On complemented copies of c₀(ω₁) in C(Kⁿ) spaces

Leandro Candido, and Piotr Koszmider. "On complemented copies of c₀(ω₁) in C(Kⁿ) spaces." Studia Mathematica 233.3 (2016): 209-226. <http://eudml.org/doc/286383>.

@article{LeandroCandido2016,
abstract = {Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kⁿ) or equivalently the n-fold injective tensor product $⊗̂^\{n\}_\{ε\}C(K)$ or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c₀(ω₁) in $⊗̂^\{n\}_\{ε\} C(K)$ under the hypothesis that C(K) contains such a copy. This is related to the results of E. Saab and P. Saab that $X ⊗̂_\{ε\} Y$ contains a complemented copy of c₀ if one of the infinite-dimensional Banach spaces X or Y contains a copy of c₀, and of E. M. Galego and J. Hagler that it follows from Martin’s Maximum that if C(K) has density ω₁ and contains a copy of c₀(ω₁), then C(K×K) contains a complemented copy of c₀(ω₁). Our main result is that under the assumption of ♣ for every n ∈ ℕ there is a compact Hausdorff space Kₙ of weight ω₁ such that C(K) is Lindelöf in the weak topology, C(Kₙ) contains a copy of c₀(ω₁), C(Kₙⁿ) does not contain a complemented copy of c₀(ω₁), while $C(Kₙ^\{n+1\})$ does contain a complemented copy of c₀(ω₁). This shows that additional set-theoretic assumptions in Galego and Hagler’s nonseparable version of Cembrano and Freniche’s theorem are necessary, as well as clarifies in the negative direction the matter unsettled in a paper of Dow, Junnila and Pelant whether half-pcc Banach spaces must be weakly pcc.},
author = {Leandro Candido, Piotr Koszmider},
journal = {Studia Mathematica},
keywords = {Banach spaces of continuous functions; injective tensor product; complemented subspaces; ostaszewski's martin's maximum; vector valued continuous functions; scattered compact spaces},
language = {eng},
number = {3},
pages = {209-226},
title = {On complemented copies of c₀(ω₁) in C(Kⁿ) spaces},
url = {http://eudml.org/doc/286383},
volume = {233},
year = {2016},
}

TY - JOUR
AU - Leandro Candido
AU - Piotr Koszmider
TI - On complemented copies of c₀(ω₁) in C(Kⁿ) spaces
JO - Studia Mathematica
PY - 2016
VL - 233
IS - 3
SP - 209
EP - 226
AB - Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kⁿ) or equivalently the n-fold injective tensor product $⊗̂^{n}_{ε}C(K)$ or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c₀(ω₁) in $⊗̂^{n}_{ε} C(K)$ under the hypothesis that C(K) contains such a copy. This is related to the results of E. Saab and P. Saab that $X ⊗̂_{ε} Y$ contains a complemented copy of c₀ if one of the infinite-dimensional Banach spaces X or Y contains a copy of c₀, and of E. M. Galego and J. Hagler that it follows from Martin’s Maximum that if C(K) has density ω₁ and contains a copy of c₀(ω₁), then C(K×K) contains a complemented copy of c₀(ω₁). Our main result is that under the assumption of ♣ for every n ∈ ℕ there is a compact Hausdorff space Kₙ of weight ω₁ such that C(K) is Lindelöf in the weak topology, C(Kₙ) contains a copy of c₀(ω₁), C(Kₙⁿ) does not contain a complemented copy of c₀(ω₁), while $C(Kₙ^{n+1})$ does contain a complemented copy of c₀(ω₁). This shows that additional set-theoretic assumptions in Galego and Hagler’s nonseparable version of Cembrano and Freniche’s theorem are necessary, as well as clarifies in the negative direction the matter unsettled in a paper of Dow, Junnila and Pelant whether half-pcc Banach spaces must be weakly pcc.
LA - eng
KW - Banach spaces of continuous functions; injective tensor product; complemented subspaces; ostaszewski's martin's maximum; vector valued continuous functions; scattered compact spaces
UR - http://eudml.org/doc/286383
ER -