isometric dilations for infinite sequences of noncommuting operators this paper develops a dilation theory for t n l an infinite sequence of noncommuting operators on a hilbert space when the matrix t1 t2 is a contraction a wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for tn i are obtained some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given this results are used to extend the sz nagy foia lifting theorem to this noncommutative setting this paper is a continuation of 5 and develops a dilation theory for an infinite sequence tn n of noncommuting operators on a hilbert space when en 1 tn tn i i is the identity on s many of the results and techniques in dilation theory for one operator 8 and also for two operators 3 4 are extended to this setting first we extend wold decomposition 8 4 to the case of an infinite sequence vn i of isometries with orthogonal final spaces in 2 we obtain a minimal isometric dilation for tn 1 by extending the schaffer construction in 6 4 using these results we give some theorems on the geometric structure of the space of the minimal isometric dilation finally we give some sufficient conditions on a sequence t7 n to be simultaneously quasi similar to a sequence rn n i of isometries on a hilbert space x with en rnrn n ix in 3 we use the above mentioned theorems to obtain the sz nagy foia5 lifting theorem 7 8 1 4 in our setting in a subsequent paper we will use the results of this paper for studying the characteristic function associated to a sequence tn lo 1 with ez 1 t t i throughout this paper a stands for the set 1 2 k k e n or the set n 1 2 received by the editors september 1 1987 1980 mathematics subject classification 1985 revision primary 47a20 secondary 47a45

tensor algebras overc correspondences representations dilations andc envelopes tensor algebras overc correspondences are noncommutative generalizations of the disk algebra they contain as special cases analytic crossed products semi crossed products and popescu s noncommutative disk algebras in this paper a dilation theorem and a commutant lifting theorem for representations of tensor algebras is proved these are used to show that for certainc correspondences thec envelopes in the sense of arveson of the tensor algebras are the cuntz pimsner algebras of the correspondences

isomorphisms of algebras associated with directed graphs given countable directed graphs g and g we show that the associated tensor algebras open image in new window g and open image in new window g are isomorphic as banach algebras if and only if the graphs g are g are isomorphic for tensor algebras associated with graphs having no sinks or no sources the graph forms an invariant for algebraic isomorphisms we also show that given countable directed graphs g g the free semigroupoid algebras open image in new window and open image in new window are isomorphic as dual algebras if and only if the graphs g are g are isomorphic in particular spatially isomorphic free semigroupoid algebras are unitarily isomorphic for free semigroupoid algebras associated with locally finite directed graphs with no sinks the graph forms an invariant for algebraic isomorphisms as well

conservation laws for equations of mixed elliptic hyperbolic and degenerate types for partial differential equations of mixed elliptic hyperbolic and degenerate types which are the euler lagrange equations for an associated lagrangian invariance with respect to changes in independent and dependent variables is investigated as are results in the classification of continuous one parameter symmetry groups for the variational and divergence symmetries conservation laws are derived via the method of multipliers the conservation laws resulting from anisotropic dilations are applied to prove uniqueness theorems for linear and nonlinear problems and the invariance under dilations of the linear part is used to derive critical exponent phenomena and to obtain localized energy estimates for supercritical problems

semi crossed products of c algebras abstract given a c algebra u and endomorphim there is an associated nonselfadjoint operator algebra z x u called the semi crossed product of u with if is an automorphim z x u can be identified with a subalgebra of the c crossed product z x u if u is commutative and is an automorphim satisfying certain conditions z x u is an operator algebra of the type studied by arveson and josephson suppose s is a locally compact hausdorff space s s is a continuous and proper map and is the endomorphim of u c0 s given by necessary and sufficient conditions on the map are given to insure that the semi crossed product z x c0 s is i semiprime ii semisimple ii strongly semisimple

duals of compact lie groups realized in the cuntz algebras and their actions on c algebras abstract as a first step towards a new duality theorem for compact groups we consider a representation category t g of a compact lie group g u d c whose objects are the tensor powers of the defining representation and whose arrows are the intertwiners we associate to t g in a natural way a c algebra og which can be identified with the fixed points of the cuntz algebra od under the natural action of g og carries a canonical endomorphism g and t g can be reconstructed from og g if g su d then og is simple we give conditions for an endomorphism of a unital c algebra a to determine an action of t su d on a such actions correspond to monomorphisms of osu d into a with natural intertwining properties

