spectral flow is the integral of one forms on the banach manifold of self adjoint fredholm operators abstract one may trace the idea that spectral flow should be given as the integral of a one form back to the 1974 vancouver icm address of i m singer our main theorem gives analytic formulae for the spectral flow along a norm differentiable path of self adjoint bounded breuer fredholm operators in a semifinite von neumann algebra these formulae have a geometric interpretation which derives from the proof namely we define a family of banach submanifolds of all bounded self adjoint breuer fredholm operators and on each submanifold define global one forms whose integral on a norm differentiable path contained in the submanifold calculates the spectral flow along this path we emphasise that our methods do not give a single globally defined one form on the self adjoint breuer fredholms whose integral along all paths is spectral flow rather as the choice of the plural forms in the title suggests we need a family of such one forms in order to confirm singer s idea the original context for this result concerned paths of unbounded self adjoint fredholm operators we therefore prove analogous formulae for spectral flow in the unbounded case as well the proof is a synthesis of key contributions by previous authors whom we acknowledge in detail in the introduction combined with an additional important recent advance in the differential calculus of functions of non commuting operators

asymptotic expansions for trace functionals abstract we obtain taylor approximations for functionals v tr f h 0 v defined on the bounded self adjoint operators where h 0 is a self adjoint operator with compact resolvent and f is a sufficiently nice scalar function relaxing assumptions on the operators made in 17 and derive estimates and representations for the remainders of these approximations

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

abelian mathrm c ast subalgebras of mathrm c ast algebras of real rank zero and inductive limit mathrm c ast algebras

numerical methods for porous medium equation by an energetic variational approach abstract we study numerical methods for porous media equation pme there are two important characteristics the finite speed propagation of the free boundary and the potential waiting time which make the problem difficult to handle based on different dissipative energy laws we develop two numerical schemes by an energetic variational approach firstly based on f log f as the total energy form of the dissipative law we obtain the trajectory equation and then construct a fully discrete scheme it is proved that the scheme is uniquely solvable on an admissible convex set by taking the advantage of the singularity of the total energy next based on 1 2 f as the total energy form of the dissipation law we construct a linear numerical scheme for the corresponding trajectory equation both schemes preserve the corresponding discrete dissipation law meanwhile under some smoothness assumption both schemes are second order convergent in space and first order convergent in time each scheme yields a good approximation for the solution and the free boundary no oscillation is observed for the numerical solution around the free boundary furthermore the waiting time problem could be naturally treated which has been a well known difficult issue for all the existing methods due to its linear nature the second scheme is more efficient

a rotational velocity correction projection method for unsteady incompressible magnetohydrodynamics equations abstract this paper provides a rotational velocity correction projection method for the unsteady incompressible magnetohydrodynamics mhd equations in this method the electrical field variable e was eliminated in this way the gauss s law can be conserved the velocity variable u and the pressure field p are solved by a rotational velocity correction projection method the optimal error estimates of the l 2 norm will be given finally we shall present some numerical results to confirm our analysis it shows that our method is effective and can conserve the gauss s law very well

asynchronous iterative water filling for gaussian frequency selective interference channels this paper considers the maximization of information rates for the gaussian frequency selective interference channel subject to power and spectral mask constraints on each link to derive decentralized solutions that do not require any cooperation among the users the optimization problem is formulated as a static noncooperative game of complete information to achieve the so called nash equilibria of the game we propose a new distributed algorithm called asynchronous iterative water filling algorithm in this algorithm the users update their power spectral density psd in a completely distributed and asynchronous way some users may update their power allocation more frequently than others and they may even use outdated measurements of the received interference the proposed algorithm represents a unified framework that encompasses and generalizes all known iterative water filling algorithms e g sequential and simultaneous versions the main result of the paper consists of a unified set of conditions that guarantee the global converge of the proposed algorithm to the unique nash equilibrium of the game

on error estimates of some higher order projection and penalty projection methods for navier stokes equations this paper is a continuation of our previous work 10 on projection methods we study first existing higher order projection schemes in the semidiscretized form for the navier stokes equations one error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent neumann boundary condition satisfied by the pressure approximations we then propose a penalty projection scheme for which we obtain improved error estimates

