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Quotients of indecomposable Banach spaces of continuous functions

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Fajardo R.

Studia Mathematica

2012

tom 212

nr 3

259-283

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.

An indecomposable Banach space of continuous functions which has small density

100%

Fajardo R.

Fundamenta Mathematicae

2009

tom 202

nr 1

43-63

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.