Amenability constants for unconditional sums of Banach algebras

TL;DR

This paper characterizes Johnson amenability for unconditional $E$-sums of Banach algebras under the uniform finite-support condition $C_E<\infty$. It proves that the sum is amenable if and only if the summands have uniformly bounded amenability constants, with a precise two-sided bound $\bigsup_i \text{AM}(A_i) \le \text{AM}(A) \le C_E^2 \bigsup_i \text{AM}(A_i)$ and shows the $C_E^2$ factor is sharp; moreover, finiteness of $C_E$ is necessary when infinitely many summands are non-zero and a bounded approximate identity exists. The work extends to weak amenability, establishing parallel two-sided estimates in the $c_0$-type regime and revealing a clear contrast with non-commutative cases (e.g., $\ell_p$ for $1<p<\infty$). The paper also connects unconditional $E$-sums to James-type $J$-sums, giving corollaries detectable inside conditional constructions, and analyzes Orlicz-heart spaces to illustrate when $C_E$ can be finite or infinite. Overall, it provides a coherent framework for transferring homological properties from summands to large direct-sum algebras and clarifies when such transfers preserve amenability and weak amenability.

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 $C_E := \sup\{\|χ_F\|_E : F \subseteq I \text{ finite}\} < \infty$, 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 \[ \sup_{i\in I}\operatorname{AM}(A_i) \;\le\; \operatorname{AM}\Bigl(\bigl(\textstyle\bigoplus_{i\in I} A_i\bigr)_{\!E}\Bigr) \;\le\; C_E^2 \sup_{i\in I}\operatorname{AM}(A_i). \] 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. 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 a two-sided estimate for weak amenability constants analogous to that for Johnson amenability.

Theorems & Definitions (75)

• Definition 2.1: Virtual diagonal and amenability constant
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4: Directed dense unions
• proof
• Lemma 2.5
• proof
• Definition 3.1
• Remark 3.2