classification ofc algebras of real rank zero and unsuspendede equivalence types abstract in this article examples are given to prove that the graded scaled ordered k group is not the complete invariant for a c algebra in the class of unital separable nuclear c algebras of real rank zero and stable rank one even for a c algebra in the subclass which consists of those real rank zero stable rank one c algebras being expressed as inductive limits of k n i 1 m n i c x n i where x n i are two dimensional finite cw complexes and n i are positive integers in the case of simple such c algebras it has been proved that the above invariant is the complete invariant by george elliott and the author these examples prove that the classification conjecture of elliott for the case of non simple real rank zero c algebras should be revised one needs extra invariants the obstruction preventing two such c algebras with the same graded scaled ordered k group from being isomorphic is that they have different unsuspended e equivalence types a refinement of kk equivalence type of c algebras due to connes and higson in this article it is proved that for the above class of inductive limit c algebras the obstruction of unsuspended e equivalence type is the only obstruction i e if two c algebras in the class are unsuspended e equivalence then they are isomorphic it is a surprise that in the case of simple such c algebras or even the case of c algebras with finitely many ideals the obstruction will disappear see section 4

error analysis for chorin s original fully discrete projection method and regularizations in space and time over twenty five years ago chorin proposed a computationally efficient method for computing viscous incompressible flow which has influenced the development of efficient modern methods and inspired much analytical work using asymptotic error analysis techniques it is now possible to describe precisely the kind of errors that are generated in the discrete solutions from this method and the order at which they occur while the expected convergence rate is seen for velocity the pressure accuracy is degraded by two effects a numerical boundary layer due to the projection step and a global error due to the alternating or parasitic modes present in the discretization of the incompressibility condition the error analysis of the projection step follows the work of e and liu and the analysis of the alternating modes is due to the author the two are combined to show the asymptotic character of the errors in the scheme regularization methods in space and time for recovering full accuracy for the computed pressure are discussed

a distributed power allocation scheme for sum rate maximization on cognitive gmacs this paper considers a distributed power allocation scheme for sum rate maximization under cognitive gaussian multiple access channels gmacs where primary users and secondary users may communicate under mutual interference with the gaussian noise formulating the problem as a standard nonconvex quadratically constrained quadratic problem qcqp provides a simple distributed method to find a solution using iterative jacobian method instead of using centralized schemes a totally asynchronous distributed power allocation for sum rate maximization on cognitive gmacs is suggested simulation results show that this distributed algorithm for power allocation converges to a fixed point and the solution achieves almost the same performance as the exhaustive search

theory of strongly irreducible operators and its applications in the matrix theory of finite dimensional space the famous jordan canonical form theorem sufficiently reveals the internal structure of matrices in the survey we will give similar jordan canonical form theorems of operator theory in a complex separable infinite dimensional hilbert space by the way we also show their applications in other mathematical branches

the banach saks p property we obtain general theorems which enable the calculation of the banach saks index set for rearrangement invariant function spaces in terms of their boyd indices for many important classes of rearrangement invariant spaces the index set is non trivial if and only if the boyd indices of the underlying space are non trivial although the latter equivalence does not hold in full generality we also study banach saks index sets defined for weakly null sequences of identically distributed independent random variables and their connections with the classical banach saks index set

a class of fully second order accurate projection methods for solving the incompressible navier stokes equations in this paper a continuous projection method is designed and analyzed the continuous projection method consists of a set of partial differential equations which can be regarded as an approximation of the navier stokes n s equations in each time interval of a given time discretization the local truncation error lte analysis is applied to the continuous projection methods which yields a sufficient condition for the continuous projection methods to be temporally second order accurate based on this sufficient condition a fully second order accurate discrete projection method is proposed a heuristic stability analysis is performed to this projection method showing that the present projection method can be stable the stability of the present scheme is further verified through numerical experiments the second order accuracy of the present projection method is confirmed by several numerical test cases

weak type commutator and lipschitz estimates resolution of the nazarov peller conjecture let m be a semi finite von neumann algebra and let f r c be a lipschitz function if a b m are self adjoint operators such that a b l1 m then

