Sur le calcul fonctionnel holomorphe de L. Waelbroeck

We construct a holomorphic functional calculus ofn variables in q-algebras in the sense of L. Waelbroeck [18] and extend to these algebras the holomorphic functional calculus of Arens-Calderon.

     

Sur un example de categorie presque-abelienne

Résumé  Si Λ est la catégorie des espaces localement convexes séparés et complets, nous montrons que la catégorie duale de Λ est une catégorie presque-abélienne et par suite, la catégorie Λ l’est aussi.

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Projective limits of weighted (LB)-spaces of continuous functions

Countable projective spectra of countable inductive limits, called (PLB)-spaces, of weighted Banach spaces of continuous functions are investigated. It is characterized when the derived projective limit functor vanishes in terms of the sequences of the weights defining the spaces. The locally convex properties of the corresponding projective limits are analyzed, too. Received: 30 January 2009     

About the properties ( V ) and ( R.D.P. ) in injective tensor products

In questo lavoro vogliamo provare che seX è uno spazio di Banach del tipoL _∞ con la proprietà (V) of Pelczynski (rispettivamente con la Reciproca Proprietà di Dunford Pettis) eY è uno spazio di Banach riflessivo alloraX⊗_ɛ Y ha la proprietà (V) (rispettivamente la Reciproca Proprietà di Dunford Pettis).

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Work partially supported by M.U.R.S.T. (40%).     

Algèbres localement uniformémentA-pseudo-convexes advertiblement complètes

We examine the structure and the spectral properties of locally uniformlyA-pseudo-convex algebras.      

Extension d'un théorème de Harte et applications aux algèbres de Jordan-Banach

In this paper we study some questions related to spectral theory in Jordan-Banach algebras. Firstly, we introduce the notion of exponential spectrum and then we extend to Jordan-Banach algebras a theorem due to Robin Harte in the associative case. Secondly, these results are used to get a theorem on spectral perturbation by inessential elements in Jordan-Banach algebras.

     

Lindelöf type of generalization of separability in Banach spaces

We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: X has CSP if each family E of closed linear subspaces of X whose intersection is the zero space contains a countable subfamily E₀ with the same intersection. All separable Banach spaces have CSP and plenty of examples of non-separable CSP spaces are provided. Connections of CSP with Marku?evi?-bases, Corson property and related geometric issues are discussed.     

Fonctions qui opèrent sur les espaces de Besov

Sans résumé

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Some Results on Best Coapproximation in L 1 (S, X)

As a counterpart to best approximation in normed linear spaces, best coapproximation was introduced by Franchetti and Furi. In this paper, we shall show that if M is a separable, coproximinal subspace of X satisfying some additional conditions, then L ¹ (S, M) is coproximinal in L ¹(S, X).      

Inexistence de structures affines sur les fibrés de Seifert

Sans résumé

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Some properties of the <Emphasis Type="Italic">ɛ</Emphasis><Subscript>∞</Subscript>-product of quotient bornological spaces

Some properties of the ? _∞-product defined in [4] are obtained by a study of a kind of isomorphism between the computation of this ? _∞-product and the ordinary ?-product of L. Schwartz [9]. The paper contains several corollaries.     

Characterization of locally convex solid topologies of lebesgue on the vector latticeC(X)

We give some sufficient and necessary conditions such that the vector lattice of continuous functions on a compact topological space, not necessary metrizable, admits a locally convex and solid topology which is Lebesgue.

