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  • supermrowkaDec2020

    Rights statement: This is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics, 381, 2021 DOI: 10.1016/j.aim.2021.107613

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    Embargo ends: 28/01/22

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A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

Research output: Contribution to journalJournal articlepeer-review

E-pub ahead of print
Article number107613
<mark>Journal publication date</mark>16/04/2021
<mark>Journal</mark>Advances in Mathematics
Volume381
Number of pages32
Publication StatusE-pub ahead of print
Early online date28/01/21
<mark>Original language</mark>English
EventFST Distinguished Visitor Programme: Visit of Piotr Koszmider - Lancaster University, Lancaster, United Kingdom
Duration: 16/02/201913/04/2019

Other

OtherFST Distinguished Visitor Programme
CountryUnited Kingdom
CityLancaster
Period16/02/1913/04/19

Abstract

We consider the closed subspace of l generated by c0 and the characteristic functions of elements of an uncountable, almost disjoint family A of infinite subsets of the natural numbers. This Banach space has the form C0(KA) for a locally compact Hausdorff space K_A that is known under many
names, including Ψ-space and Isbell-Mrówka space.

We construct an uncountable, almost disjoint family A such that the algebra of all bounded linear operators on C0(KA) is as small as possible in the precise sense that every bounded linear operator on C0(KA) is the sum of a scalar multiple of the identity and an operator that factors through c0 (which in this case is equivalent to having separable range). This implies that C0(KA) has the
fewest possible decompositions: whenever C0(KA) is written as the direct sum of two infinite-dimensional Banach spaces X and Y, either X is isomorphic to C0(KA) and Y to c0, or vice versa. These results improve previous work of the first named author in which an extra set-theoretic hypothesis was required. We also discuss the consequences of these results for the algebra of all bounded linear operators on our Banach space C0(KA) concerning the lattice of closed ideals, characters and automatic continuity of homomorphisms.

To exploit the perfect set property for Borel sets as in the classical construction of an almost disjoint family by Mrówka, we need to deal with N x N matrices rather than with the usual partitioners of an almost disjoint family. This noncommutative setting requires new ideas inspired by the theory of compact and weakly compact operators and the use of an extraction principle due to van Engelen, Kunen and Miller concerning Borel subsets of the square.

Bibliographic note

This is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics, 381, 2021 DOI: 10.1016/j.aim.2021.107613