On complemented copies of the space c(0) in spaces C-p(X x Y)

0563648 - MÚ 2023 RIV IL eng J - Článek v odborném periodiku
Kąkol, Jerzy - Marciszewski, W. - Sobota, D. - Zdomskyy, L.
On complemented copies of the space c(0) in spaces C-p(X x Y).
Israel Journal of Mathematics. Roč. 250, č. 1 (2022), s. 139-177. ISSN 0021-2172. E-ISSN 1565-8511
Grant CEP: GA ČR(CZ) GF20-22230L
Institucionální podpora: RVO:67985840
Klíčová slova: Hausdorff space * Banach space * Tychonoff space
Obor OECD: Pure mathematics
Impakt faktor: 1, rok: 2022
Způsob publikování: Omezený přístup
https://doi.org/10.1007/s11856-022-2334-2

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 c0. We extend this theorem to the class of Cp-spaces, that is, we prove that for all infinite Tychonoff spaces X and Y the space Cp (X × Y) of continuous functions on X × Y endowed with the pointwise topology contains either a complemented copy of ℝω or a complemented copy of the space (c0)p = {(xn)n∈ω ∈ ℝω: xn → 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 Cp(X × X) does not contain a complemented copy of (c0)p. As a corollary to the first result, we show that for all infinite Tychonoff spaces X and Y the space Cp(X × Y) is linearly homeomorphic to the space Cp(X × Y) × ℝ, although, as proved earlier by Marciszewski, there exists an infinite compact space X such that Cp(X) cannot be mapped onto Cp(X) × ℝ by a continuous linear surjection. This provides a positive answer to a problem of Arkhangel’ski for spaces of the form Cp(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 Ck(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: ℝω, (c0)p or c0.
Trvalý link: https://hdl.handle.net/11104/0335549