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

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F22%3A00563648" target="_blank" >RIV/67985840:_____/22:00563648 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1007/s11856-022-2334-2" target="_blank" >https://doi.org/10.1007/s11856-022-2334-2</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s11856-022-2334-2" target="_blank" >10.1007/s11856-022-2334-2</a>

Result language

angličtina

Original language name

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

Original language description

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.

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GF20-22230L" target="_blank" >GF20-22230L: Banach spaces of continuous and Lipschitz functions</a><br>

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2022

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Israel Journal of Mathematics

ISSN

0021-2172

e-ISSN

1565-8511

Volume of the periodical

250

Issue of the periodical within the volume

1

Country of publishing house

IL - THE STATE OF ISRAEL

Number of pages

39

Pages from-to

139-177

UT code for WoS article

000839567400005

EID of the result in the Scopus database

2-s2.0-85135791255