Quelques remarques sur les opérateurs et les q-algèbres de Banach.
Quotients of Banach spaces and surjectively universal spaces
We characterize those classes 𝓒 of separable Banach spaces for which there exists a separable Banach space Y not containing ℓ₁ and such that every space in the class 𝓒 is a quotient of Y.
Quotients of Banach Spaces with the Daugavet Property
We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak* analogue. We introduce and study analogues of narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of L₁[0,1] by an ℓ₁-subspace need not have the Daugavet property. The latter answers in the negative a question posed to us by A. Pełczyński.
Quotients of Continuous Convex Functions on Nonreflexive Banach Spaces
On each nonreflexive Banach space X there exists a positive continuous convex function f such that 1/f is not a d.c. function (i.e., a difference of two continuous convex functions). This result together with known ones implies that X is reflexive if and only if each everywhere defined quotient of two continuous convex functions is a d.c. function. Our construction also gives a stronger version of Klee's result concerning renormings of nonreflexive spaces and non-norm-attaining functionals.
Quotients with a shrinking basis
We present a simple proof of a theorem that yields as a corollary a result of Valdivia that sharpens an old result of Johnson and Rosenthal.
Reflexive Banach spaces without equivalent norms which are uniformly convex or uniformly differentiable in every direction
Reflexive spaces and numerical radius attaining operators.
In this note we deal with a version of James' Theorem for numerical radius, which was already considered in [4]. First of all, let us recall that this well known classical result states that a Banach space satisfying that all the (bounded and linear) functionals attain the norm, has to be reflexive [16].
Regulated domains and Bergman type projections.
Relations d'ordre dans les Banach : quelques applications à l'analyse
Renormalizations of Banach and locally convex algebras
Renorming and operators.
Renorming Concerning Mazur's Intersection Property of Balls for Weakly Compact Convex Sets.
Renorming in the Space of Bochner Intgrable Functions.
Renormings of c 0 and the minimal displacement problem
The aim of this paper is to show that for every Banach space (X, || · ||) containing asymptotically isometric copy of the space c0 there is a bounded, closed and convex set C ⊂ X with the Chebyshev radius r(C) = 1 such that for every k ≥ 1 there exists a k-contractive mapping T : C → C with [...] for any x ∊ C.
Representable Banach Spaces and Uniformly Gateaux-Smooth Norms
It is proved that a representable non-separable Banach space does not admit uniformly Gâteaux-smooth norms. This is true in particular for C(K) spaces where K is a separable non-metrizable Rosenthal compact space.
Representing trees as relatively compact subsets of the first Baire class
Schroeder-Bernstein Quintuples for Banach Spaces
Let X and Y be two Banach spaces, each isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain necessary and sufficient conditions on the quintuples (p,q,r,s,t) in ℕ for X to be isomorphic to Y whenever ⎧, ⎨ ⎩ . Such quintuples are called Schroeder-Bernstein quintuples for Banach spaces and they yield a unification of the known decomposition...
Smooth renormings of the Lebesgue-Bochner function space L¹(μ,X)
We show that, if μ is a probability measure and X is a Banach space, then the space L¹(μ,X) of Bochner integrable functions admits an equivalent Gâteaux (or uniformly Gâteaux) smooth norm provided that X has such a norm, and that if X admits an equivalent Fréchet (resp. uniformly Fréchet) smooth norm, then L¹(μ,X) has an equivalent renorming whose restriction to every reflexive subspace is Fréchet (resp. uniformly Fréchet) smooth.
Smoothness in Banach spaces. Selected problems.
This is a short survey on some recent as well as classical results and open problems in smoothness and renormings of Banach spaces. Applications in general topology and nonlinear analysis are considered. A few new results and new proofs are included. An effort has been made that a young researcher may enjoy going through it without any special pre-requisites and get a feeling about this area of Banach space theory. Many open problems of different level of difficulty are discussed. For the reader...
Snarked sums of Banach spaces.