Sailing over three problems of Koszmider

Sánchez, Félix Cabello; Castillo, Jesús M. F.; Marciszewski, Witold; Plebanek, Grzegorz; Salguero-Alarcón, Alberto

doi:10.48550/arxiv.1910.07273

Cited by 1 publication

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“…This answers a question raised in [27], and together with [5], [34] and [8] it completes the list of solutions to the problems left open in [27].…”

Section: Introductionsupporting

confidence: 63%

“…[8]) that there are 2 c nonisomorphic Banach spaces of the form C 0 (K A ). Recently F. Cabello Sánchez, J. Castillo, W. Marciszewski, G. Plebanek, A. Salguero-Alarcón showed in [8] that assuming MA and the negation of CH any two Banach spaces of the form C 0 (K A ) for almost disjoint families A of the same uncountable cardinality smaller than c are isomorphic. In particular they are isomorphic to their squares.…”

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confidence: 99%

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

Koszmider¹,

Laustsen²

2020

Preprint

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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 bounded linear operator on C 0 (K A ) is the sum of a scalar multiple of the identity and an operator that factors through c 0 (which in this case is equivalent to having separable range). This implies that C 0 (K A ) has the fewest possible decompositions: whenever C 0 (K A ) is written as the direct sum of two infinite-dimensional Banach spaces X and Y, either X is isomorphic to C 0 (K A ) and Y to c 0 , or vice versa. These results improve previous work of the first named author in which an extra set-theoretic hypothesis was required.To exploit the perfect set property for Borel sets as in the classical construction of an almost disjoint family of Mrówka we need to deal with N×N-matrices rather than with the usual partitioners. 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 F. van Engelen, K. Kunen and A. Miller concerning Borel subsets of the square.

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“…This answers a question raised in [27], and together with [5], [34] and [8] it completes the list of solutions to the problems left open in [27].…”

Section: Introductionsupporting

confidence: 63%

mentioning

confidence: 99%