Existence of Feller Cocycles on a C*-Algebra

The quantum stochastic differential equation is considered on a unital C^*-algebra, with separablenoise dimension space. Necessary conditions on the matrix ofbounded linear maps for the existence of a completely positivecontractive solution are shown to be sufficient. It is knownthat for completely positive contraction processes, k satisfiessuch an equation if and only if k is a regular Markovian cocycle.‘Feller’ refers to an invariance condition analogousto probabilistic terminology if the algebra is thought of asa non-commutative topological space. 2000 Mathematics SubjectClassification 81S25, 46L07, 46L53, 47D06.     

Uniform Convergence of Schr?dinger Cocycles over Bounded Toeplitz Subshift

For locally constant cocycle defined on an aperiodic subshift, Damanik and Lenz proved that if the subshift satisfies a certain condition (B), then the cocycle is uniform. For any simple Toeplitz subshift, we proved that the corresponding Schr?dinger cocycle is uniform, although it does not satisfy condition (B) in general. In this paper, we study bounded Toeplitz subshift. In general, it does not satisfy condition (B); and it contains non-simple case, which make us cannot use Chebishev polynomial. By a combination of trace formula and avalanche principle, we prove that for any bounded Toeplitz subshift, the corresponding Schr?dinger cocycle is also uniform.     

The Schwinger cocycle on algebras with unbounded operators

In this article, we describe a class of algebras with unbounded operators on which the Schwinger cocycle extends. For this, we replace a space of bounded operators commonly used in the literature by some space of (maybe unbounded) tame operators, in particular by spaces of pseudo-differential operators, acting on the space of sections of a vector bundle E→M. We study some particular examples which we hope interesting or instructive. The case of classical and log-polyhomogeneous pseudo-differential operators is studied, because it carries other cocycles, defined with renormalized traces of pseudo-differential operators, that are some generalizations of the Khesin-Kravchenko-Radul cocycle. The present construction furnishes a simple proof of an expected result: The cohomology class of these cocycles are the same as cohomology class of the Schwinger cocycle. When M=S¹, we show that the Schwinger cocycle is non-trivial on many algebras of pseudo-differential operators (these operators need not to be classical or bounded). These two results complete the work and extend the results of a previous work [J.-P. Magnot, Renormalized traces and cocycles on the algebra of S¹-pseudo-differential operators, Lett. Math. Phys. 75 (2) (2006) 111-127]. When dim(M)>1, we furnish a new example of sign operator which could suggest that the framework of pseudo-differential operators is not adapted to all the cases. On this example, we have to work on some algebras of tame operators, in order to show that the Schwinger cocycle has a non-vanishing cohomology class.     

Novikov-Morse theory for dynamical systems

The present paper contains an interpretation and generalization of Novikov's theory for Morse type inequalities for closed 1-forms in terms of concepts from Conley's theory for dynamical systems. We introduce the concept of a flow carrying a cocycle , (generalized) -flow for short, where is a continuous cocycle in bounded Alexander-Spanier cohomology theory. Gradient-like flows can then be characterized as flows carrying a trivial cocycle. We also define -Morse-Smale flows that allow the existence of “cycles” in contrast to the usual Morse-Smale flows. -flows without fixed points carry not only a cocycle, but a cohomology class, in the sense of [8], and we shall deduce a vanishing theorem for generalized Novikov numbers in that situation. By passing to a suitable cover of the underlying compact polyhedron adapted to the cocycle , we construct a so-called -Morse decomposition for an -flow. On this basis, we can use the Conley index to derive generalized Novikov-Morse inequalitites, extending those of M. Farber [12]. In particular, these inequalities include both the classical Morse type inequalities (corresponding to the case when is a coboundary) as well as the Novikov type inequalities ( when is a nontrivial cocycle). Received: 26 June 2001 / Accepted: 15 January 2002 / Published online: 6 August 2002     

Bounded Solutions of n-Cocycle And Related Equations on Amenable Semigroups

The cocycle functional equation, originating in group theory and playing a role in such areas as cohomology, polyhedral algebra, and information theory, has a long and rich history. Cocycles of higher orders have been introduced in cohomology theory. This paper presents the bounded solutions of cocycle equations of all orders on amenable semigroups. Some related functional equations are treated also. These results generalize some recent results of Pales and Szekelyhidi.     

