Amenability constants for unconditional sums of Banach algebras
Tomasz Kania, Jerzy K\k{a}kol

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 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 \[ \sup_{i\in I}\operatorname{AM}(A_i) \;\le\; \operatorname{AM}\Bigl(\bigl(\textstyle\bigoplus_{i\in I} A_i\bigr)_{\!E}\Bigr) \;\le\; C_E^2 \sup_{i\in I}\operatorname{AM}(A_i). \] We show that the factor is sharp by exhibiting finite-dimensional examples where equality holds. We further prove that finiteness of is…
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Banach Space Theory · Holomorphic and Operator Theory
