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

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 $⟨{\mu }_n:n\in \mathbb{N}⟩$ of normalized signed measures with finite supports which converges to $0$ with respect to the weak* topology of the dual Banach space $C(K\times L{)}^{\ast }$. 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 $c_0$.

7 pages, 1 open problem! arXiv admin note: text overlap with arXiv:2009.07552