The Banach Space B(l2) is Primary

We prove that if A is an injective operator system on l² andP is a completely bounded projection on A then either PA or(I–P)A is completely boundedly isomorphic to A. We alsoprove that if B(l²) is linearly homeomorphic to X Y then eitherX or Y is linearly homeomorphic to B(l²). Current address: Merton College, Oxford 0X1 4JD     

Direct Sums of Operator Spaces

It is proved that if X and Y are operator spaces such that everycompletely bounded operator from X into Y is completely compactand Z is a completely complemented subspace of X Y, then thereexists a completely bounded automorphism : X Y X Y with completelybounded inverse such that Z = X₀ Y₀, where X₀ and Y₀ are completelycomplemented subspaces of X and Y, respectively. If X and Yare homogeneous, the existence is proved of such a under aweaker assumption that any operator from X to Y is strictlysingular. An upper estimate is obtained for ||||_cb||^–1||_cbif X and Y are separable homogeneous Hilbertian operator spaces.Also proved is the uniqueness of a ‘completely unconditional’basis in X Y if X and Y satisfy certain conditions.     

Ergodic reduction of random products of two-by-two matrices

We consider a random product of two-by-two matrices of determinant one over an abstract dynamical system. When the two Lyapunov exponents are distinct, Oseledets’ theorem asserts that the matrix cocycle is cohomologous to a diagonal matrix cocycle. When they are equal, we show that the cocycle is conjugate to one of three cases: a rotation matrix cocycle, an upper triangular matrix cocycle, or a diagonal matrix cocycle modulo a rotation by π/2.     

A Variational Method For The Existence of Bounded Solutions of a Sublinear Forced Oscillator

We prove that, for every bounded and measurable forcing p(t),the differential equation has bounded solutions with arbitrarily large amplitude. In generalit is not possible to say that all solutions are bounded, asshown by an example due to Littlewood. The proof is based on a variational method which can be seenas a dual version of Nehari's method for boundary value problemson compact intervals. 2000 Mathematics Subject Classification34C11, 34B14, 49J35.     

On a question of Eremenko concerning escaping components of entire functions

Let f be an entire function with a bounded set of singular values,and suppose furthermore that the postsingular set of f is bounded.We show that every component of the escaping set I(f) is unbounded.This provides a partial answer to a question of Eremenko.     

Frames and the Feichtinger conjecture

We show that the conjectured generalization of the Bourgain-Tzafriri restricted-invertibility theorem is equivalent to the conjecture of Feichtinger, stating that every bounded frame can be written as a finite union of Riesz basic sequences. We prove that any bounded frame can at least be written as a finite union of linearly independent sequences. We further show that the two conjectures are implied by the paving conjecture. Finally, we show that Weyl-Heisenberg frames over rational lattices are finite unions of Riesz basic sequences.

     

Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations

We prove that the admissibility of any pair of vector-valued Schäffer function spaces (satisfying a very general technical condition) implies the existence of a “no past” exponential dichotomy for an exponentially bounded, strongly continuous cocycle (over a semiflow). Roughly speaking the class of Schäffer function spaces consists in all function spaces which are invariant under the right-shift and therefore our approach addresses most of the possible pairs of admissible spaces. Complete characterizations for the exponential dichotomy of cocycles are also obtained. Moreover, we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the classical concept defined by Sacker and Sell (1994) in [23]. Our definition of exponential dichotomy follows partially the definition given by Chow and Leiva (1996) in [4] in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to O. Perron (1930) [14], J. Daleckij and M. Krein (1974) [7], J.L. Massera and J.J. Schäffer (1966) [11], N. van Minh, F. Räbiger and R. Schnaubelt (1998) [26].     

