On complemented copies of the space c0 in spaces Cp(X × Y)

Cembranos and Freniche proved that for every two infinite compact Hausdorff spaces X and Y the Banach space C ( X × Y ) of continuous real-valued functions on X × Y endowed with the supremum norm contains a complemented copy of the Banach space c 0 . We extend this theorem to the class of C p -spaces, that is, we prove that for all infinite Tychonoff spaces X and Y the space C p ( X × Y ) of continuous functions on X × Y endowed with the pointwise topology contains either a complemented copy of ℝ ω or a complemented copy of the space ( c 0 ) p = {( x n ) n ∈ ω ∈ ℝ ω : x n → 0}, both endowed with the product topology. We show that the latter case holds always when X × Y is pseudocompact. On the other hand, assuming the Continuum Hypothesis (or even a weaker set-theoretic assumption), we provide an example of a pseudocompact space X such that C p ( X × X ) does not contain a complemented copy of ( c 0 ) p . As a corollary to the first result, we show that for all infinite Tychonoff spaces X and Y the space C p ( X × Y ) is linearly homeomorphic to the space C p ( X × Y ) × ℝ, although, as proved earlier by Marciszewski, there exists an infinite compact space X such that C p ( X ) cannot be mapped onto C p ( X ) × ℝ by a continuous linear surjection. This provides a positive answer to a problem of Arkhangel’ski for spaces of the form C p ( X × Y ). Another corollary—analogous to the classical Rosenthal-Lacey theorem for Banach spaces C ( X ) with X compact and Hausdorff—asserts that for every infinite Tychonoff spaces X and Y the space C k ( X × Y ) of continuous functions on X × Y endowed with the compact-open topology admits a quotient map onto a space isomorphic to one of the following three spaces: ℝ ω , ( c 0 ) p or c 0 .