a description of it in terms of noncommutative coordinates is given

crossed products of c correspondences by amenable group actions abstract let x a be a c correspondence with a continuous action by an amenable locally compact group g we will show that if x g a g is the induced c correspondence and is the induced action on o x then o x g o x g as applications we obtain b abadie takai duality for crossed products by hilbert c modules preprint arxiv 0709 1122v1 proposition 4 5 and a kumjian d pask c algebras of directed graphs and group actions ergodic theory dynam systems 19 6 1999 1503 1519 corollary 2 5

linear quadratic differential games saddle point and riccati differential equation zhang siam j control optim 43 2005 pp 2157 2165 recently established the equivalence between the finiteness of the open loop value of a two player zero sum linear quadratic lq game and the finiteness of its open loop lower and upper values in this paper we complete and sharpen the results of zhang for the finiteness of the lower value of the game by providing a set of necessary and sufficient conditions that emphasizes the feasibility condition 0 0 is a solution of the open loop lower value of the game for the zero initial state then we show that under the assumption of an open loop saddle point in the time horizon 0 t for all initial states there is an open loop saddle point in the time horizon s t for all initial times s 0 le s t and all initial states at time s from this we get an optimality principle and adapt the invariant embedding approach to construct the decoupling symmetrical matrix function p s and show that it is an h 1 0 t solution of the matrix riccati differential equation thence an open loop saddle point in 0 t yields closed loop optimal strategies for both players furthermore a necessary and sufficient set of conditions for the existence of an open loop saddle point in 0 t for all initial states is the convexity concavity of the utility function and the existence of an h 1 0 t symmetrical solution to the matrix riccati differential equation as an illustration of the cases where the open loop lower upper value of the game is infty infty we work out two informative examples of solutions to the riccati differential equation with and without blow up time

compact group actions on c algebras an application of non commutative duality abstract we first prove that for a coaction of a compact group on a c algebra a the largest liminal resp postliminal ideal of a is invariant under the coaction as a consequence of this and an earlier characterization by the authors of the ideals of a crossed product algebra which are invariant under the dual coaction we answer affirmatively a question of landstad s and olesen s specifically we prove that if is an action of a compact group g on a c algebra a and if the fixed point algebra a is liminal resp postliminal then so is the crossed product algebra g a

equivariant deformations of homogeneous spaces abstract c algebraic deformations of homogeneous spaces g are constructed by completing dense subspaces of c 0 g in a different multiplication and c norm these deformations are equivariant in the sense that they still carry a natural action of g by left translation the motivating examples are the noncommutative heisenberg manifolds of rieffel but the construction given here is based on the authors twisted dual group algebras which are equivariant deformations of c 0 g the procedure is general enough to give other recently studied families of noncommutative spaces and some interesting new examples of simple c algebras

generalized fixed point algebras and square integrable group actions we analyze rieffel s construction of generalized fixed point algebras in the setting of group actions on hilbert modules let g be a locally compact group acting on a c algebra b we construct a hilbert module f over the reduced crossed product of g and b using a pair e r where e is an equivariant hilbert module over b and r is a dense subspace of e with certain properties the generalized fixed point algebra is the c algebra of compact operators on f any hilbert module over the reduced crossed product arises by this construction for a pair e r that is unique up to isomorphism a necessary condition for the existence of r is that e be square integrable the consideration of square integrable representations of abelian groups on hilbert space shows that this condition is not sufficient and that different choices for r may yield different generalized fixed point algebras if b is proper in kasparov s sense there is a unique r with the required properties thus the generalized fixed point algebra only depends on e

morita rieffel equivalence and spectral theory for integrable automorphism groups of c algebras abstract given a c dynamical system a g we discuss conditions under which subalgebras of the multiplier algebra m a consisting of fixed points for are morita rieffel equivalent to ideals in the crossed product of a by g in case g is abelian we also develop a spectral theory giving a necessary and sufficient condition for to be equivalent to the dual action on the cross sectional c algebra of a fell bundle in our main application we show that a proper action of an abelian group on a locally compact space is equivalent to a dual action

pull backs of algebras and crossed products by certain diagonal actions let g be a locally compact group and p q t a principal g bundle if a is a c algebra with primitive ideal space t the pull back p a of a along p is the balanced tensor product co q c t a if 3 g aut a consists of c t module automorphisms and y g aut co q is the natural action then the automorphism group y 0x 3 of co q 0 a respects the balancing and induces the diagonal action p 3 of g on p a we discuss some examples of such actions and study the crossed product p a x p g we suggest a substitute d for the fixed point algebra prove p a x g is strongly morita equivalent to d and investigate the structure of d in various cases in particular we ask when d is strongly morita equivalent to a sometimes but by no means always and investigate the case where a has continuous trace let b be a c algebra and g a locally compact group acting on b as a strongly continuous automorphism group a our goal here is to study the crossed product c algebra b x g for two classes of diagonal actions for which the induced action of g on b is free the first class includes actions of the form y 0x3 on b co q 0 a where p q t is a principal g bundle y is the dual action of g on co q and 3 g aut a is an action of g on another c algebra a we also consider diagonal actions on algebras which are the pull backs of another algebra a along a principal bundle p q t if a is a c algebra with primitive ideal space t then the pull back p a is the balanced tensor product co q cb t a when 3 g aut a consists of c t module automorphisms the product action ty 0 3 preserves the balancing and the diagonal action p 3 is by definition the induced action on p a in general if f x y and q prima y are continuous then cb y acts on co x by composition with f and on a by composition with q and the daunshofmann theorem we can therefore define the pull back f a of a along f as the c algebraic tensor product co x cb y a the reason for the name is that when a is the algebra of sections of some c bundle e over y there is a natural isomorphism of f a onto the algebra of sections of the pull back f e in 1 we discuss this and other basic properties of pull backs and give some evidence to show they are likely to be of interest in particular we show that if g is abelian and a g aut a is locally unitary in the sense of 18 then the crossed product received by the editors october 20 1983 and in revised form february 15 1984 1980 mathematics subject classification primary 46l40 46l55

deformation of operator algebras by borel cocycles abstract assume that we are given a coaction of a locally compact group g on a c algebra a and a t valued borel 2 cocycle on g motivated by the approach of kasprzak to rieffel s deformation we define a deformation a of a among other properties of a we show that a k l 2 g is canonically isomorphic to a g g this together with a slight extension of a result of echterhoff et al implies that for groups satisfying the baum connes conjecture the k theory of a remains invariant under homotopies of

deformation quantization for actions of qpd abstract the main objective of this article is to develop the theory of deformation of c algebras endowed with a group action from the perspective of non formal equivariant quantization this program initiated in 4 aims to extend rieffel s deformation theory 27 for more general groups than r d in 4 we have constructed such a theory for a class of non abelian lie groups in the present article we study the somehow opposite situation of abelian but non lie groups more specifically we construct here a deformation theory of c algebras endowed with an action of a finite dimensional vector space over a non archimedean local field of characteristic different from 2 at the root of our construction stands the p adic version of the weyl quantization introduced by haran 12 and further extended by bechata 1 and unterberger 34

proper actions fixed point algebras and naturality in nonabelian duality abstract suppose a locally compact group g acts freely and properly on a locally compact hausdorff space x and let be the induced action on c 0 x we consider a category in which the objects are c dynamical systems a g for which there is an equivariant homomorphism of c 0 x into the multiplier algebra m a rieffel has shown that such systems are proper and saturated and hence have a generalized fixed point algebra a which is morita equivalent to a r g we show that the assignment a a is functorial and that rieffel s morita equivalence is natural in a suitable sense we then use our results to prove a categorical version of landstad duality which characterizes crossed products by coactions and to prove that mansfield imprimitivity for crossed products by homogeneous spaces is natural

induced representations of crossed products by coactions abstract let a m a c r g be a coaction of a locally compact group g on a c algebra a then for any closed normal amenable subgroup h of g we define a coaction a m a c r g h of g h on a we present dense subalgebras of the crossed products a g and a g h and use these to induce representations of a g h to representations of a g we then formulate an imprimitivity theorem for the induction process that is we classify the induced representations we also investigate the continuity of the induction process and are able to prove rieffel s conjecture that the dual action to a coaction is proper and saturated

some results on two person zero sum linear quadratic differential games considered in this paper are the two person zero sum linear quadratic differential games it is shown that the value of the game exists if and only if both the upper and lower values exist as a consequence we prove that another necessary and sufficient condition for the existence of the values of the game is the existence of an open loop open loop saddle point an example is also given in which the lower value exists but the upper value does not exist

incorporate intelligence into an ecological system an adaptive fuzzy control approach in this paper we incorporate intelligence into kolmogorov s ecological system that is we propose a fuzzy system to combine heuristic rules into the design of nonlinear adaptive controller where the heuristic rules are in the form of fuzzy if then rules by using lyapunov synthesis approach we attain an adaptive law adjusting the free parameters in the controller such that the closed loop system follows a desired trajectory although there exist uncertainty and chaos in komogorov s model application shows that an adaptive fuzzy control is a powerful tool to pursuit sustainable development of ecology

large population lqg games involving a major player the nash certainty equivalence principle we consider linear quadratic gaussian lqg games with a major player and a large number of minor players the major player has a significant influence on others the minor players individually have negligible impact but they collectively contribute mean field coupling terms in the individual dynamics and costs to overcome the dimensionality difficulty and obtain decentralized strategies the so called nash certainty equivalence methodology is applied the control synthesis is preceded by a state space augmentation via a set of aggregate quantities giving the mean field approximation subsequently within the population limit the lqg game is decomposed into a family of limiting two player games as each is locally seen by a representative minor player next when solving these limiting two player games we impose certain interaction consistency conditions such that the aggregate quantities initially assumed coincide with the ones replicated by the closed loop of a large number of minor players this procedure leads to a set of decentralized strategies for the original lqg game which is an varepsilon nash equilibrium

optimal fuzzy control of a poisoning pest model in this paper optimal fuzzy control of a poisoning pest model is considered optimal control has been successfully applied to many biological models we give a rule based fuzzy control for the poisoning pest model by using the pontryagin minimum principle the optimal fuzzy control function can be worked out by a computer with the membership function of the fuzzy sets of the universe of discourse the control parameters of the poisoning pest model will be suitable for practical applications

weakly proper group actions mansfield s imprimitivity and twisted landstad duality alcides buss and siegfried echterhoffabstract using the theory of weakly proper actions of locally compactgroups recently developed by the authors we give a uni ed proof of both re duced and maximal versions of mans eld s imprimitivity theorem and obtaina general version of landstad s duality theorem for twisted group coactions as one application we obtain the stabilization trick for arbitrary twisted coac tions showing that every twisted coaction is morita equivalent an in atedcoaction

metric aspects of noncommutative homogeneous spaces for a closed cocompact subgroup of a locally compact group g given a compact abelian subgroup k of g and a homomorphism k g satisfying certain conditions landstad and raeburn constructed equivariant noncommutative deformations c g of the homogeneous space g generalizing rieffel s construction of quantum heisenberg manifolds we show that when g is a lie group and g is connected given any norm on the lie algebra of g the seminorm on c g induced by the derivation map of the canonical g action defines a compact quantum metric furthermore it is shown that this compact quantum metric space depends on continuously with respect to quantum gromov hausdorff distances

global attractivity and oscillations in a periodic delay logistic equation abstract sufficient conditions are obtained for the global attractivity of a positive periodic solution of the delay logistic equation dx t dt r t x t 1 x t n k t where r and k are positive periodic functions of period and n is a positive integer sufficient conditions are also obtained for all solutions to be oscillatory about k

crossed products by actions which are locally unitary on the stabilisers let a g be a c dynamical system with g abelian and hausdorff we investigate the ideal structure of the crossed product a g under the hypothesis that the stabiliser subgroups for the action of g on vary continuously we discuss a new notion of locally trivial g space for such actions and dually actions which are locally unitarily implemented on the stabiliser groups our main result asserts that when is locally unitary in this sense and is a locally trivial g space a g is a locally trivial space

continuous fields of c algebras coming from group cocycles and actions recently i have been attempting to formulate a suitable c algebraic framework for the subject of deformation quantization 3 19 continuous fields of c algebras provide one of the key elements for this framework the main examples of deformation quantizations which i have constructed up to now in this c algebra framework come from letting either cocycles on groups or actions of groups vary it has thus become necessary to show that one obtains in this way fields of c aigebras that are indeed continuous since this material is of a general nature and can be useful in other situations 4 6 11 13 it has seemed appropriate to give a separate exposition of it in the present article section 1 of this article contains a review of the published results on continuous fields which we will need as well as a discussion of the fact that the approach which we will take involves treating upper and lower semi continuity separately in sect 2 we discuss the continuity of fields of c algebras which arise from varying cocycles on groups while in sect 3 we do the same for actions which vary

pointwise unitary automorphism groups abstract let a g be a separable c dynamical system in which g is abelian and a has continuous trace the action is said to pointwise unitary if for each in a there is a covariant representation u of the system a g it is shown that for pointwise unitary actions the spectrum a g of the crossed product is a locally compact hausdorff space and the restriction map res which takes u in a g to maps a g onto a furthermore res induces a homeomorphism of a g g onto a these results lead to a characterization of crossed products a g with continuous trace when a has continuous trace and g acts freely on a this happens precisely when the action of g on a is also proper this extends green s characterization of free and proper transformation groups finally it is shown that the results have applications to the problem of determining the dual topology on a locally compact group

the primitive ideal space of two step nilpotent group c algebras abstract let n be a two step nilpotent locally compact second countable group having center z and quotient a n z we study the jacobson topology on the primitive ideal space prim c n of the group c algebra of n we are able to describe this topology in terms of convergence of subgroup representation pairs as used by the first author in an earlier work under appropriate conditions on n we are able to describe prim c n globally as the quotient of a principal bundle over modulo an equivalence relation determined entirely by the group structure we use this second result to compute the primitive ideal spaces of several examples including all finitely generated non torsion two step nilpotent discrete groups of rank less than or equal to five applications of our methods to more general central twisted crossed products are discussed

linear quadratic stochastic differential games open loop and closed loop saddle points in this paper we consider a linear quadratic stochastic two person zero sum differential game the controls for both players are allowed to appear in both drift and diffusion of the state equation the weighting matrices in the performance functional are not assumed to be definite non singular the existence of an open loop saddle point is characterized by the existence of an adapted solution to a linear forward backward stochastic differential equation with constraints together with a convexity concavity condition and the existence of a closed loop saddle point is characterized by the existence of a regular solution to a riccati differential equation it turns out that there is a significant difference between open loop and closed loop saddle points also it is found that there is an essential feature that prevents a linear quadratic optimal control problem from being a special case of linear quadratic two person zero sum differential games

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

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

induced c algebras and landstad duality for twisted coactions suppose n is a closed normal subgroup of a locally compact group g a coaction e a m a c n of n on a c algebra a can be inflated to a coaction s of g on a and the crossed product a x g is then isomorphic to the induced c algebra ind a xe n we prove this and a natural generalization in which a xf n is replaced by a twisted crossed product a x g n g in case g is abelian we recover a theorem of olesen and pedersen we then use this to extend the landstad duality of the first author to twisted crossed products and give several applications in particular we prove that if l n d g n 1 is topologically trivial but not necessarily split as a group extension then every twisted crossed product a x g n g is isomorphic to a crossed product of the form a x n

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

k groups of c algebras deformed by actions of rd abstract we show that the topological k groups of a c algebra deformed by an action of r d are isomorphic to those of the original c a1gebra we do this by exhibiting the deformed algebra as a suitable generalized fixed point algebra this also shows that the deformed algebra is nuclear if the original one is

linear quadratic uncertain differential game with application to resource extraction problem uncertain differential game investigates interactive decision making of players over time and the system dynamics is described by an uncertain differential equation this paper goes further to study the two player zero sum uncertain differential game in order to guarantee the saddle point nash equilibrium a max min theorem is provided furthermore when the system dynamics is described by a linear uncertain differential equation and the performance index function is quadratic the existence of saddle point nash equilibrium is obtained via the solvability of a corresponding riccati equation finally a resource extraction problem is analyzed by using the theory proposed in this paper