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

• Autorzy: Leandro Candido , Piotr Koszmider
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46E15: Banach spaces of continuous, differentiable or analytic functions
• 54B10: Product spaces
• 03E65: Other hypotheses and axioms
• 54G12: Scattered spaces
• 46B25: Classical Banach spaces in the general theory
• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2016
• Tom: 233
• Numer: 3
• Strony: 209-226

Daty

wydano

2016

Twórcy

autor

Leandro Candido

• Instituto de Ciência e Tecnologia, Universidade Federal de São Paulo Campus São José dos Campos - Parque Tecnológico, Avenida Cesare Monsueto Giulio Lattes, 1211, 12231-280, São José dos Campos, SP, Brazil

autor

Piotr Koszmider

• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland

Identyfikatory

DOI

10.4064/sm8181-4-2016