On the simplicial cohomology of Banach operator algebras

We investigate the simplicial cohomology of certain Banach operator algebras. The two main examples considered are the Banach algebra of all bounded operators on a Banach space and its ideal of approximable operators. Sufficient conditions will be given forcing Banach algebras of this kind to be simplicially trivial.

Remarks on common hypercyclic vectors

We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator T on a complex Fréchet space X and a set ? ? R+ × C which is not of zero three-dimensional Lebesgue measure, the family {a T + b I : (a, b) ? ?} has no common hypercyclic vectors. This allows to answer negatively questions raised by Godefroy and Shapiro and by Aron. We also prove a sufficient condition for a family of scalar multiples of a given operator on a complex Fréchet space to have a common hypercyclic vector. It allows to show that if D = {z ? C : | z | < 1} and f ? H8 (D) is non-constant, then the family {z Mf{star operator} : b- 1 < | z | < a- 1} has a common hypercyclic vector, where Mf : H2 (D) ? H2 (D), Mf f = f f, a = inf {| f (z) | : z ? D} and b = sup {| f (z) | : | z | ? D}, providing an affirmative answer to a question by Bayart and Grivaux. Finally, extending a result of Costakis and Sambarino, we prove that the family {a Tb : a, b ? C {set minus} {0}} has a common hypercyclic vector, where Tb f (z) = f (z - b) acts on the Fréchet space H (C) of entire functions on one complex variable.

Disjoint mixing operators

Chan and Shapiro showed that each (non-trivial) translation operator acting on the Fréchet space of entire functions endowed with the topology of locally uniform convergence supports a universal function of exponential type zero. We show the existence of d-universal functions of exponential type zero for arbitrary finite tuples of pairwise distinct translation operators. We also show that every separable infinite-dimensional Fréchet space supports an arbitrarily large finite and commuting disjoint mixing collection of operators. When this space is a Banach space, it supports an arbitrarily large finite disjoint mixing collection of C0-semigroups. We also provide an easy proof of the result of Salas that every infinite-dimensional Banach space supports arbitrarily large tuples of dual d-hypercyclic operators, and construct an example of a mixing Hilbert space operator T so that (T,T2) is not d-mixing.

The C*-algebra of ax+b-like groups

We describe the C*-algebras of " ax+b" -like groups in terms of algebras of operator fields defined over their dual spaces.

Sets of multiplicity and closable multipliers on group algebras

We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L-2(G)) of bounded linear operators on L-2 (G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E subset of G is a set of multiplicity if and only if the set E* = {(s,t) is an element of G x G : ts(-1) is an element of E} is a set of operator multiplicity. Analogous results are established for M-1-sets and M-0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function psi : G -> C defines a closable multiplier on the reduced C*-algebra G(r)*(G) of G if and only if Schur multiplication by the function N(psi): G x G -> C, given by N(psi)(s, t) = psi(ts(-1)), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L-2(G). Similar results are obtained for multipliers on VN(C).

Ideals of A(G) and bimodules over maximal abelian selfadjoint algebras

This paper is concerned with weak⁎ closed masa-bimodules generated by A(G)-invariant subspaces of VN(G). An annihilator formula is established, which is used to characterise the weak⁎ closed subspaces of B(L2(G)) which are invariant under both Schur multipliers and a canonical action of M(G) on B(L2(G)) via completely bounded maps. We study the special cases of extremal ideals with a given null set and, for a large class of groups, we establish a link between relative spectral synthesis and relative operator synthesis.

Local operator multipliers and positivity

We establish an unbounded version of Stinespring's Theorem and a lifting result for Stinespring representations of completely positive modular maps defined on the space of all compact operators. We apply these results to study positivity for Schur multipliers. We characterise positive local Schur multipliers, and provide a description of positive local Schur multipliers of Toeplitz type. We introduce local operator multipliers as a non-commutative analogue of local Schur multipliers, and characterise them extending both the characterisation of operator multipliers from [16] and that of local Schur multipliers from [27]. We provide a description of the positive local operator multipliers in terms of approximation by elements of canonical positive cones.