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

operator algebras and the conjugacy of transformations abstract a family of algebras is associated with each continuous map on a compact hausdorff space these algebras are called conjugacy algebras it is shown that two continuous maps are conjugate if and only if some conjugacy algebra of one is isomorphic to some conjugacy for the other numerous examples of conjugacy algebras are provided

a characterization of operator algebras abstract an operator algebra is a uniformly closed algebra of bounded operators on a hilbert space in this paper we give a characterization of unital operator algebras in terms of their matricial norm structure more precisely if a is an l matricially normed space and also an algebra with a completely contractive multiplication and an identity of norm 1 then there is a completely isometric isomorphism of a onto a unital operator algebra indeed the multiplication on a need not be assumed to be associative for this conclusion to follow examples are given to show that the condition on the identity is necessary it follows from the above that the quotient of an operator algebra by a closed two sided ideal with the natural matricial structure is again an operator algebra up to complete isometric isomorphism

a general implicit splitting for stabilizing numerical simulations of ito stochastic differential equations in this paper we describe a general 2nd order accurate weak sense procedure for stabilizing monte carlo simulations of ito stochastic differential equations the splitting procedure includes explicit runge kutta heun methods semi implicit methods and the trapezoidal rule we prove the semi implicit method of ottinger stochastic processes in polymeric fluids springer verlag berlin 1996 for stabilizing simulations in the presence of nonlinear drift coefficients

on the stability of internal waves the extended kdv equation ut uux u2ux uxxx 0 is widely used as a model describing internal waves in ideal fluids the equation admits a family of negative and positive solitary waves c these solitary waves exhibit the typical broadening effect seen in internal waves it is shown here that all solitary wave solutions of the extended kdv equation are orbitally stable the proof of stability is based on the general theory of grillakis et al 1987 j funct anal 74 160 for equations of the form ut je u which have two conserved integrals e u and v u a spectral analysis of the linear operator reduces the question of orbital stability to the question of whether the scalar function d c e c cv c is convex to prove the stability an explicit calculation showing the convexity is performed

semicrossed products of operator algebras and their c envelopes abstract let a be a unital operator algebra and let be an automorphism of a that extends to a automorphism of its c envelope c env a we introduce the isometric semicrossed product a is z and we show that c env a is z c env a z in contrast the c envelope of the familiar contractive semicrossed product a z may not equal c env a z our main tool for calculating c envelopes for semicrossed products is a new concept the relative semicrossed product of an operator algebra by an endomorphism as an application of our theory we show that if t x is the tensor algebra of a c correspondence x a and is a extendible automorphism of t x that fixes the diagonal elementwise then the contractive semicrossed product satisfies c env t x z o x z where o x denotes the cuntz pimsner algebra of x a this extends the main result of davidson and katsoulis 2010 6

c algebras and equivalences for c correspondences abstract we study several notions of shift equivalence for c correspondences and the effect that these equivalences have on the corresponding pimsner dilations among others we prove that non degenerate regular full c correspondences which are shift equivalent have strong morita equivalent pimsner dilations we also establish that the converse may not be true these results settle open problems in the literature in the context of c algebras we prove that if two non degenerate regular full c correspondences are shift equivalent then their corresponding cuntz pimsner algebras are strong morita equivalent this generalizes results of cuntz and krieger and muhly tomforde and pask as a consequence if two subshifts of finite type are eventually conjugate then their cuntz krieger algebras are strong morita equivalent our results suggest a natural analogue of the shift equivalence problem in the context of c correspondences even though we do not resolve the general shift equivalence problem we obtain a positive answer for the class of imprimitivity bimodules

c envelopes of semicrossed products by lattice ordered abelian semigroups abstract a semicrossed product is a non selfadjoint operator algebra encoding the action of a semigroup on an operator or c algebra we prove that when the positive cone of a discrete lattice ordered abelian group acts on a c algebra the c envelope of the associated semicrossed product is a full corner of a crossed product by the whole group by constructing a c cover that itself is a full corner of a crossed product and computing the shilov ideal we obtain an explicit description of the c envelope this generalizes a result of davidson fuller and kakariadis from z n to the class of all discrete lattice ordered abelian groups

finitely correlated representations of product systems of c correspondences over nk abstract we study isometric representations of product systems of correspondences over the semigroup n k which are minimal dilations of finite dimensional fully coisometric representations we show the existence of a unique minimal cyclic coinvariant subspace for all such representations the compression of the representation to this subspace is shown to be a complete unitary invariant for a certain class of graph algebras the nonself adjoint wot closed algebra generated by these representations is shown to contain the projection onto the minimal cyclic coinvariant subspace this class includes free semigroup algebras this result extends to a class of higher rank graph algebras which includes higher rank graphs with a single vertex

