TY - JOUR

T1 - A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

AU - Koszmider, Piotr

AU - Laustsen, Niels

N1 - 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

PY - 2021/4/16

Y1 - 2021/4/16

N2 - 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 manynames, 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 thefewest 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.

AB - 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 manynames, 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 thefewest 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.

KW - Banach space of continuous functions

KW - compact Hausdorff space

KW - Psi-space

KW - almost disjoint family

KW - bounded linear operator

KW - closed operator ideal

KW - partitioner

KW - perfect set property

U2 - 10.1016/j.aim.2021.107613

DO - 10.1016/j.aim.2021.107613

M3 - Journal article

VL - 381

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 107613

T2 - FST Distinguished Visitor Programme

Y2 - 16 February 2019 through 13 April 2019

ER -