An indecomposable Banach space of continuous functions which has small density

Abstract

Abstract. Using the method of forcing we construct a model for ZFC where CH does not hold and where there exists a connected compact topological space K of weight ω1 < 2 ω such that every operator on the Banach space of continuous functions on K is multiplication by a continuous function plus a weakly compact operator. In particular, the Banach space of continuous functions on K is indecomposable.1. Introduction. In Banach space theory, several questions about complemented subspaces have been asked. Recall that a closed subspace Y of a Banach space X is complemented in X if there exists a closed subspace Z of X such that X = Y ⊕ Z, where ⊕ means direct sum. For many years it remained an open problem if every infinite-dimensional Banach space X has infinite-dimensional closed subspaces Y and Z such that X = Y ⊕ Z. When it occurs we say that X is decomposable. Since decompositions of Banach spaces are given by projections, indecomposable Banach spaces are related to the property of having few operators, in some sense.In 1993 Gowers and Maurey [GM] constructed the first example of an indecomposable Banach space. Moreover, that space is hereditarily indecomposable, i.e., all its closed subspaces are indecomposable.All operators on the space constructed by Gowers and Maurey have the form cI +S, where I is the identity operator, c ∈ R and S is strictly singular, i.e., the restriction of S to no infinite-dimensional closed subspace is an isomorphism onto its range.

A Banach space in which every injective operator is surjective

“…We also prove (see 2.3, 2.4 and the remarks after it) that a Banach space of the form C(K) where all injective operators are automorphisms cannot be one of the known constructions as in [19,25,20,10,22].…”

There is no bound on sizes of indecomposable Banach spaces

“…Note that by the classification of separable Banach spaces of the form C(K) due to Milutin, Bessaga and Pe lczyński ( [35]) indecomposable C(K)s must be nonseparable. On the other hand it is consistently possible to obtain indecomposable C(K)s, with few operators of densities strictly smaller than continuum ( [10]). It should be also added that the classes of strictly singular and weakly compact operators coincide for C(K) spaces ( [30]).…”

Suprema of continuous functions on connected spaces

“…If we take t = 1 2 and V an open neighborhood of t which intersects supp(f n ) for finitely many n's, the set (K 0 ×V )∩K 1 intersects supp(f n ) for finitely many n's, since f n (x, t) =f n (t).…”

“…For brevity, we will call a n = 1 2 n+1 and b n = 3 2 n+2 , for all n ∈ N. Let K 0 be the extension of [0, 1] by (g n ) n∈N . Define K = K 0 ∪({0}× [1,2]) and, for each n ∈ N, consider the function f n : K −→ [0, 1] given by f n (x, t) = t, if x ∈ (a n+1 , a n ) 0, otherwise. 1, 2]).…”

“…Extensions by continuous functions were applied in several counterexamples in the theory of Banach spaces of the form C(K), as it is shown in [4], [2] and [3]. Alternative ways of adding suprema in connected spaces were developed in [9] -using Wallman representation, which generalizes the Stone representation for connected lattices -and [6] -using ranges of Stone spaces of Boolean algebras.…”