On complemented copies of c(0)(omega(1)) in c(k-n) spaces

Abstract

Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(K-n) or equivalently the n-fold injective tensor product (circle times) over cap C-n(epsilon)(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 c(0)(omega(1)) in (circle times) over cap C-n(epsilon)(K) under the hypothesis that C(K) contains an isomorphic copy of c(0)(omega(1)). This is related to the results of E. Saab and P. Saab that X (circle times) over cap Y-epsilon contains a complemented copy of c(0), if one of the infinite dimensional Banach spaces X or Y contains a copy of c(0) and of E. M. Galego and J. Hagler that it follows from Martin's Maximum that if C(K) has density omega(1) and contains a copy of c(0)(omega(1)), then C(KXK) contains a complemented copy c(0)(omega(1)). Our main result is that under the assumption of (sic) for every n is an element of N there is a compact Hausdorff space K-n of weight omega(1) such that C(K) is Lindelof in the weak topology, C(K-n) contains a copy of c(0)(omega(1)), C(K-n(n)) does not contain a complemented copy of c(0)(omega(1)) while C(K-n(n+1)) does contain a complemented copy of c(0)(omega(1)). 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.