Closed ideals in L(X) and L(X^*) when X contains certain copies of ℓ_p and c₀

2015/07/12 by Ben Wallis, Wallis, Ben
Mathematics · #46B20 #Advanced Banach Space Theory #Advanced Topics in Algebra #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46B20

paper · pdf · doi:10.48550/arxiv.1507.03241

28 pages

arxiv created 2015/07/12 · openalex publication_date 2015/07/12 · arxiv updated 2015/07/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Suppose X is a real or complexified Banach space containing a complemented copy of ℓ_p, p∈(1,2), and a copy (not necessarily complemented) of either ℓ_q, q∈(p,∞), or c₀. Then L(X) and L(X^*) each admit continuum many closed ideals. If in addition q≥ p', (1)/(p)+(1)/(p')=1, then the closed ideals of L(X) and L(X^*) each fail to be linearly ordered. We obtain additional results in the special cases of L(ℓ₁⊕ℓ_q) and L(ℓ_p⊕ c₀), 1<p<2<q<∞.

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• Ideals in L(L₁)
• Banach spaces for which the space of operators has 2^{\mathfrak c} closed ideals
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• The algebra of bounded linear operators on ℓ_p⊕ℓ_q has infinitely many closed ideals