Products of Commuting Boolean Algebras of Projections and Banach Space Geometry

New criteria and Banach spaces are presented (for example, GL-spacesand Banach spaces with property () that ensure that the Booleanalgebra generated by a pair of bounded, commuting Boolean algebrasof projections is itself bounded. The notion of R-boundednessplays a fundamental role. It is shown that the strong operatorclosure of any R-bounded Boolean algebra of projections is necessarilyBade complete. Also, for a Dedekind -complete Banach latticeE, the Boolean algebra consisting of all band projections inE is R-bounded if and only if E has finite cotype. In this situation,every bounded Boolean algebra of projections in E is R-boundedand has a Bade complete strong closure. 2000 Mathematics SubjectClassification 46B20, 47L10 (primary), 46B42, 47B40, 47B60 (secondary).     

On the Mahler measure in several variables

If the total degree of a polynomial in n 2 variables of dimensionn is bounded by a double exponential function in n, then weshow that its Mahler measure is bounded from below by an absoluteconstant greater than 1.     

Chracters on Weighted Amenable Groups

In this paper we prove that for every weight on an amenablegroup there is always a continuous bounded character on thatgroup. Thus we may assume that any weight on an amenable groupis always greater than 1. Using a result of N. Grønbæk[1], this implies that the only amenable weighted group algebrasare up to isomorphism L¹(G)for some amenable group G. A Hahn–Banachtype generalisation is given for the extension of bounded charactersand examples are given showing that the assumption of amenabilityis necessary.     

A Note on a Maximal Function Over Arbitrary Sets of Directions

Let M_S be the universal maximal operator over unit vectors ofarbitrary directions. This operator is not bounded in L²(R²).We consider a sequence of operators over sets of finite equidistributeddirections converging to M_S. We provide a new proof of N. Katz'sbound for such operators. As a corollary, we deduce that M_Sis bounded from some subsets of L² to L². These subsets arecomposed of positive functions whose Fourier transforms havea logarithmic decay or which are supported on a disc. 1991 MathematicsSubject Classification 42B25.     

The spine of a Fourier-Stieltjes algebra

We define the spine A *(G) of the Fourier–Stieltjes algebraB (G) of a locally compact group G. This algebra encodes informationabout much of the fine structure of B (G), particularly informationabout certain homomorphisms and idempotents. We show that A *(G) is graded over a certain semi-lattice, thatof non-quotient locally precompact topologies on G. We computethe spine's spectrum G*, which admits a semi-group structure.We discuss homomorphisms from A *(G) to B (H) where H is anotherlocally compact group; and we show that A *(H) contains theimage of every completely bounded homomorphism from the Fourieralgebra A (H) of any amenable group G. We also show that A *(G)contains all of the idempotents in B (G). Finally, we computeexamples for vector groups, abelian lattices, minimally almostperiodic groups and the (ax + b)-group; and we explore the complexityof A *(G) for the discrete rational numbers and free groups.     

On Convergence Properties of Algorithms for Unconstrained Minimization

Suppose F is a convex function on R" for which there is a sequenceof points on which the function values are bounded below andthe gradients converge to zero. Is it possible that F is unboundedbelow? The answer, perhaps surprisingly, is yes for n > 1.     

Homological Properties of Fully Bounded Noetherian Rings

Let R be a fully bounded Noetherian ring of finite global dimension.Then we prove that K dim (R) gldim (R). If, in addition, Ris local, in the sense that R/J(R) is simple Artinian, thenwe prove that R is Auslander-regular and satisfies a versionof the Cohen–Macaulay property. As a consequence, we showthat a local fully bounded Noetherian ring of finite globaldimension is isomorphic to a matrix ring over a local domain,and a maximal order in its simple Artinian quotient ring.