resolution of peller s problem concerning koplienko neidhardt trace formulae a formula for the norm of a bilinear schur multiplier acting from the cartesian product s 2 s 2 of two copies of the hilbert schmidt classes into the trace class s 1 is established in terms of linear schur multipliers acting on the space s 1 of all compact operators using this formula we resolve peller s problem on koplienko neidhardt trace formulae namely we prove that there exist a twice continuously differentiable function f with a bounded second derivative a self adjoint unbounded operator a and a self adjoint operator b 2 s 2 such that f a b f a d dt f a tb t 0 2 s 1

finite difference schemes for incompressible flows in the velocity impulse density formulation we consider finite difference schemes based on the impulse density variable we show that the original velocity impulse density formulation of oseledets is marginally ill posed for the inviscid flow and this has the consequence that some ordinarily stable numerical methods in other formulations become unstable in the velocity impulse density formulation we present numerical evidence of this instability we then discuss the construction of stable finite difference schemes by requiring that at the numerical level the nonlinear terms be convertible to similar terms in the primitive variable formulation finally we give a simplified velocity impulse density formulation which is free of these complications and yet retains the nice features of the original velocity impulse density formulation with regard to the treatment of boundary we present numerical results on this simplified formulation for the driven cavity flow on both the staggered and non staggered grids

spectral flow and dixmier traces abstract we obtain general theorems which enable the calculation of the dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi finite von neumann algebra our results have several applications we deduce a formula for the chern character of an odd l 1 summable breuer fredholm module in terms of a hochschild 1 cycle we explain how to derive a wodzicki residue for pseudo differential operators along the orbits of an ergodic r n action on a compact space x finally we give a short proof of an index theorem of lesch for generalised toeplitz operators

an unconstrained hamiltonian formulation for incompressible fluid flow the equations governing the time evolution of an ideal fluid in material coordinates are expressed as an unconstrained canonical hamiltonian system the incompressibility of the flow is consequent upon certain first integrals of the motion the variable conjugate to the configuration field is not the usual linear momentum but is instead a quantity that is related to linear momentum through an auxiliary scalar field whose time derivative is the pressure the definition of the hamiltonian involves a minimization with respect to this auxiliary field the method of derivation may be generally applied to obtain unconstrained hamiltonian descriptions of lagrangian field equations subject to pointwise constraints

a numerical method for the incompressible navier stokes equations based on an approximate projection in this method we present a fractional step discretization of the time dependent incompressible navier stokes equations the method is based on a projection formulation in which we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields our treatment of the diffusion convection step uses a specialized second order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high reynolds number in contrast to conventional projection type discretizations that impose a discrete form of the divergence free constraint we only approximately impose the constraint i e the velocity field we compute is not exactly divergence free the approximate projection is computed using a conventional discretization of the laplacian and the resulting linear system is solved using conventional multigrid methods numerical examples are presented to validate the second order convergence of the method for euler finite reynolds number and stokes flow a second example illustrating the behavior of the algorithm on an unstable shear layer is also presented

the best constants for operator lipschitz functions on schatten classes suppose that f is a lipschitz function on r with kfklip 1 let a be a bounded self adjoint operator on a hilbert space h let p 1 and suppose that x b h is an operator such that the commutator a x is contained in the schatten class sp it is proved by the last two authors that then also f a x sp and there exists a constant cp independent of x and f such that k f a x kp cpk a x kp the main result of this paper is to give a sharp estimate for cp in terms of p namely we show that cp p 2 p 1 in particular this gives the best estimates for operator lipschitz inequalities we treat this result in a more general setting this involves commutators of n self adjoint operators a1 an for which we prove the analogous result the case described here in the abstract follows as a special case

projection method i convergence and numerical boundary layers this is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations the purpose of these papers is to provide a thorough understanding of the

the dixmier trace and asymptotics of zeta functions abstract we obtain general theorems which enable the calculation of the dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup we prove our results in a general semi finite von neumann algebra we find for p 1 that the asymptotics of the zeta function determines an ideal strictly larger than l p on which the dixmier trace may be defined we also establish stronger versions of other results on dixmier traces and zeta functions

stability and convergence of efficient navier stokes solvers via a commutator estimate for strong solutions of the incompressible navier stokes equations in bounded domains with velocity specified at the boundary we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure these schemes require no solution of stationary stokes systems nor any compatibility between velocity and pressure spaces to ensure an inf sup condition and are representative of a class of highly efficient computational methods that have recently emerged the proofs are simple based upon a new sharp estimate for the commutator of the laplacian and helmholtz projection operators this allows us to treat an unconstrained formulation of the navier stokes equations as a perturbed diffusion equation c 2006 wiley periodicals inc

