On complementability of $c_0$ in spaces $C(K\times L)$

TL;DR

This paper proves that for infinite compact spaces K and L, the space C(K×L) contains a complemented copy of c_0 by constructing explicit measures that converge weak* to zero, extending classical results.

Contribution

It provides a constructive proof using probabilistic methods that generalizes the classical theorem on complementability of c_0 in C(K×L).

Findings

Constructs explicit measures converging weak* to zero.

Generalizes classical complementability theorem.

Uses elementary probabilistic methods for proof.

Abstract

Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces $K$ and $L$ the product $K\times L$ admits a sequence $\langle\mu_n\colon n\in\mathbb{N}\rangle$ of normalized signed measures with finite supports which converges to $0$ with respect to the weak* topology of the dual Banach space $C(K\times L)^*$. Our approach is completely constructive -- the measures $\mu_n$ are defined by an explicit simple formula. We also show that 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 space $c_0$.