     

Asymptotique des nombres de Betti,invariants $l^2$ et laminations

Let K be a finite simplicial complex. We are interested in the asymptotic behavior of the Betti numbers of a sequence of finite sheeted covers of $K$, when normalized by the index of the covers. W. Lück, has proved that for regular coverings, these sequences of numbers converge to the $l^2$ Betti numbers of the associated (in general infinite) limit regular cover of K. In this article we investigate the non regular case. We show that the sequences of normalized Betti numbers still converge. But this time the good limit object is no longer the associated limit cover of K, but a lamination by simplicial complexes. We prove that the limits of sequences of normalized Betti numbers are equal to the $l^2$ Betti numbers of this lamination. Even if the associated limit cover of K is contractible, its $l^2$ Betti numbers are in general different from those of the lamination. We construct such examples. We also give a dynamical condition for these numbers to be equal. It turns out that this condition is equivalent to a former criterion due to M. Farber. We hope that our results clarify its meaning and show to which extent it is optimal. In a second part of this paper we study non free measure-preserving ergodic actions of a countable group $\Gamma$ on a standard Borel probability space. Extending group-theoretic similar results of the second author, we obtain relations between the $l^{2}$ Betti numbers of $\Gamma$ and those of the generic stabilizers. For example, if $b_1^{(2)} (\Gamma ) \neq 0$, then either almost each stabilizer is finite or almost each stabilizer has an infinite first $l^2$ Betti number.

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Asymptotique des nombres de Betti, invariants $l^2$ et laminations

     

Various Slicing Indices on Banach Spaces

We give a short and direct proof for the computation of the Szlenk index of the C(K) spaces, when K is a countable compact space and determine their Lavrientiev indices. We also compute the Szlenk index of certain C(α) spaces, where α is an uncountable ordinal. Finally, we show that if the Szlenk index of a Banach space is ω (first infinite ordinal), then its weak*-dentability index is at most ω² and that this estimate is optimal. The first author was supported by the grants: Institutional Research Plan AV0Z10190503, A100190502, GA ČR 201/04/0090.     

A representation theorem for maxitive measures

A maxitive measure is a nonnegative function η on a σ-algebra Σ and such that η(U_j A_j ) = sup_j η(A_j) for all countable disjoint families of sets (A_j) in Σ. A representation theorem for such measures is established, and next applied to represent Köthe function M-spaces as L_∞-spaces.     

Operators of Fourier type <Emphasis Type="Italic">p</Emphasis> with respect to some subgroups of a locally compact Abelian group

It is shown that all Banach space operators which have Fourier type p(1 < p 2) with respect to a second countable locally compact abelian group G also have Fourier type p with respect to every closed discrete subgroup H of G. The same statement holds for any closed subgroup H of G when p = 2. Also shown as a corollary is that Fourier type 2 operators have Walsh type 2. Received: 10 June 2002     

On generating countable sets of endomorphisms

Sierpiski proved that every countable set of mappings on an infinite set X is contained in a 2-generated subsemigroup of the semigroup of all mappings on X. In this paper we prove that every countable set of endomorphisms of an algebra which has an infinite basis (independent generating set) is contained in a 2-generated subsemigroup of the semigroup of all endomorphisms of .     

Chain conditions and weak topologies

We study conditions on Banach spaces close to separability. We say that a topological space is pcc if every point-finite family of open subsets of the space is countable. For a Banach space E, we say that E is weakly pcc if E, equipped with the weak topology, is pcc, and we also consider a weaker property: we say that E is half-pcc if every point-finite family consisting of half-spaces of E is countable. We show that E is half-pcc if, and only if, every bounded linear map E→c₀(ω₁) has separable range. We exhibit a variety of mild conditions which imply separability of a half-pcc Banach space. For a Banach space C(K), we also consider the pcc-property of the topology of pointwise convergence, and we note that the space Cp(K) may be pcc even when C(K) fails to be weakly pcc. We note that this does not happen when K is scattered, and we provide the following example:

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There exists a non-metrizable scattered compact Hausdorff space K with C(K) weakly pcc.

     

Advertible complétude et structure deQ-algèbre

We give different characterizations of algebra types (Q-property, advertible completeness, MackeyQ-property) in terms of the relation ρ<-β and sequentiality notions in locally pseudo-convex algebras.

     

Weak limits of the extreme dual ball in JB*-triples

We study weak limits of the extreme points, ∂ _e (E ^* ₁), of the dual ball of a JB*-triple, E. We show that all such weak limits, except possibly the zero functional, are weak sequential limits and we discuss implications for the structure of E. Received: 9 April 2001