a dixmier trace formula for the density of states a version of connes trace formula allows to associate a measure on the essential spectrum of a schrodinger operator with bounded potential in solid state physics there is another celebrated measure associated with such operators the density of states in this paper we demonstrate that these two measures coincide we show how this equality can be used to give explicit formulae for the density of states in some circumstances

open string theory and planar algebras in this paper we show that abstract planar algebras are algebras over the topological operad of moduli spaces of stable maps with lagrangian boundary conditions which in the case of the projective line are described in terms of real rational functions these moduli spaces appear naturally in the formulation of open string theory on the projective line we also show two geometric ways to obtain planar algebras from real algebraic geometry one based on string topology and one on gromov witten theory in particular through the well known relation between planar algebras and subfactors these results establish a connection between open string theory real algebraic geometry and subfactors of von neumann algebras

on the classification of simple inductive limit c algebras ii the isomorphism theorem in this article it is proved that the invariant consisting of the scaled ordered k group and the space of tracial states together with the natural pairing between them is a complete invariant for the class of unital simple c algebras which can be expressed as the inductive limit of a sequence a 1 to a 2 to cdots to a n to cdots with a n bigoplus i 1 t n p n i m n i c x n i p n i where xn i is a compact metrizable space and pn i is a projection in m n i c xn i for each n and i and the spaces xn i are of uniformly bounded finite dimension note that the c algebras in the present class are not assumed to be of real rank zero as they were in eg2

non commutative spheres iii irrational rotations leta be the irrational rotation algebra i e thec algebra generated by two unitariesu v satisfyingvu e 2 i uv with irrational and consider the fixed point subalgebrab under the flip automorphismu u 1 v v 1 we prove thatb is an af algebra

the local index formula in semifinite von neumann algebras i spectral flow abstract we generalise the local index formula of connes and moscovici to the case of spectral triples for a subalgebra a of a general semifinite von neumann algebra in this setting it gives a formula for spectral flow along a path joining an unbounded self adjoint breuer fredholm operator affiliated to the von neumann algebra to a unitarily equivalent operator our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is almost a b b cocycle in the cyclic cohomology of a

spectral shift function of higher order this paper resolves affirmatively koplienko s sib mat zh 25 62 71 1984 conjecture on existence of higher order spectral shift measures moreover the paper establishes absolute continuity of these measures and thus existence of the higher order spectral shift functions a spectral shift function of order n is the function n n h v such that operatorname tr biggl f h v sum k 0 n 1 frac 1 k frac d k dt k bigl f h tv bigr bigg t 0 biggr int mathbb r f n t eta n t dt for every sufficiently smooth function f where h is a self adjoint operator defined in a separable hilbert space and v is a self adjoint operator in the n th schatten von neumann ideal sn existence and summability of 1 and 2 were established by krein mat sb 33 597 626 1953 and koplienko sib mat zh 25 62 71 1984 respectively whereas for n 2 the problem was unresolved we show that n h v exists integrable and vert eta n vert l 1 mathbb r leq c n vert v vert s n n for some constant cn depending only on n our results for n rely on estimates for multiple operator integrals obtained in this paper our method also applies to the general semi finite von neumann algebra setting of the perturbation theory

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

the gauge uzawa finite element method part i the navier stokes equations the gauge uzawa fem is a new first order fully discrete projection method which combines advantages of both the gauge and uzawa methods within a variational framework a time step consists of a sequence of d 1 poisson problems d being the space dimension thereby avoiding both the incompressibility constraint as well as dealing with boundary tangential derivatives as in the gauge method this allows for a simple finite element discretization in space of any order in both two and three dimensions this first part introduces the method for the navier stokes equations of incompressible fluids and shows unconditional stability and error estimates for both velocity and pressure via a variational approach under realistic regularity assumptions several numerical experiments document performance of the gauge uzawa fem and compare it with other projection methods

