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 $\langle \mu _n\colon n\in \mathbb {N}\rangle$ 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 [Proc. Amer. Math. Soc. 91 (1984), pp. 556–558] and Freniche [Math. Ann. 267 (1984), pp. 479–486] 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 $c_0$.