Twisted properties of banach spaces

Abstract

If P, Q are two linear topological properties, say that a Banach space X has the property P-by-Q (or is a P-by-Q space) if X has a subspace Y with property P such that the corresponding quotient X/Y has property Q. The choices P, Q ∈ {separable, reflexive} lead naturally to some new results and new proofs of old results concerning weakly compactly generated Banach spaces. For example, every extension of a subspace of L 1 (0, 1) by a WCG space is WCG. They also give a simple new example of a Banach space property which is not a 3-space property but whose dual is a 3-space property.

Nonseparable C(K)-spaces can be twisted when K is a finite height compact

“…Proof. The proof of (1) appears in [10]: According to Reif [24], since the space C(K) is WCG it admits a Markusevič basis (x γ , f γ ) γ ∈ Γ, for which it can be assumed that (f γ ) is bounded. It is possible to define a dense-range operator T : C(K) → c 0 (Γ) as T (f ) = (f γ (f )).…”

Three-Space Problems for Polynomial Properties in Banach Spaces

“…We cannot resist mentioning that in [14] it is shown that given a non-separable Banach space Z there exists an extension of c 0 by Z that is not isomorphic to the product c 0 È Z. Sobczyk's theorem yields that the nonseparability assumption is necessary.…”

Twisted sums with 𝐶(𝐾) spaces

“…It is known that if Γ is uncountable, then Ext(c 0 (Γ), c 0 ) = {0}; this is essentially contained in one proof of the fact that c 0 is uncomplemented in ∞ ; see also [1], [19, p. 260] and [13, §3]. It was noted in [17,Theorem 3.4] that if X is any non-separable WCG-space, then Ext(X, c 0 ) = {0}, and this settles the case when K is an Eberlein compact; similar arguments can be used for Corson compact spaces. At the other extreme, if K is extremally disconnected, then C(K) contains a complemented ∞ and Ext( ∞ , c 0 ) = {0} was shown in [12].…”