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

Leandro Candido ; Piotr Koszmider

Studia Mathematica, Tome 233 (2016), p. 209-226 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

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Publié le : 2016-01-01

Zbl 06602797

EUDML-ID : urn:eudml:doc:286383