TY - JOUR

T1 - Uniqueness of algebra norm on quotients of the algebra of bounded operators on a Banach space

AU - Arnott, Max

AU - Laustsen, Niels Jakob

PY - 2024/10/15

Y1 - 2024/10/15

N2 - We show that the quotient algebra B ( X ) / I has a unique algebra norm for every closed ideal I of the Banach algebra B ( X ) of bounded operators on X, where X denotes any of the following Banach spaces: • ( ⨁ n ∈ N ℓ 2 n ) c 0 or its dual space ( ⨁ n ∈ N ℓ 2 n ) ℓ 1 , • ( ⨁ n ∈ N ℓ 2 n ) c 0 ⊕ c 0 ( Γ ) or its dual space ( ⨁ n ∈ N ℓ 2 n ) ℓ 1 ⊕ ℓ 1 ( Γ ) for an uncountable cardinal number Γ, • 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.

AB - We show that the quotient algebra B ( X ) / I has a unique algebra norm for every closed ideal I of the Banach algebra B ( X ) of bounded operators on X, where X denotes any of the following Banach spaces: • ( ⨁ n ∈ N ℓ 2 n ) c 0 or its dual space ( ⨁ n ∈ N ℓ 2 n ) ℓ 1 , • ( ⨁ n ∈ N ℓ 2 n ) c 0 ⊕ c 0 ( Γ ) or its dual space ( ⨁ n ∈ N ℓ 2 n ) ℓ 1 ⊕ ℓ 1 ( Γ ) for an uncountable cardinal number Γ, • 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.

KW - Banach algebra of bounded operators

KW - Banach space

KW - Quantitative factorization of bounded operators

KW - Uniqueness of algebra norm

U2 - 10.1016/j.jfa.2024.110559

DO - 10.1016/j.jfa.2024.110559

M3 - Journal article

VL - 287

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 8

M1 - 110559

ER -