Ternary Weakly Amenable C*-algebras and JB*-triples

TL;DR

This paper investigates the concept of ternary weak amenability in C*-algebras and JB*-triples, establishing new results about which structures possess this property and extending the understanding of derivations in these algebraic contexts.

Contribution

It introduces the notion of ternary weak amenability for Jordan Banach triples and characterizes this property for commutative C*-algebras and specific operator algebras.

Findings

Commutative C*-algebras are ternary weakly amenable.

B(H) and K(H) are not ternary weakly amenable unless H is finite dimensional.

Study extends to Jordan Banach triples, including commutative JB*-triples and Cartan factors.

Abstract

A well known result of Haagerup from 1983 states that every C*-algebra A is weakly amenable, that is, every (associative) derivation from A into its dual is inner. A Banach algebra B is said to be ternary weakly amenable if every continuous Jordan triple derivation from B into its dual is inner. We show that commutative C*-algebras are ternary weakly amenable, but that B(H) and K(H) are not, unless H is finite dimensional. More generally, we inaugurate the study of weak amenability for Jordan Banach triples, focussing on commutative JB*-triples and some Cartan factors.