Amenability constants for unconditional sums of Banach algebras

TL;DR

This paper investigates the amenability and weak amenability of unconditional sums of Banach algebras, establishing bounds and conditions under which these properties are preserved or fail, with applications to various sequence space sums.

Contribution

It provides new bounds for amenability constants of unconditional sums of Banach algebras and characterizes when these sums are amenable or weakly amenable, including sharpness of bounds and necessary conditions.

Findings

Amenability of sums is equivalent to uniform boundedness of summand amenability constants.

Established two-sided bounds for amenability and weak amenability constants.

Demonstrated that certain bounds are sharp with explicit finite-dimensional examples.

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 \[ \sup_{i\in I}\operatorname{AM}(A_i) \;\le\; \operatorname{AM}\Bigl(\bigl(\textstyle\bigoplus_{i\in I} A_i\bigr)_{\!E}\Bigr) \;\le\; C_E^2 \sup_{i\in I}\operatorname{AM}(A_i). \] 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.