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

Abstract

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∈Nℓ2n)c0 and its dual, X=(⨁n∈Nℓ2n)ℓ1,•X=(⨁n∈Nℓ2n)c0⊕c0(Γ)and its dual, X=(⨁n∈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 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.