L’incidence d’un dispositif de soutien en gestion de classe sur les pratiques disciplinaires et le sentiment d’efficacité d’enseignants débutants

Résumé Cette étude quasi expérimentale consistait à élaborer et à mettre à l’essai une mesure de soutien à l’intention d’enseignants débutants ainsi qu’à évaluer l’efficacité de celle-ci. L’une des particularités de cette mesure, appelée Dispositif de soutien en gestion de classe, était qu’elle était centrée essentiellement sur le développement de la compétence à gérer la classe. L’application du dispositif, échelonnée sur une année scolaire, portait sur une trentaine d’enseignants débutants œuvrant au primaire, en milieu défavorisé, à Montréal. Basé sur les trois phases du modèle théorique d’Archambault et Chouinard (2003), le dispositif se déclinait selon trois cycles de formation : l’établissement du fonctionnement de la classe, le maintien de celui-ci et le soutien à la motivation scolaire, ainsi que l’intervention pour résoudre des problèmes de comportement. Chaque cycle commençait par une journée de formation et d’appropriation (JFA) durant laquelle il y avait présentation d’un contenu théorique puis des ateliers d’appropriation. Par la suite, les enseignants effectuaient des mises en pratique dans leur classe. Pour terminer le cycle, un autre type de rencontre, la rencontre de suivi (RS), servait entre autres à objectiver la pratique. L’aspect original de cette mesure de soutien était que la première rencontre de formation était offerte une semaine avant la rentrée scolaire. Sur le thème « Commencer l’année du bon pied en gestion de classe », cette journée avait pour objectif de soutenir les enseignants débutants dans l’installation du fonctionnement de leur classe. L’efficacité du dispositif a été évaluée sur la base de trois dimensions : l’établissement et le maintien de l’ordre et de la discipline, le sentiment d’efficacité personnelle ainsi que la motivation professionnelle. Les perceptions du groupe d’enseignants débutants ayant pris part aux activités du dispositif (n = 27) ont été comparées à celles d’un groupe témoin (n = 44). Les participants avaient, en moyenne, 2,9 années d’expérience et leur âge variait de 23 à 56 ans. Les données ont été recueillies à l’aide d’un questionnaire auto rapporté rempli en deux temps, soit au deuxième et au huitième mois de l’année scolaire. Les scores des enseignants débutants du dispositif ont augmenté dans le temps pour l’ensemble des variables à l’étude. De plus, les analyses de variance à mesures répétées ont révélé que le dispositif a eu une triple incidence positive, attestée par des effets d’interaction. Les enseignants débutants engagés dans la démarche ont connu une augmentation de leur capacité à implanter les règles de classe, de leur sentiment d’efficacité personnelle à gérer les situations d’apprentissage et de leur motivation professionnelle. En effet, alors que, au début de l’étude, ils rapportaient des scores significativement inférieurs à ceux du groupe témoin, à la fin, les scores étaient équivalents. Les résultats ont aussi montré que les participants du groupe expérimental se distinguaient en affichant un meilleur sentiment d’efficacité à faire apprendre leurs élèves. L’étude nous apprend également que le sentiment d’efficacité personnelle à faire face aux problèmes de comportement et la capacité à gérer les comportements se sont renforcés de façon significative dans le temps chez l’ensemble des enseignants débutants. Finalement, aucun changement significatif n’a été détecté pour deux des huit variables à l’étude : le sentiment d’efficacité personnelle à avoir un effet sur le comportement des élèves et l’application des règles de classe. En définitive, ces résultats sont encourageants. Ils montrent l’enrichissement professionnel que les enseignants débutants peuvent retirer lorsqu’ils sont soutenus adéquatement. Nous croyons que la journée de formation portant sur l’installation du fonctionnement de la classe, avant la rentrée scolaire, a joué un rôle central dans les succès vécus par les enseignants débutants participants. C’est pourquoi nous recommandons ce type de formation assorti d’un suivi à long terme, où d’autres composantes entrent en jeu, afin de nourrir le sentiment d’efficacité personnelle et la motivation professionnelle des nouveaux enseignants.

