Amenability constants for unconditional sums of Banach algebras

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 $C_E$.