An Elementary Proof Of The Josefson-Nissenzweig Theorem For Banach Spaces C(KxL)

TL;DR

This paper provides an elementary combinatorial proof of the Josefson-Nissenzweig theorem for Banach spaces of continuous functions on product spaces, offering new inequalities and explicit descriptions of complemented subspaces isomorphic to c_0.

Contribution

It introduces a simplified, elementary proof of the theorem, replacing probabilistic methods with combinatorial calculus, and extends classical results with explicit inequalities and subspace descriptions.

Findings

Derived inequalities for measures μ_n on product spaces

Explicitly described complemented subspaces isomorphic to c_0

Generalized classical theorems to broader classes of Banach spaces

Abstract

In [8] probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, have been used to show that for every infinite compact spaces K and L there exists a sequence $(\mu_n)$ of normalized signed measures on $K\times L$ with finite supports which converges to $0$ with respect to the weak topology of the dual Banach space $C(K\times L).$ In this paper, we return to this construction, limiting ourselves only to elementary combinatorial calculus. The main efects of this construction are additional information about the measures $\mu_n$, this is particularly clearly seen (among the others) in the resulting inequalities $$\frac{1}{2\sqrt{\pi}}\frac{1}{\sqrt{n}} <\sup_{A\times B\subset X\times Y} |\mu_n(A\times B)|<\frac{2}{\sqrt{\pi}}\frac{1}{\sqrt{n}},$$ $n\in\mathbb{N}$, with $\mu_n(f) \to_n 0$ for every $f\in C(X \times Y);$ where X and Y are arbitrary Tychonoff spaces containing infinite compact subsets, respectively. As an application we explicitly describe for Banach spaces $C(X\times Y)$ some complemented subspaces isomorphic to $c_0$. This result generalizes the classical theorem of Cembranos and Freniche, which states that for every infinite compact spaces K and L, the Banach space $C(K\times L)$ contains a complemented copy of the Banach space $c_0.$