Amenability Constants for Unconditional Sums of Banach Algebras

Abstract

We study Johnson amenability for unconditional direct sums of Banach algebras. Given a family ${\left(A_i\right)}_{i\in I}$ of Banach algebras and a Banach sequence lattice $E$ on $I$, the $E$-sum ${\left({⨁}_{i\in I},A_i\right)}_E$ carries a natural Banach algebra structure via coordinatewise multiplication. Under the hypothesis that , 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

As applications, we recover the classical formula $\mathrm{AM}\left(c_0,-,{⨁}_{i\in I},A_i\right)={sup}_{i\in I}\mathrm{AM}\left(A_i\right)$ for arbitrary (possibly uncountable) index sets, extend it to weighted $c_0$-spaces, and characterize 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 , 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 (), we establish two-sided estimates for weak amenability constants with constants depending only on $C_E$.