the hochschild class of the chern character for semifinite spectral triples abstract we provide a proof of connes formula for a representative of the hochschild class of the chern character for p summable spectral triples our proof is valid for all semifinite von neumann algebras and all integral p 1 we employ the minimum possible hypotheses on the spectral triples

differentiation of operator functions in non commutative lp spaces the principal results in this paper are concerned with the description of differentiable operator functions in the non commutative lp spaces 1 p associated with semifinite von neumann algebras for example it is established that if f r r is a lipschitz function then the operator function f is g teaux differentiable in l2 m for any semifinite von neumann algebra m if and only if it has a continuous derivative furthermore if f r r has a continuous derivative which is of bounded variation then the operator function f is g teaux differentiable in any lp m 1 p

simple nuclear c algebras of tracial topological rank one we give a classification theorem for unital separable nuclear c algebras with tracial rank no more than one let a and b be two unital separable simple nuclear c algebras with tr a tr b 1 which satisfy the universal coefficient theorem we show that a b if and only if there is an order and unit preserving isomorphism 0 1 2 k0 a k0 a 1a k1 a t a k0 b k0 b 1b k1 b t b where 2 1 x 0 x for each x k0 a and t b

efficient power allocation schemes for nonconvex sum rate maximization on gaussian cognitive mac this letter considers a sum rate maximizing power allocation problem under the gaussian cognitive multiple access channel mac where primary users and secondary users may communicate under mutual interference formulating the problem as a standard nonconvex quadratically constrained quadratic problem qcqp provides a simple method to find a solution using semidefinite relaxation sdr simulation results show that the solution achieves almost the same performance of the exhaustive search within polynomial time

on error estimates of projection methods for navier stokes equations first order schemes in this paper projection methods or fractional step methods are studied in the semi discretized form for the navier stokes equations in a two or three dimensional bounded domain error estimates for the velocity and the pressure of the classical projection scheme are established via the energy method a modified projection scheme which leads to improved error estimates is also proposed

asymptotically unitary equivalence and asymptotically inner automorphisms let c be a unital ah algebra and let a be a unital separable simple c algebra with tracial rank zero suppose that phi 1 phi 2 colon c to a are two unital monomorphisms we show that there is a continuous path of unitaries u t colon t in 0 infty of a such that lim t to infty u t phi 1 a u t phi 2 a quad hbox rm for all quad a in c if and only if phi 1 phi 2 in kk c a tau circ phi 1 tau circ phi 2 for all tau in t a and the rotation map tilde eta phi 1 phi 2 associated with phi 1 and phi 2 is zero in particular an automorphism alpha on a unital separable simple c algebra a in cal n with tracial rank zero is asymptotically inner if and only if alpha rm id a quad rm in quad kk a a and the rotation map tilde eta phi 1 phi 2 is zero let a be a unital ah algebra not necessarily simple and let alpha in aut a be an automorphism as an application we show that the associated crossed product a rtimes alpha bbb z can be embedded into a unital simple af algebra if and only if a admits a strictly positive alpha invariant tracial state

the optimal range of the calderon operator and its applications abstract we identify the optimal range of the calderon operator and that of the classical hilbert transform in the class of symmetric quasi banach spaces further consequences of our approach concern the optimal range of the triangular truncation operator operator lipschitz functions and commutator estimates in ideals of compact operators

type ii non commutative geometry i dixmier trace in von neumann algebras abstract we define the notion of connes von neumann spectral triple and consider the associated index problem we compute the analytic chern connes character of such a generalized spectral triple and prove the corresponding local formula for its hochschild class this formula involves the dixmier trace for ii von neumann algebras in the case of foliations we identify this dixmier trace with the corresponding measured wodzicki residue

integration on locally compact noncommutative spaces abstract we present an ab initio approach to integration theory for nonunital spectral triples this is done without reference to local units and in the full generality of semifinite noncommutative geometry the main result is an equality between the dixmier trace and generalised residue of the zeta function and heat kernel of suitable operators we also examine definitions for integrable bounded elements of a spectral triple based on zeta function heat kernel and dixmier trace techniques we show that zeta functions and heat kernels yield equivalent notions of integrability which imply dixmier traceability

