Ternary weakly amenable \(C^\ast\)-algebras and \(JB^\ast\)-triples (Q2871301)

A Banach algebra \(A\) is said to be ternary weakly amenable if every continuous Jordan triple derivation from \(A\), considered as a Jordan-Banach triple system, into its dual is inner. In this paper, the authors prove that commutative \(C^{*}\)-algebras are ternary weakly amenable. They also prove that neither \(B(H)\), the bounded linear operators on a Hilbert space, nor \(K(H)\), the compact operators, are ternary weakly amenable unless \(H\) is finite-dimensional. By contrast, every \(C^{*}\)-algebra is weakly amenable [\textit{U. Haagerup}, Invent. Math. 74, 305--319 (1983; Zbl 0529.46041)]. Once stablished that weak amenability and ternary weak amenability do not coincide in general, the authors conclude with the initial steps in the study of ternary weak amenability for some Cartan factors.