X\times Y), sequences of measures, and ultrafilter

The 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.