Twisted sums of c0 and C(K)-spaces: A solution to the CCKY problem

The complemented subspace problem for $C(K)$-spaces: A counterexample

“…2 . If we put P n = S n \ Q n then (ϕ n (P n )) n∈ω and (ϕ n (S n \ P n )) n∈ω are both convergent to 1 2 . Therefore, we only need to consider an infinite…”

“…By the open mapping theorem, i(Y ) is a closed subspace of Z so that Z/i(Y ) is isomorphic to X. We refer the reader to [3] for a fully detailed exposition on the topic and to [1] for a discussion of twisted sums of c 0 and C-spaces.…”

“…It seems, however, that the quotient space X in Theorem 6.1 is not isomorphic to a complemented subspace of C(L). 1 Namely, one can find a complemented copy of c 0 inside X. Then X = c 0 ⊕ Y for some closed subspace Y of X, and assuming C(L) ≃ c 0 ⊕ X, we would have isomorphic relations…”

“…Aleksandrov-Urysohn compacta K A have found various applications in Banach space theory, see for instance [1,4,9]. In [15] we used K A -spaces to prove that (consistently) there is a Banach space X which is not a C-space, but it contains a copy of c 0 so that X/c 0 = c 0 (c).…”

Twisted sums of $c_0(I)$

“…The nature and properties of such twisted sums depend on set-theoretic assumptions: in [23] it is shown that, under [CH], there exist 2 ℵ 1 non-isomorphic C(K A ) spaces generated by families of size ℵ 1 ; while in [5] it is shown that, under [MA + ℵ 1 < c] all such spaces are isomorphic. And in [2] it is shown that Ext(C(K A ), c 0 ) = 0 under [CH] while [23] proves that Ext(C(K A ), c 0 ) = 0 provided A is of size ℵ 1 under [MA + ℵ 1 < c]. All this make us think that time is ripe to undertake a classification of such twisted sum spaces.…”