semicrossed products generated by two commuting automorphisms abstract in this paper we study the semicrossed product of a finite dimensional c algebra for two types of z 2 actions and identify them with matrix algebras of analytic functions in two variables we look at the connections with semicrossed by z actions

crossed products by endomorphisms and reduction of relations in relative cuntz pimsner algebras abstract starting from an arbitrary endomorphism of a unital c algebra a we construct a crossed product it is shown that the natural construction depends not only on the c dynamical system a but also on the choice of an ideal j orthogonal to ker the article gives an explicit description of the internal structure of this crossed product and in particular discusses the interrelation between relative cuntz pimsner algebras and partial isometric crossed products we present a canonical procedure that reduces any given c correspondence to the smallest c correspondence yielding the same relative cuntz pimsner algebra as the initial one in the context of crossed products this reduction procedure corresponds to the reduction of c dynamical systems and allows us to establish a coincidence between relative cuntz pimsner algebras and crossed products introduced

operator algebras and subproduct systems arising from stochastic matrices we study subproduct systems in the sense of shalit and solel arising from stochastic matrices on countable state spaces and their associated operator algebras we focus on the non self adjoint tensor algebra and viselter s generalization of the cuntz pimsner c algebra to the context of subproduct systems suppose that x and y are arveson stinespring subproduct systems associated to two stochastic matrices over a countable set and let t x and t y be their tensor algebras we show that every algebraic isomorphism from t x onto t y is automatically bounded furthermore t x and t y are isometrically isomorphic if and only if x and y are unitarily isomorphic up to a automorphism of l when is finite we prove that t x and t y are algebraically isomorphic if and only if there exists a similarity between x and y up to a automorphism of l moreover we provide an explicit description of the cuntz pimsner algebra o x in the case where is finite and the stochastic matrix is essential

crossed products of operator algebras applications of takai duality abstract let g be a partially ordered abelian group with haar measure let a g be a dynamical system and let a be the associated semicrossed product using takai duality we establish a stable isomorphism a s a k g ad g where k g denotes the compact operators in the csl algebra alg l g and denotes the right regular representation of g we also show that there exists a complete lattice isomorphism between the invariant ideals of a and the ad invariant ideals of a k g using takai duality we also continue our study of the radical for the crossed product of an operator algebra and we solve open problems stemming from the earlier work of the authors among others we show that the crossed product of a radical operator algebra by a compact abelian group is a radical operator algebra we also show that the permanence of semisimplicity fails for crossed products by r a final section of the paper is devoted to the study of radically tight dynamical systems i e dynamical systems a g for which the identity rad a g rad a g persists a broad class of such dynamical systems is identified

dilating covariant representations of the non commutative disc algebras let be an isometric automorphism of the non commutative disc algebra an for n 2 we show that every contractive covariant representation of an dilates to a unitary covariant representation of on hence the c envelope of the semicrossed product an z is on z

the ideal structure of certain nonselfadjoint operator algebras soit x un systeme dynamique localement compact et z x co x la sous algebre close en norme du produit croise zx co x engendre par les puissances non negatives de dans le cas homeomorphisme on determine la structure ideale de ces algebres dans le cas ou agit librement

isomorphisms between topological conjugacy algebras a family of algebras which we call topological conjugacy algebras is associated with each proper continuous map on a locally compact hausdorff space assume that i xi xi is a continuous proper map on a locally compact hausdorff space xi for i 1 2 we show that the dynamical systems x1 1 and x2 2 are conjugate if and only if some topological conjugacy algebra of x1 1 is isomorphic as an algebra to some topological conjugacy algebra of x2 2 this implies as a corollary the complete classification of the semicrossed products c0 x z which was previously considered by arveson and josephson 3 peters 19 hadwin and hoover 8 and power 21 we also obtain a complete classification of all semicrossed products of the form a d z where a d denotes the disc algebra and d d a continuous map which is analytic on the interior in this case a surprising dichotomy appears in the classification scheme which depends on the fixed point set of we also classify more general semicrossed products of uniform algebras