Dunford-Pettis properties, Hilbert spaces and projective tensor products

We study complete continuity properties of operators onto ℓ2 and prove several results in the Dunford–Pettis theory of JB∗-triples and their projective tensor products, culminating in characterisations of the alternative Dunford–Pettis property for where E and F are JB∗-triples.

Sufficiency of Favard's condition for a class of band-dominated operators on the axis

The purpose of this paper is to show that, for a large class of band-dominated operators on $\ell^\infty(Z,U)$, with $U$ being a complex Banach space, the injectivity of all limit operators of $A$ already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at infinity of $A$, which, on the other hand, is often equivalent to the Fredholmness of $A$. As a consequence, for operators $A$ in the Wiener algebra, we can characterize the essential spectrum of $A$ on $\ell^p(Z,U)$, regardless of $p\in[1,\infty]$, as the union of point spectra of its limit operators considered as acting on $\ell^p(Z,U)$.

Fourier transform, null variety, and Laplacian's eigenvalues

We consider a quantity κ(Ω)—the distance to the origin from the null variety of the Fourier transform of the characteristic function of Ω. We conjecture, firstly, that κ(Ω) is maximised, among all convex balanced domains of a fixed volume, by a ball, and also that κ(Ω) is bounded above by the square root of the second Dirichlet eigenvalue of Ω. We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between κ(Ω) and the eigenvalues of the Laplacians.

Spectral analysis of diffusions with jump boundary

In this paper we consider one-dimensional diffusions with constant coefficients in a finite interval with jump boundary and a certain deterministic jump distribution. We use coupling methods in order to identify the spectral gap in the case of a large drift and prove that there is a threshold drift above which the bottom of the spectrum no longer depends on the drift. As a corollary to our result we are able to answer two questions concerning elliptic eigenvalue problems with non-local boundary conditions formulated previously by Iddo Ben-Ari and Ross Pinsky.

The full infinite dimensional moment problem on semi-algebraic sets of generalized functions

We consider a generic basic semi-algebraic subset S of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on S, namely to be the moment sequence of a finite measure concentrated on S. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of Rd. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes the support of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider the set of all Radon measures and the set of all the measures having bounded Radon–Nikodym density w.r.t. the Lebesgue measure.

Uniqueness of signature for simple curves

We propose a topological approach to the problem of determining a curve from its iterated integrals. In particular, we prove that a family of terms in the signature series of a two dimensional closed curve with finite p-variation, 1≤p<2, are in fact moments of its winding number. This relation allows us to prove that the signature series of a class of simple non-smooth curves uniquely determine the curves. This implies that outside a Chordal SLEκ null set, where 0<κ≤4, the signature series of curves uniquely determine the curves. Our calculations also enable us to express the Fourier transform of the n-point functions of SLE curves in terms of the expected signature of SLE curves. Although the techniques used in this article are deterministic, the results provide a platform for studying SLE curves through the signatures of their sample paths.

On the Fucik spectrum of the Laplacian on a torus

We study the Fucik spectrum of the Laplacian on a two-dimensional torus T(2). Exploiting the invariance properties of the domain T(2) with respect to translations we obtain a good description of large parts of the spectrum. In particular, for each eigenvalue of the Laplacian we will find an explicit global curve in the Fucik spectrum which passes through this eigenvalue; these curves are ordered, and we will show that their asymptotic limits are positive. On the other hand, using a topological index based on the mentioned group invariance, we will obtain a variational characterization of global curves in the Fucik spectrum; also these curves emanate from the eigenvalues of the Laplacian, and we will show that they tend asymptotically to zero. Thus, we infer that the variational and the explicit curves cannot coincide globally, and that in fact many curve crossings must occur. We will give a bifurcation result which partially explains these phenomena. (C) 2008 Elsevier Inc. All rights reserved.