"Complemented subspaces of function spaces $C_p(X\times Y)$, sequences of measures, and ultrafilters"
Marciszewski, WitoldThe result of Schachermayer and Cembranos asserts that, for a compact space $K$, the Banach space $C(K)$ of continuous real valued maps on $K$, contains a complemented copy of the Banach space $c_0$ if and only if $K$ admits a sequence of regular Borel measures which is weak* convergent, but not weakly convergent. Cembranos and Freniche proved that, for infinite compact spaces $K$ and $L$, $C(K\times L)$ always contains a complemented copy of $c_0$. We extend this theorem by exploring spaces $C_p(X)$ with the pointwise topology. We prove that, for all infinite Tikhonov spaces $X$ and $Y$, the space $C_p(X\times Y)$ either contains a complemented copy of the countable product of real lines $R^\omega$, or contains a complemented copy of $c_0$ endowed with the pointwise topology. Assuming the continuum hypothesis, we construct a pseudocompact space X such that $C_p(X\times X)$ does not contain a complemented copy of $c_0$. Our techniques use sequences of finitely supported measures and some special ultrafilters on $\omega$. This is a joint research with Jerzy Kakol, Damian Sobota, and Lyubomyr Zdomskyy.