Let $X$ be a non-reflexive real Banach space. Then for each norm $|·|$ from a dense set of equivalent norms on $X$ (in the metric of uniform convergence on the unit ball of $X$), there exists a three-point set that has no Chebyshev center in $(X,|·\left|\right)$. This result strengthens theorems by Davis and Johnson, van Dulst and Singer, and Konyagin.

This work deals with various questions concerning Fourier multipliers on $L^p$, Schur multipliers on the Schatten class $S^p$ as well as their completely bounded versions when $L^p$ and $S^p$ are viewed as operator spaces. For this purpose we use subsets of ℤ enjoying the non-commutative Λ(p)-property which is a new analytic property much stronger than the classical Λ(p)-property. We start by studying the notion of non-commutative Λ(p)-sets in the general case of an arbitrary discrete group before turning to the...

Given an equibounded (₀)-semigroup of linear operators with generator A on a Banach space X, a functional calculus, due to L. Schwartz, is briefly sketched to explain fractional powers of A. Then the (modified) K-functional with respect to $(X,D\left({(-A)}^{\alpha }\right))$, α > 0, is characterized via the associated resolvent R(λ;A). Under the assumption that the resolvent satisfies a Nikolskii type inequality, ${\left|\right|\lambda R(\lambda ;A)f\left|\right|}_Y{\le c\phi (1/\lambda )\left|\right|f\left|\right|}_X$, for a suitable Banach space Y, an Ulyanov inequality is derived. This will be of interest if one has good control...

We study the notion of fractional $L^p$-differentiability of order $s\in (0,1)$ along vector fields satisfying the Hörmander condition on ${ℝ}^n$. We prove a modified version of the celebrated structure theorem for the Carnot-Carathéodory balls originally due to Nagel, Stein and Wainger. This result enables us to demonstrate that different $W^{s,p}$-norms are equivalent. We also prove a local embedding $W^{1,p}\subset W^{s,q}$, where q is a suitable exponent greater than p.

Let K be a compact Hausdorff space, $C_p\left(K\right)$ the space of continuous functions on K endowed with the pointwise convergence topology, D ⊂ K a dense subset and $t_p\left(D\right)$ the topology in C(K) of pointwise convergence on D. It is proved that when $C_p\left(K\right)$ is Lindelöf the $t_p\left(D\right)$-compact subsets of C(K) are fragmented by the supremum norm of C(K). As a consequence we obtain some Namioka type results and apply them to prove that if K is separable and $C_p\left(K\right)$ is Lindelöf, then K is metrizable if, and only if, there is a countable and dense...

Recent work has studied the fragmentability and σ-fragmentability properties of Banach spaces. Here examples are given that justify the definitions that have been used. The fragmentability and σ-fragmentability properties of the spaces ${\ell }^{\infty }$ and ${\ell }_c^{\infty }\left(\Gamma \right)$, with Γ uncountable, are determined.

We prove that if a Banach space X admits a Lipschitz β-smooth bump function, then (X ∗ , weak ∗ ) is fragmented by a metric, generating a topology, which is stronger than the τβ -topology. We also use this to prove that if X ∗ admits a Lipschitz Gateaux-smooth bump function, then X is sigma-fragmentable.

The purpose of this note is two-fold: firstly, to give a new and interesting result concerning separate and joint continuity, and secondly, to give a stream-lined (and self-contained) proof of the fact that "tame" CHART groups are topological groups.

Associated with every vector measure m taking its values in a Fréchet space X is the space L1(m) of all m-integrable functions. It turns out that L1(m) is always a Fréchet lattice. We show that possession of the AL-property for the lattice L1(m) has some remarkable consequences for both the underlying Fréchet space X and the integration operator f → ∫ f dm.

In some recent papers ([1],[2],[3],[4]) we have investigated some general spectral properties of a multiplier defined on a commutative semi-simple Banach algebra. In this paper we expose some aspects concerning the Fredholm theory of multipliers.