Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support
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© 2016 American Mathematical Society. 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 nonzero values. We classify all infinitedimensional 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.
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Johnson, W. B., Kania, T., & Schechtman, G.
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Johnson, William BKania, TomaszSchechtman, Gideon
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