the index formula and the spectral shift function for relatively trace class perturbations we compute the fredholm index index da of the operator da d dt a on l2 r h associated with the operator path a t t where af t a t f t for a e t r and appropriate f l2 r h via the spectral shift function a a associated with the pair a a of asymptotic operators a a on the separable complex hilbert space h in the case when a t is generally an unbounded relatively trace class perturbation of the unbounded self adjoint operator a we derive a formula an extension of a formula due to pushnitski relating the spectral shift function a a for the pair a a and the corresponding spectral shift function h2 h1 for the pair of operators h2 h1 dada da da in this relative trace class context h2 h1 1 1 2 1 2 a a d 2 1 2for a e 0 this formula is then used to identify the fredholm index of da with 0 a a in addition we prove that index da coincides with the spectral flow spflow a t t of the family a t t r and also relate it to the fredholm perturbation determinant for the pair a a index da spflow a t t 0 a a 1lime 0im ln deth a iei a iei 1 0 h2 h1 with the choice of the branch of ln deth on c such that limim z ln deth a zi a zi 1 0 we also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta invariant

the classification of separable simple c algebras which are inductive limits of continuous trace c algebras with spectrum homeomorphic to the closed interval 0 1 abstract a classification is given of certain separable nuclear c algebras not necessarily of real rank zero namely the class of separable simple c algebras which are inductive limits of continuous trace c algebras whose building blocks have spectrum homeomorphic to the closed interval 0 1 or to a disjoint union of copies of this space also the range of the invariant is calculated

operator lipschitz functions in schatten von neumann classes this paper resolves a number of problems in the perturbation theory of linear operators linked with the 45 year old conjecure of m g kre n in particular we prove that every lipschitz function is operator lipschitz in the schatten von neumann ideals s 1 0 such that left f a f b right alpha leqslant c alpha left f right text lip 1 left a b right alpha where f is a lipschitz function with left f right text lip 1 mathop sup limits lambda ne mu lambda mu in mathbb r left frac f left lambda right f left mu right lambda mu right infty left cdot right alpha is the norm is s and a and b are self adjoint linear operators such that a b in s alpha

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

writing on dirty paper corresp a channel with output y x s z is examined the state s sim n 0 qi and the noise z sim n 0 ni are multivariate gaussian random variables i is the identity matrix the input x in r n satisfies the power constraint l n sum i 1 n x i 2 leq p if s is unknown to both transmitter and receiver then the capacity is frac 1 2 ln 1 p n q nats per channel use however if the state s is known to the encoder the capacity is shown to be c ast frac 1 2 ln 1 p n independent of q this is also the capacity of a standard gaussian channel with signal to noise power ratio p n therefore the state s does not affect the capacity of the channel even though s is unknown to the receiver it is shown that the optimal transmitter adapts its signal to the state s rather than attempting to cancel it

bmo estimates for non commutative vector valued lipschitz functions abstract we construct markov semi groups t and associated bmo spaces on a finite von neumann algebra m and obtain results for perturbations of commutators and non commutative lipschitz estimates in particular we prove that for any a m self adjoint and f r r lipschitz there is a markov semi group t such that for x m f a x bmo m t c a b s f a x we obtain an analogue of this result for more general von neumann valued functions f r n n by imposing hormander mikhlin type assumptions on f in establishing these result we show that markov dilations of markov semi groups have certain automatic continuity properties we also show that markov semi groups of double operator integrals admit standard and reversed markov dilations

double operator integrals this paper is concerned with perturbation formulae of the form f a f b lp m k a b lp m with k 0 being a constant depending on p and f only where f is a real valued lipschitz function and a b are self adjoint measurable operators affiliated with a semifinite von neumann algebra m such that the difference a b belongs to lp m 1 p in order to treat the situation where the von neumann algebra m is not necessarily hyperfinite we first develop an integration theory with respect to finitely additive spectral measures in a banach space applied to product measures this integration theory may be considered as an abstract version of the double operator integrals due to birman and solomyak to describe the class of integrable functions we employ our recent study of multiplier theory in umd spaces our perturbation formulae extend those of davies and birman solomyak for the case when m is a hyperfinite i factor i e for the schatten p classes we also discuss analogous perturbation results in the setting of symmetric operator spaces associated with m

