Amenability Constants for Unconditional Sums of Banach Algebras

We study Johnson amenability for unconditional direct sums of Banach algebras. Given a family of Banach algebras and a Banach sequence lattice on , the ‐sum carries a natural Banach algebra structure via coordinatewise multiplication. Under the hypothesis that , we prove that this ‐sum is amenable if and only if the amenability constants of the summands are uniformly bounded, and we establish the two‐sided estimate We show that the factor is sharp by exhibiting finite‐dimensional examples where equality holds. We further prove that finiteness of is necessary whenever infinitely many summands are non‐zero and the sum admits a bounded approximate identity. As applications, we recover the classical formula for arbitrary (possibly uncountable) index sets, extend it to weighted ‐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 ‐sums. Finally, we investigate weak amenability of ‐sums. We prove that weak amenability passes to summands, that ‐sums of commutative weakly amenable algebras are weakly amenable, and—contrasting sharply with the Johnson amenability picture—that for , the ‐sum of infinitely many copies of a non‐commutative weakly amenable algebra fails to be weakly amenable. In the ‐type regime (), we establish two‐sided estimates for weak amenability constants with constants depending only on .