Quotients of indecomposable Banach spaces of continuous functions

Abstract

Assuming ♦, we construct a connected compact topological space K such that for every closed L ⊂ K the Banach space C(L) has few operators, in the sense that every operator on C(L) is multiplication by a continuous function plus a weakly compact operator. In particular, C(K) is indecomposable and has continuum many non-isomorphic indecomposable quotients, and K does not contain a homeomorphic copy of βN.Moreover, assuming CH we construct a connected compact K where C(K) has few operators and K contains a homeomorphic copy of βN. Introduction.A Banach space X is called indecomposable if there are no infinite-dimensional subspaces Y and Z such that X = Y ⊕ Z.The first example of an indecomposable Banach space is due to Gowers and Maurey [GM]. Their space is hereditarily indecomposable, which means that all its subspaces are also indecomposable. Ferenczi [Fe] modified Gowers and Maurey's construction in order to obtain a quotient hereditarily indecomposable Banach space, which means that all its quotients are hereditarily indecomposable.We may ask similar questions about Banach spaces of the form C(K), i.e., the space of all real continuous functions on a compact topological space K normed by the supremum. The first indecomposable C(K) was built by Koszmider [Ko1], using the concept of few operators.Since C(K) always contains an isomorphic copy of c 0 , it cannot be hereditarily indecomposable. By a result of Lacey and Morris [LM], every C(K) either contains a complemented copy of c 0 or has l 2 as quotient. So, C(K) cannot be quotient indecomposable, i.e., it always has a decomposable quotient.Although it is not possible to make the properties of indecomposability or having few operators hereditary to all quotients, one may ask if it is possible

Suprema of continuous functions on connected spaces

“…In [3], lemmas 3.6 and 3.8 provide another way of obtaining connectedness in the extension by modifying slightly the functions f n . Lemma 2.6 ([3], 3.6 and 3.8).…”

“…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.…”

“…For this, we need, eventually, to go to a subsequence (as in [4]) or to modify the functions (as in [3]). …”

Extensões conexas e espaços de Banach C(K) com poucos operadores

“…Assumindo diamante (♦), um princípio combinatório mais forte que CH, Rogério Fajardo, em [Fa2], constrói um compacto Hausdorff e conexo K tal que, para todo fechado L ⊆ K, C(L) possui poucos operadores, no sentido de que todo operador linear e contínuo em C(L)é multiplicação fraca. Em particular, o espaço K construído por Fajardo responde positivamente ao problema de Efimov, sobre a existência de um espaço compacto que não contém cópia de βN e nem sequências convergentes não triviais.…”

“…No capítulo 3, unindo as técnicas presentes em [Fa2] e [Fa3] construímos, assumindo ♦, uma família (K ξ ) ξ<2 (2 ω ) de espaços conexos e hereditariamente Koszmider tais que C(K ξ ) e C(K η ) são essencialmente incomparáveis, para todo ξ = η. Como espaços hereditariamente Koszmider não possuem cópia de βN nem sequências convergentes não trivias, cada K ξ construído responde positivamente ao problema de Efimov. No capítulo 4, apresentamos um método alternativo de construção via forcing de espaços C(K) com poucos operadores com a propriedade de que C(L) também tem poucos operadores, para todo fechado L ⊆ K. Em particular, o compacto K obtido via forcing também responde positivamente ao problema de Efimov.…”