Extension operators and twisted sums of c0 and C(K) spaces

Abstract

We investigate the following problem posed by Cabello Sanchéz, Castillo, Kalton, and Yost:Let K be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of c0 and C(K), i.e., does there exist a Banach space X containing a non-complemented copy Y of c0 such that the quotient space X/Y is isomorphic to C(K)?Using additional set-theoretic assumptions we give the first examples of compact spaces K providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either K is the Cantor cube 2ω1 or K is a separable scattered compact space of height 3 and weight ω1, then every twisted sum of c0 and C(K) is trivial.We also construct nontrivial twisted sums of c0 and C(K) for K belonging to several classes of compacta. Our main tool is an investigation of pairs of compact spaces K⊆L which do not admit an extension operator C(K)→C(L).