Amenability constants for unconditional sums of Banach algebras

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 $({⨁}_{i\in I}{A}_i{)}_{\text{\hspace{0.17em}⁣}E}$ carries a natural Banach algebra structure via coordinatewise multiplication. Under the hypothesis that $C_E:=\sup \{\mathrm{\parallel }{\chi }_F{\mathrm{\parallel }}_E:F\subseteq I\text{ finite}\}<\mathrm{\infty }$, 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 $$\underset{i\in I}{\sup }\mathrm{AM}(A_i)\le \mathrm{AM}\left(\right({⨁}_{i\in I}{A}_i{)}_{\text{\hspace{0.17em}⁣}E})\le C_E^2{\sup }_{i\in I}\mathrm{AM}(A_i)\mathrm{.}$$ 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 $1<p<\mathrm{\infty }$, the ${\mathrm{\ell }}_p$-sum of infinitely many copies of a non-commutative weakly amenable algebra fails to be weakly amenable. In the $c_0$-type regime ($C_E<\mathrm{\infty }$), we establish a two-sided estimate for weak amenability constants analogous to that for Johnson amenability.