2020
DOI: 10.48550/arxiv.2003.03832
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A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

Abstract: We consider the closed subspace of ℓ∞ generated by c 0 and the characteristic functions of elements of an uncountable, almost disjoint family A of infinite subsets of N. This Banach space has the form C 0 (K A ) for a locally compact Hausdorff space K A that is known under many names, such as Ψ-space and Isbell-Mrówka space.We construct an uncountable, almost disjoint family A such that the Banach algebra of all bounded linear operators on C 0 (K A ) is as small as possible in the precise sense that every boun… Show more

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Cited by 2 publications
(3 citation statements)
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“…We obtain the following corollary for Banach spaces of continuous functions, which can be viewed as a strengtening of the first part of [16,Proposition 44]. Corollary 3.24.…”
Section: Auxiliary Results and The Proofs Of Theorem A And Bmentioning
confidence: 95%
See 1 more Smart Citation
“…We obtain the following corollary for Banach spaces of continuous functions, which can be viewed as a strengtening of the first part of [16,Proposition 44]. Corollary 3.24.…”
Section: Auxiliary Results and The Proofs Of Theorem A And Bmentioning
confidence: 95%
“…(iii) By [16,Theorem 2], there is an uncountable almost disjoint family A ⊆ [N] ω such that B(C 0 (K A )) has a character, where K A is the Isbell-Mrówka space corresponding to A. Consequently by [12,Lemma 2.2] the Banach space C 0 (K A ) does not have the SHAI property. On the one hand, as X A ∼ = C 0 (K A ), it follows that X A does not have the SHAI property either.…”
Section: Auxiliary Results and The Proofs Of Theorem A And Bmentioning
confidence: 99%
“…A construction within ZFC is given in [18]. A possible explanation for the scarcity of Banach spaces X whose closed ideals of operators have been classified, especially among classical spaces, is that recent research has shown that in many cases B(X) has 2 c closed ideals, where c denotes the cardinality of the contiuum.…”
Section: Introductionmentioning
confidence: 99%