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

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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 εC(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 εC(K) under the hypothesis that C(K) contains an isomorphic copy of c₀(ω₁). 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 c₀(ω₁). The 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(Kₙⁿ⁺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.