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Abstract

The main result of the note is a characterization of 1-amenability of Banach algebras of approximable operators for a class of Banach spaces with 1-unconditional bases in terms of a new basis property. It is also shown that amenability and symmetric amenability are equivalent concepts for Banach algebras of approximable operators, and that a type of Banach space that was long suspected to lack property $\mathbb A$ has in fact the property. Some further ideas on the problem of whether or not amenability (in this setting) implies property $\mathbb A$ are discussed.