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Closed ideals of operators on and complemented subspaces of banach spaces of functions with countable support

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Published
  • William B. Johnson
  • Tomasz Kania
  • Gideon Schechtman
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<mark>Journal publication date</mark>10/2016
<mark>Journal</mark>Proceedings of the American Mathematical Society
Issue number10
Volume144
Number of pages15
Pages (from-to)4471-4485
Publication StatusPublished
Early online date25/04/16
<mark>Original language</mark>English

Abstract

Let λ be an infinite cardinal number and let ℓc ∞(λ) denote the subspace of ℓ∞(λ) consisting of all functions that assume at most countably many non-zero values. We classify all infinite-dimensional complemented subspaces of ℓc ∞(λ), proving that they are isomorphic to ℓc ∞(κ) for some cardinal number κ. Then we show that the Banach algebra of all bounded linear operators on ℓc ∞(λ) or ℓ∞(λ) has the unique maximal ideal consisting of operators through which the identity operator does not factor. Using similar techniques, we obtain an alternative to Daws’ approach description of the lattice of all closed ideals of B(X), where X = c0(λ) or X = ℓp(λ) for some p ∈ [1,∞), and we classify the closed ideals of B(ℓc ∞(λ)) that contains the ideal of weakly compact operators. © 2016 American Mathematical Society.

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