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

Johnson, William B. and Kania, Tomasz and Schechtman, Gideon (2016) Closed ideals of operators on and complemented subspaces of banach spaces of functions with countable support. Proceedings of the American Mathematical Society, 144 (10). pp. 4471-4485. ISSN 0002-9939

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.