Banach spaces of continuous functions with few operators

Koszmider, Piotr

doi:10.1007/s00208-004-0544-z

Cited by 73 publications

(246 citation statements)

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“…Despite some progress in this direction, e.g. showing it for arbitrary compact topological groups ( [36]) it turned out only recently that there are C(K)s not isomorphic to C(L)s or L totally disconnected ( [25], for further references see [28]) and that such Ks can be relatively nice like in [2]. Hence we cannot assume in the isomorphic theory that all Banach spaces C(K) are given by totally disconnected compact Ks.…”

Section: That L Is Nonmetrizable and Totally Disconnected?mentioning

confidence: 99%

On the problem of compact totally disconnected reflection of nonmetrizability

Koszmider

2016

Topology and its Applications

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Abstract. We construct a ZFC example of a nonmetrizable compact space K such that every totally disconnected closed subspace L ⊆ K is metrizable. In fact, the construction can be arranged so that every nonmetrizable compact subspace may be of fixed big dimension. Then we focus on the problem if a nonmetrizable compact space K must have a closed subspace with a nonmetrizable totally disconnected continuous image. This question has several links with the the structure of the Banach space C(K), for example, by Holsztyński's theorem, if K is a counterexample, then C(K) contains no isometric copy of a nonseparable Banach space C(L) for L totally disconnected. We show that in the literature there are diverse consistent counterexamples, most eliminated by Martin's axiom and the negation of the continuum hypothesis, but some consistent with it. We analyze the above problem for a particular class of spaces. OCA+MA however, implies the nonexistence of any counterexample in this class but the existence of some other absolute example remains open.

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Section: That L Is Nonmetrizable and Totally Disconnected?mentioning

confidence: 99%

On the problem of compact totally disconnected reflection of nonmetrizability

Koszmider

2016

Topology and its Applications

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“…In [4], Koszmider constructed a totally disconnected compact Hausdorff space K such that every operator on C(K) is a multiplication by continuous function plus a weakly compact operator. The space K was obtained as a Stone space of a Boolean algebra constructed by transfinite induction, where each A α+1 is the Boolean algebra generated by A α and the supremum of an antichain in A α .…”

Section: Introductionmentioning

confidence: 99%

“…Hence, S(B) preserves suprema of functions of this kind, i.e., which are characteristic functions of basic clopen sets. For the general case, it follows from Lemma 4.1 of [4] that (C (S(B)), π) preserves suprema.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, the best approach to add suprema in C(K) for K totally disconnected is via Boolean algebras. If K is connected, Koszmider, in [4], introduced the definition of extension by continuous functions (see definition 2.3) to obtain a compactum L ⊂ K × [0, 1] which adds supremum of a given pairwise disjoint sequence in (f n ) n∈N in C 1 (K). The function π is the standard projection on K. We use the notation K((f n ) n∈N ) for the extension of K by (f n ) n∈N .…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Suprema of continuous functions on connected spaces

Barbeiro

Fajardo

2016

São Paulo J. Math. Sci.

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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.

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