# Amenability constants for unconditional sums of Banach algebras

Source: [Amenability constants for unconditional sums of Banach algebras](<https://www.sota2.com/research/paper/6965d16b6343fa78c83233c8>)

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Data updated: 2026-07-03T05:00:04.966Z

Authors: Tomasz Kania, Jerzy K\\k{a}kol

Published: 2026-01-10T20:35:59.000Z

## Abstract

We study Johnson amenability for unconditional direct sums of Banach algebras. Given a family $(A\_i)\_{i\\in I}$ of Banach algebras and a Banach sequence lattice $E$ on~$I$, the $E$-sum $\\bigl(\\bigoplus\_{i\\in I} A\_i\\bigr)\_{\\!E}$ carries a natural Banach algebra structure via coordinatewise multiplication. Under the hypothesis that $C\_E := \\sup\\{\\\|\\chi\_F\\\|\_E: F\\subseteq I\\text{ finite}\\}\<\\infty$, we prove that this $E$-sum is amenable if and only if the amenability constants of the summands are uniformly bounded, and we establish the two-sided estimate \\\[ \\sup\_{i\\in I}\\text{AM}(A\_i) \\;\\le\\; \\text{AM}\\Bigl(\\bigl(\\textstyle\\bigoplus\_{i\\in I} A\_i\\bigr)\_{\\!E}\\Bigr) \\;\\le\\; C\_E^2\\,\\sup\_{i\\in I}\\text{AM}(A\_i). \\\] We show that the factor $C\_E^2$ is sharp by exhibiting finite-dimensional examples where equality holds. We further prove that finiteness of $C\_E$ is necessary whenever infinitely many summands are non-zero and the sum admits a bounded approximate identity.   As applications, we recover the classical formula $\\text{AM}\\bigl(c\_0\\text{-}\\bigoplus\_{i\\in I} A\_i\\bigr) = \\sup\_{i\\in I}\\text{AM}(A\_i)$ for arbitrary (possibly uncountable) index sets, extend it to weighted $c\_0$-spaces, and characterise amenability for Orlicz sequence algebra sums. We also record how these unconditional criteria give obstructions within the conditional framework of James-type $J$-sums.   Finally, we investigate weak amenability of $E$-sums. We prove that weak amenability passes to summands, that $E$-sums of commutative weakly amenable algebras are weakly amenable, and--contrasting sharply with the Johnson amenability picture--that for $1 \< p \< \\infty$, the $\\ell\_p$-sum of infinitely many copies of a non-commutative weakly amenable algebra fails to be weakly amenable. In the $c\_0$-type regime ($C\_E \< \\infty$), we establish two-sided estimates for weak amenability constants with constants depending only on $C\_E$.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2601.06680>)

[PDF](<https://arxiv.org/pdf/2601.06680.pdf>)