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Closed ideals in the algebra of compact-by-approximable operators

I describe various examples of non-trivial closed ideals of the compact-by-approximable algebra 𝔄X=:𝒦(X)/𝒜(X) for Banach spaces X that fail to have the approximation property. Here 𝒦⁢(X) is the algebra of compact operators on X and 𝒜(X)=:ℱ⁢(X)¯ is the uniform closure of the bounded finite rank operators ℱ⁢(X). The quotient algebra 𝔄X is a highly mysterious radical Banach algebra.Sample results: (i) if X has cotype 2, Y has type 2, 𝔄X≠{0} and 𝔄Y≠{0}, then 𝔄X⊕Y has at least 2 (and in some cases even up to 8) closed ideals, (ii) there are closed subspaces X⊂ℓp for 4<p<∞ such that 𝔄X contains a non-trivial closed ideal, (iii) there is a closed subspace Y⊂c0 where 𝔄Y contains two countable non-comparable chains of closed ideals, (iv) there is a Banach space Z such that 𝔄Z contains an uncountable lattice of closed ideal having the reverse order structure of the power set (𝒫⁢(ℕ),⊂).

The talk is based on [1] and [2] which were motivated by questions of Garth Dales. Relevant earlier examples were obtained by Wojtyński (1978) and Shulman & Turovskii (2011) by completely different methods.

REFERENCES

[1] Hans-Olav Tylli & Henrik Wirzenius: Closed ideals in the algebra of compact-by-approximable operators, J. Funct. Anal. 282 (2022), 109328

[2] Henrik Wirzenius: Quotient algebras of Banach operator ideals related to non-classical approximation properties, arXiv:2202.11500