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On complemented copies of the space c(0) in spaces C-p(X x Y)
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SYSNO ASEP 0563648 Druh ASEP J - Článek v odborném periodiku Zařazení RIV J - Článek v odborném periodiku Poddruh J Článek ve WOS Název On complemented copies of the space c(0) in spaces C-p(X x Y) Tvůrce(i) Kąkol, Jerzy (MU-W) SAI, RID, ORCID
Marciszewski, W. (PL)
Sobota, D. (PL)
Zdomskyy, L. (AT)Zdroj.dok. Israel Journal of Mathematics. - : Magnes press - ISSN 0021-2172
Roč. 250, č. 1 (2022), s. 139-177Poč.str. 39 s. Jazyk dok. eng - angličtina Země vyd. IL - Izrael Klíč. slova Hausdorff space ; Banach space ; Tychonoff space Vědní obor RIV BA - Obecná matematika Obor OECD Pure mathematics CEP GF20-22230L GA ČR - Grantová agentura ČR Způsob publikování Omezený přístup Institucionální podpora MU-W - RVO:67985840 UT WOS 000839567400005 EID SCOPUS 85135791255 DOI 10.1007/s11856-022-2334-2 Anotace 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. Pracoviště Matematický ústav Kontakt Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Rok sběru 2023 Elektronická adresa https://doi.org/10.1007/s11856-022-2334-2
Počet záznamů: 1