Uniqueness of algebra norm on quotients of the algebra of bounded operators on a Banach space

Arnott, Max and Laustsen, Niels Jakob (2024) Uniqueness of algebra norm on quotients of the algebra of bounded operators on a Banach space. Journal of Functional Analysis: 110559. ISSN 0022-1236 (In Press)

Abstract

dedicatoryIn memoriam: H. G. Dales (1944–2022) We show that for each of the following Banach spaces X, the quotient algebra B ( X ) / I has a unique algebra norm for every closed ideal I of B ( X ) : • X = ( ⨁ n ∈ N ℓ 2 n ) c 0 and its dual, X = ( ⨁ n ∈ N ℓ 2 n ) ℓ 1 , • X = ( ⨁ n ∈ N ℓ 2 n ) c 0 ⊕ c 0 ( Γ ) and its dual, X = ( ⨁ n ∈ N ℓ 2 n ) ℓ 1 ⊕ ℓ 1 ( Γ ) ,for an uncountable cardinal number Γ, • X = C 0 ( K A ) , the Banach space of continuous functions vanishing at infinity on the locally compact Mrówka space K A induced by an uncountable, almost disjoint family A of infinite subsets of N , constructed such that C 0 ( K A ) admits “few operators”. Equivalently, this result states that every homomorphism from B ( 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 B ( X ) ∖ I with control over the norms of the operators used in the factorization. These quantitative factorization results may be of independent interest.