### $\u2102$-convexity in infinite-dimensional Banach spaces and applications to Kergin interpolation.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

The main result of the note is a characterization of 1-amenability of Banach algebras of approximable operators for a class of Banach spaces with 1-unconditional bases in terms of a new basis property. It is also shown that amenability and symmetric amenability are equivalent concepts for Banach algebras of approximable operators, and that a type of Banach space that was long suspected to lack property 𝔸 has in fact the property. Some further ideas on the problem of whether or not amenability (in...

Let 1 ≤ p < ∞, $={\left(X\u2099\right)}_{n\in \mathbb{N}}$ be a sequence of Banach spaces and ${l}_{p}\left(\right)$ the coresponding vector valued sequence space. Let $={\left(X\u2099\right)}_{n\in \mathbb{N}}$, $={\left(Y\u2099\right)}_{n\in \mathbb{N}}$ be two sequences of Banach spaces, $={\left(V\u2099\right)}_{n\in \mathbb{N}}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator ${M}_{}:{l}_{p}\left(\right)\to {l}_{q}\left(\right)$ by ${M}_{}\left({\left(x\u2099\right)}_{n\in \mathbb{N}}\right):={\left(V\u2099\left(x\u2099\right)\right)}_{n\in \mathbb{N}}$. We give necessary and sufficient conditions for ${M}_{}$ to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞.

A Banach space X with a Schauder basis is defined to have the restricted quotient hereditarily indecomposable property if X/Y is hereditarily indecomposable for any infinite-codimensional subspace Y with a successive finite-dimensional decomposition on the basis of X. The following dichotomy theorem is proved: any infinite-dimensional Banach space contains a quotient of a subspace which either has an unconditional basis, or has the restricted quotient hereditarily indecomposable property.

We prove that the symmetric convexified Tsirelson space is of weak cotype 2 but not of cotype 2.

We prove that, for a compact metric space X not reduced to a point, the existence of a bilinear mapping ⋄: C(X) × C(X) → C(X) satisfying ||f⋄g|| = ||f|| ||g|| for all f,g ∈ C(X) is equivalent to the uncountability of X. This is derived from a bilinear version of Holsztyński's theorem [3] on isometries of C(X)-spaces, which is also proved in the paper.

We provide a complex version of a theorem due to Bednar and Lacey characterizing real ${L}_{1}$-preduals. Hence we prove a characterization of complex ${L}_{1}$-preduals via a complex barycentric mapping.