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Baire classes of affine vector-valued functions

Ondřej F. K. Kalenda, Jiří Spurný (2016)

Studia Mathematica

We investigate Baire classes of strongly affine mappings with values in Fréchet spaces. We show, in particular, that the validity of the vector-valued Mokobodzki result on affine functions of the first Baire class is related to the approximation property of the range space. We further extend several results known for scalar functions on Choquet simplices or on dual balls of L₁-preduals to the vector-valued case. This concerns, in particular, affine classes of strongly affine Baire mappings, the...

Baire classes of complex L 1 -preduals

Pavel Ludvík, Jiří Spurný (2015)

Czechoslovak Mathematical Journal

Let X be a complex L 1 -predual, non-separable in general. We investigate extendability of complex-valued bounded homogeneous Baire- α functions on the set ext B X * of the extreme points of the dual unit ball B X * to the whole unit ball B X * . As a corollary we show that, given α ∈ [ 1 , ω 1 ) , the intrinsic α -th Baire class of X can be identified with the space of bounded homogeneous Baire- α functions on the set ext B X * when ext B X * satisfies certain topological assumptions. The paper is intended to be a complex counterpart to the same authors’...

Banach algebra techniques in the theory of arithmetic functions

Lutz G. Lucht (2008)

Acta Mathematica Universitatis Ostraviensis

For infinite discrete additive semigroups X ⊂ [ 0 , ∞ ) we study normed algebras of arithmetic functions g : X → ℂ endowed with the linear operations and the convolution. In particular, we investigate the problem of scaling the mean deviation of related multiplicative functions for X = log ℕ . This involves an extension of Banach algebras of arithmetic functions by introducing weight functions and proving a weighted inversion theorem of Wiener type in the frame of Gelfand’s theory of commutative Banach algebras.

Banach spaces widely complemented in each other

Elói Medina Galego (2013)

Colloquium Mathematicae

Suppose that X and Y are Banach spaces that embed complementably into each other. Are X and Y necessarily isomorphic? In this generality, the answer is no, as proved by W. T. Gowers in 1996. However, if X contains a complemented copy of its square X², then X is isomorphic to Y whenever there exists p ∈ ℕ such that X p can be decomposed into a direct sum of X p - 1 and Y. Motivated by this fact, we introduce the concept of (p,q,r) widely complemented subspaces in Banach spaces, where p,q and r ∈ ℕ. Then,...

Biorthogonal systems in Banach spaces

Michael A. Coco (2004)

Studia Mathematica

We give biorthogonal system characterizations of Banach spaces that fail the Dunford-Pettis property, contain an isomorphic copy of c₀, or fail the hereditary Dunford-Pettis property. We combine this with previous results to show that each infinite-dimensional Banach space has one of three types of biorthogonal systems.

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