on the arazy conjecture concerning schur multipliers on schatten ideals abstract we confirm in the affirmative the conjecture made by arazy in 2 suppose that f is a continuously differentiable function on 1 1 such that f t c t t 1 1 for some constant c 0 and 0 let j j 1 l r for some 0 r if 0 p and 1 q satisfy 1 p r 1 q then the matrix f j f k j k j k 1 is a schur multiplier from the schatten class s q into s p we also consider a general version of this conjecture for arbitrary quasi banach ideals of compact operators

error estimates for semi discrete gauge methods for the navier stokes equations the gauge formulation of the navier stokes equations for incompressible fluids is a new projection method it splits the velocity u a in terms of auxiliary nonphysical variables a and and replaces the momentum equation by a heat like equation for a and the incompressibility constraint by a diffusion equation for this paper studies two time discrete algorithms based on this splitting and the backward euler method for a with explicit boundary conditions and shows their stability and rates of convergence for both velocity and pressure the analyses are variational and hinge on realistic regularity requirements on the exact solution and data both neumann and dirichlet boundary conditions are in principle admissible for but a compatibility restriction for the latter is uncovered which limits its applicability

convergence of gauge method for incompressible flow a new formulation a gauge formulation of the incompressible navier stokes equations in terms of an auxiliary field a and a gauge variable u a was proposed recently by e and liu this paper provides a theoretical analysis of their formulation and verifies the computational advantages we discuss the implicit gauge method which uses backward euler or crank nicolson in time discretization however the boundary conditions for the auxiliary field a are implemented explicitly vertical extrapolation the resulting momentum equation is decoupled from the kinematic equation and the computational cost is reduced to solving a standard heat and poisson equation moreover such explicit boundary conditions for the auxiliary field a will be shown to be unconditionally stable for stokes equations for the full nonlinear navier stokes equations the time stepping constraint is reduced to the standard cfl constraint t x c we also prove first order convergence of the gauge method when we use mac grids as our spatial discretization the optimal error estimate for the velocity field is also obtained

noncommutative kothe duality using techniques drawn from the classical theory of rearrangement invariant banach function spaces we develop a duality theory in the sense of kothe for symmetric banach spaces of measurable operators affiliated with a semifinite von neumann algebra equipped with a distinguished trace a principal result of the paper is the identification of the kothe dual of a given banach space of measurable operators in terms of normality

frechet differentiability of sp norms abstract one of the long standing questions in the theory of schatten von neumann ideals of compact operators is whether their norms have the same differentiability properties as the norms of their commutative counterparts we answer this question in the affirmative a key technical observation underlying our proof is a discovery of connection between this question and recent affirmative resolution of l s koplienko s conjecture concerning existence of higher order spectral shift functions

lipschitz continuity of the absolute value and riesz projections in symmetric operator spaces a principal result of the paper is that ifeis a symmetric banach function space on the positive half line with the fatou property then for all semifinite von neumann algebras m the absolute value mapping is lipschitz continuous on the associated symmetric operator spacee m with lipschitz constant depending only oneif and only ifehas non trivial boyd indices it follows that if m is any von neumann algebra then the absolute value map is lipschitz continuous on the corresponding haageruplp space provided 1 p

quantum differentiability of essentially bounded functions on euclidean space abstract we investigate the properties of the singular values of the quantised derivatives of essentially bounded functions on r d with d 1 the commutator i sgn d 1 m f of an essentially bounded function f on r d acting by pointwise multiplication on l 2 r d and the sign of the dirac operator d acting on c 2 d 2 l 2 r d is called the quantised derivative of f we prove the condition that the function x f x 2 d 1 f x 2 d f x 2 d 2 x r d being integrable is necessary and sufficient for the quantised derivative of f to belong to the weak schatten d class this problem has been previously studied by rochberg and semmes and is also explored in a paper of connes sullivan and telemann here we give new and complete proofs using the methods of double operator integrals furthermore we prove a formula for the dixmier trace of the d th power of the absolute value of the quantised derivative for real valued f when x f x 2 d is integrable there exists a constant c d 0 such that for every continuous normalised trace on the weak trace class l 1 we have sgn d 1 m f d c d r d f x 2 d d x