stabilizing simulations of complex stochastic representations for quantum dynamical systems path integral representations of quantum dynamics can often be formulated as stochastic differential equations sdes in a series of papers corney and drummond 2004 phys rev lett 93 260401 deuar and drummond 2001 comput phys commun 142 442 5 drummond and gardnier 1980 j phys a math gen 13 2353 68 gardiner and zoller 2004 quantum noise a handbook of markovian and non markovian quantum stochastic methods with applications to quantum optics springer series in synergetics 3rd edn berlin springer and gilchrist et al 1997 phys rev a 55 3014 32 and their collaborators have derived sdes from coherent states representations for density matrices computationally these sdes are attractive because they seem simple to simulate they can be quite unstable however in this paper we consider some of the instabilities and propose a few remedies particularly because the variances of the simulated paths typically grow exponentially the processes become de localized in relatively short times hence the issues of boundary conditions and stable integration methods become important we use the bose einstein hamiltonian as an example our results reveal that it is possible to significantly extend integration times and show the periodic structure of certain functionals

a class of c algebras generalizing both graph algebras and homeomorphism c algebras i fundamental results we introduce a new class of c algebras which is a generalization of both graph algebras and homeomorphism c algebras this class is very large and also very tractable we prove the so called gauge invariant unique ness theorem and the cuntz krieger uniqueness theorem and compute the k groups of our algebras

sufficient conditions for stability of solitary wave solutions of model equations for long waves abstract we consider solitary wave solutions of model equations for long waves that feature a general form of linear dispersion sufficient conditions for the non linear stability of such solutions are derived these conditions are shown to obtain for the korteweg de vries equation and certain of its generalizations such as the benjamin ono equation and the intermediate long wave equation

circle actions on c algebras partial automorphisms and a generalized pimsner voiculescu exact sequence abstract we introduce a method to study c algebras possessing an action of the circle group from the point of view of their internal structure and their k theory under relatively mild conditions our structure theorem shows that any c algebra where an action of the circle is given arises as the result of a construction that generalizes crossed products by the group of integers such a generalized crossed product construction is carried out for any partial automorphism of a c algebra where by a partial automorphism we mean an isomorphism between two ideals of the given algebra our second main result is an extension to crossed products by partial automorphisms of the celebrated pimsner voiculescu exact sequence for k groups the representation theory of the algebra arising from our construction is shown to parallel the representation theory for c dynamical systems in particular we generalize several of the main results relating to regular and covariant representations of crossed products

hamiltonian long wave expansions for free surfaces and interfaces the theory of internal waves between two bodies of immiscible fluid is important both for its interest to ocean engineering and as a source of numerous interesting mathematical model equations which exhibit nonlinearity and dispersion in this paper we derive a hamiltonian formulation of the problem of a dynamic free interface with rigid lid upper boundary conditions and of a free surface and a free interface this latter situation occurring more commonly in experiment and in nature from the formulation we develop a hamiltonian perturbation theory for the long wave limits and we carry out a systematic analysis of the principal long wave scaling regimes this analysis provides a uniform treatment of the classical works of peters and stoker 28 benjamin 3 4 ono 26 and many others our considerations include the boussinesq and kdv regimes over finite depth fluids the benjamin ono regimes in the situation in which one fluid layer is infinitely deep and the intermediate long wave regimes in addition we describe a novel class of scaling regimes of the problem in which the amplitude of the interface disturbance is of the same order as the mean fluid depth and the characteristic small parameter corresponds to the slope of the interface our principal results are that we highlight the discrepancies between the case of rigid lid and of free surface upper boundary conditions which in some circumstances can be significant motivated by the recent results of choi and camassa 6 7 we also derive novel systems of nonlinear dispersive long wave equations in the large amplitude small slope regime our formulation of the dynamical free surface free interface problem is shown to be very effective for perturbation calculations and as well it holds promise as a basis for numerical simulations c 2000 wiley periodicals inc

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

on c algebras associated with c correspondences abstract we study c algebras arising from c correspondences which were introduced by the author we prove the gauge invariant uniqueness theorem and obtain conditions for our c algebras to be nuclear exact or satisfy the universal coefficient theorem we also obtain a 6 term exact sequence of k groups involving the k groups of our c algebras

residually finite dimensional operator algebras abstract we study non selfadjoint operator algebras that can be entirely understood via their finite dimensional representations in contrast with the elementary matricial description of finite dimensional c algebras in the non selfadjoint setting we show that an additional level of flexibility must be allowed motivated by this peculiarity we consider a natural non selfadjoint notion of residual finite dimensionality we identify sufficient conditions for the tensor algebra of a c correspondence to enjoy this property to clarify the connection with the usual self adjoint notion we investigate the residual finite dimensionality of the minimal and maximal c covers associated to an operator algebra

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely