Closed ideals of operators on the Tsirelson and Schreier spaces

Beanland, Kevin and Kania, Tomasz and Laustsen, Niels (2020) Closed ideals of operators on the Tsirelson and Schreier spaces. Journal of Functional Analysis, 279 (8): 108668. ISSN 0022-1236

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Abstract

Let B(X) denote the Banach algebra of bounded operators on X, where X is either Tsirelson's Banach space or the Schreier space of order n for some natural number n. We show that the lattice of closed ideals of B(X) has a very rich structure; in particular B(X) contains at least continuum many maximal ideals. Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection P_N∈B(X) corresponding to each non-empty subset N of natural numbers. A closed ideal of B(X) is spatial if it is generated by P_N for some set N. We can now state our main conclusions as follows: - the family of spatial ideals lying strictly between the ideal of compact operators and B(X) is non-empty and has no minimal or maximal elements; - for each pair of spatial ideals I and J such that I is properly contained in J, there is a family {Γ_α : α∈Δ}, where the index set Δ has the cardinality of the continuum, such that Γ_α is an uncountable chain of spatial ideals, each lying strictly between I and J, lies stric∪Γ_α is a closed ideal that is not spatial, and the ideal L+M is dense in J whenever α,β∈Δ are distinct and L∈Γ_α, M∈Γ_β.