Bref survol du théorème de non-plongement de J. Cheeger et B. Kleiner pour le groupe d’Heisenberg dans $L^1$.

We introduce the notion of projective generator on a given Banach space. Weakly countably determined and dual spaces with the Radon Nikodým property have projective generators. If a Banach space has projective generator, then it admits a projective resolution of the identity. When a Banach space and its dual both have a projective generator then the space admits a shrinking resolution of the identity. These results include previous ones of Amir and Lindenstrauss, John and Zizler, Gul?ko, Vaak, Tacon,...

Tauberian operators, which appeared in response to a problem in summability [GaW, KW] have found application in several situations: factorization of operators [DFJP], preservation of isomorphic properties of Banach spaces [N, NR], equivalence between the Radon-Nikodym property and the Krein-Milman property [Sch], and generalized Fredholm operators [Ta, Y].This paper is a survey of the main properties and applications of Tauberian operators.

Nous montrons que $h^2(𝔻,L^1\left(𝕋\right))$ admet une norme équivalente $LUR,$ ce qui répond négativement à une question de Dowling, Hu et Smith. Puis nous obtenons une propriété de stabilité des opérateurs de Radon-Nikodym analytique. Motivés par l’identification entre $h^p(𝔻,X)$ et $V{B}^p(𝕋,X)$ où $X$ est un espace de Banach, pour un groupe abélien compact métrisable $G$, son dual $\Gamma$, et ${\Lambda }_2\subset {\Lambda }_1\subset \Gamma$, nous prouvons que, si l’espace $V{B}_{{\Lambda }_1}^p(G,X)/V{B}_{{\Lambda }_2}^p(G,X)$ a la propriété $Kadec-Klee-{\beta }^{\prime }-\omega$, alors il coincïde avec $L_{{\Lambda }_1}^p(G,X)/L_{{\Lambda }_2}^p(G,X),$$...$

We prove that a continuous image of a Radon-Nikodým compact of weight less than b is Radon-Nikodým compact. As a Banach space counterpart, subspaces of Asplund generated Banach spaces of density character less than b are Asplund generated. In this case, in addition, there exists a subspace of an Asplund generated space which is not Asplund generated and which has density character exactly b.

For a Banach space $E$ and a probability space $(X,𝒜,\lambda )$, a new proof is given that a measure $\mu :𝒜\to E$, with $\mu \ll \lambda$, has RN derivative with respect to $\lambda$ iff there is a compact or a weakly compact $C\subset E$ such that ${\left|\mu \right|}_C:𝒜\to [0,\infty ]$ is a finite valued countably additive measure. Here we define ${\left|\mu \right|}_C\left(A\right)=sup\{{\sum }_k|\langle \mu \left(A_k\right),f_k\rangle \left|\right\}$ where $\left\{A_k\right\}$ is a finite disjoint collection of elements from $𝒜$, each contained in $A$, and $\left\{f_k\right\}\subset E^{\text{'}}$ satisfies ${sup}_k\left|f_k\left(C\right)\right|\le 1$. Then the result is extended to the case when $E$ is a Frechet space.

A family of compact spaces containing continuous images of Radon-Nikod’ym compacta is introduced and studied. A family of Banach spaces containing subspaces of Asplund generated (i.e., GSG) spaces is introduced and studied. Further, for a continuous image of a Radon-Nikod’ym compact $K$ we prove: If $K$ is totally disconnected, then it is Radon-Nikod’ym compact. If $K$ is adequate, then it is even Eberlein compact.

The Radon-Nikodým property was introduced to describe those Banach spaces X for which all operators acting between L1 and X have a representation function. These spaces can be characterized in terms of martingales, as those spaces in which every uniformly bounded martingale converges. In the present work we study some classes of operators defined upon their behaviour with respect to the convergence of such martingales. We prove that an operator preserves the non-convergence of uniformly bounded...

We get a characterization of PCP in Banach spaces with shrinking basis. Also, we prove that the Radon-Nikodym and Krein-Milman properties are equivalent for closed, convex and bounded subsets of some Banach spaces with shrinking basis.

We show that if $P_k$ is a boundedly complete, unconditional Schauder decomposition of a Banach space X, then X is weakly sequentially complete whenever $P_{k}X$ is weakly sequentially complete for each k ∈ ℕ. Then through semi-embeddings, we give a new proof of Lewis’s result: if one of Banach spaces X and Y has an unconditional basis, then X ⊗̂ Y, the projective tensor product of X and Y, is weakly sequentially complete whenever both X and Y are weakly sequentially complete.