On the three space property for C(K)-spaces

Twisted sums of $c_0(I)$

“…As can be seen for instance in [13,Theorem 1.2.4], M 1 (K) is a convex Eberlein compact. Thus, by [28] there is an exact sequence 0…”

“…However, Ext(ℓ ∞ , c 0 ) = 0, as shown in [4, §2.1]. Another question that stems from [28] is whether the spaces Z(K, Σ) are subspaces of ℓ ∞ . This is of course evident if K is separable, since in that case C(K) is a subspace of ℓ ∞ and "being a subspace of ℓ ∞ " is a three-space property.…”

“…Twisted sums of c 0 and C(K) that are not C-spaces. The purpose of [28] was to construct, for every Eberlein compactum K of weight c a twisted sum of c 0 and C(K) which is not a C-space. Let us recall the key points of the construction.…”

“…• First, Theorem 3.4 in [28] produces, under p = c, a very special almost disjoint family A of subsets of natural numbers and size c. Its existence relies on the fact that, since K is Eberlein, M 1 (K) is a sequentially compact space and…”

“…• Spaces of continuous functions C(K A ) over the Stone compactum of a Boolean algebra generated by an uncountable almost disjoint family A of subsets of N. • When |I| = c, the spaces constructed in [28] (under some form of Martin's axiom), whose main feature is that they are not isomorphic to any C(K)-space. • Products and c 0 -sums of both.…”