classification of simple c algebras inductive limits of matrix algebras over one dimensional spaces abstract in this article we will give a complete classification of simple c algebras which can be written as inductive limits of algebras of the form a n i 1 k n m n i c x n i where x n i are arbitrary variable one dimensional compact metrizable spaces the results unify and generalize the previous results for the case x n i s 1 and for the case of x n i being trees we obtain our classification results by reducing the case of general one dimensional spaces to the case of circles the techniques in this paper play important roles in the study of the case of higher dimensional spaces

non commutative banach function spaces in this paper we survey some aspects of the theory of non commutative banach function spaces that is spaces of measurable operators associated with a semi finite von neumann algebra these spaces are also known as non commutative symmetric spaces the theory of such spaces emerged as a common generalization of the theory of classical commutative rearrangement invariant banach function spaces in the sense of w a j luxemburg and a c zaanen and of the theory of symmetrically normed ideals of bounded linear operators in hilbert space in the sense of i c gohberg and m g krein these two cases may be considered as the two extremes of the theory in the first case the underlying von neumann algebra is the commutative algebra l on some measure space with integration as trace in the second case the underlying von neumann algebra is b the algebra of all bounded linear operators on a hilbert space with standard trace important special cases of these non commutative spaces are the non commutative l p spaces which correspond in the commutative case with the usual l p spaces on a measure space and in the setting of symmetrically normed operator ideals they correspond to the schatten p classes mathfrak s p

on inductive limits of matrix algebras over the two torus it will be shown in this paper that certain real rank zero c algebras which are inductive limits of c algebras of the form i m k i c inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i can be expressed as inductive limits of c algebras of the form i m k i c s 1 in particular if both a and b are of real rank zero and are inductive limits of c algebras of the form i m k i c s 1 then also a b is an inductive limit of c algebras of the form i m k i c s 1 hence a b can be classified by its k theory this is a key step in the general classification theory of inductive limit c algebras

on unbounded p summable fredholm modules abstract we prove that odd unbounded p summable fredholm modules are also bounded p summable fredholm modules this is the odd counterpart of a result of a connes for the case of even fredholm modules the approach we use is via estimates of the form d d0 lp m c d d0 1 2 where t t 1 t2 1 2 d0 d 0 is an unbounded linear operator affiliated with a semifinite von neumann algebra m d d0 is a bounded self adjoint linear operator from m and 1 d20 1 2 lp m where lp m is a non commutative lp space associated with m it follows from our results that if p 1 then d d0 belongs to the space lp m

a complete set of numerical invariants for a subfactor abstract we show that certain numerical invariants associated naturally to a subfactor planar algebra constitute a complete family in the sense of determining the isomorphism class of the subfactor planar algebra in the course of the proof we show also that planar algebra isomorphisms of subfactor planar algebras can always be chosen to be preserving this latter statement generalises the fact that hopf algebra isomorphisms of finite dimensional kac algebras can be chosen to be preserving

projection method iii spatial discretization on the staggered grid in e liu siam j numer anal 1995 we studied convergence and the structure of the error for several projection methods when the spatial variable was kept continuous we call this the semi discrete case in this paper we address similar questions for the fully discrete case when the spatial variables are discretized using a staggered grid we prove that the numerical solution in velocity has full accuracy up to the boundary despite the fact that there are numerical boundary layers present in the semi discrete solutions

at structure of ah algebras with the ideal property and torsion free k theory abstract let a be an ah algebra that is a is the inductive limit c algebra of a 1 1 2 a 2 2 3 a 3 a n with a n i 1 t n p n i m n i c x n i p n i where x n i are compact metric spaces t n and n i are positive integers and p n i m n i c x n i are projections suppose that a has the ideal property each closed two sided ideal of a is generated by the projections inside the ideal as a closed two sided ideal suppose that sup n i dim x n i this condition can be relaxed to a certain condition called very slow dimension growth in this article we prove that if we further assume that k a is torsion free then a is an approximate circle algebra or an a t algebra that is a can be written as the inductive limit of b 1 b 2 b n where b n i 1 s n m n i c s 1 one of the main technical results of this article called the decomposition theorem is proved for the general case i e without the assumption that k a is torsion free this decomposition theorem will play an essential role in the proof of a general reduction theorem where the condition that k a is torsion free is dropped in the subsequent paper gong et al preprint 31 of course in that case in addition to space s 1 we will also need the spaces t ii k t iii k and s 2 as in gong 2002 29