2023/08/22 by Max Arnott, Arnott, Max, Niels Jakob Laustsen +1 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2308.11586
openalex publication_date 2023/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that for each of the following Banach spaces~X, the quotient algebra \mathscrB(X)/\mathscrI has a unique algebra norm for every closed ideal \mathscrI of \mathscrB(X)\colon - X= (\bigoplusn∈\Nℓ2n)c0 and its dual, X= (\bigoplusn∈\Nℓ2n)ℓ1, - X= (\bigoplusn∈\Nℓ2n)c0⊕ c0(Γ) and its dual, X= (\bigoplusn∈\Nℓ2n)ℓ1⊕ℓ1(Γ), for an uncountable cardinal number~Γ, - X = C0(KA), the Banach space of continuous functions vanishing at infinity on the locally compact Mrówka space~KA induced by an uncountable, almost disjoint family~A of infinite subsets of~ℕ, constructed such that C0(KA) admits "few operators". Equivalently, this result states that every homomorphism from~\mathscrB(X) into a Banach algebra is continuous and has closed range. The key step in our proof is to show that the identity operator on a suitably chosen Banach space factors through every operator in \mathscrB(X)∖\mathscrI with control over the norms of the operators used in the factorization. These quantitative factorization results may be of independent interest.