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

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.

We construct, under Axiom ♦, a family (C(K ξ )) ξ<2 (2 ω ) of indecomposable Banach spaces with few operators such that every operator from C(K ξ ) into C(Kη) is weakly compact, for all ξ = η. In particular, these spaces are pairwise essentially incomparable.Assuming no additional set-theoretic axiom, we obtain this result with size 2 ω instead of 2 (2 ω ) .

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Let K be a compact Hausdorff space and let (f n ) n∈N be a pairwise disjoint sequence of continuous functions from K into [0, 1]. We say that a compact space L adds supremum of (f n ) n∈N in K if there exists a continuous surjection π : L −→ K such that there exists sup{f n • π : n ∈ N} in C(L). Moreover, we expect that L preserves suprema of disjoint continuous functions which already existed in C(K). Namely, if sup{g n : n ∈ N} exists in C(K), we must haveThis paper studies the preservation of connectedness in extensions by continuous functions -a technique developed by Piotr Koszmider to add suprema of continuous functions on Hausdorff connected compact spaces -proving the following results:(1) If K is a metrizable and locally connected compactum, then any extension of K by continuous functions is connected (but it may be not locally connected).(2) There exists a disconnected extension of a metrizable connected compactum K.