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.
Highlights
Two fundamental questions concerning derivations from a Banach algebra A into a Banach A-bimodule M are: Is an everywhere defined derivation automatically continuous? Are all continuous derivations inner? If not, can every continuous derivation be approximated by inner derivations? One can ask the same questions in the setting of Jordan Banach algebras, and more generally for Jordan Banach triple systems
In the context of C∗-algebras, automatic continuity results were initiated by Kaplansky before 1950 and culminated in the following series of results: Every derivation from a C∗-algebra into itself is continuous
Is a linear mapping D : A → M such that D(a ◦ b) = a ◦ Db + Da ◦ b, where ◦ denotes both the product in the Jordan algebra and the module action. (Jordan Banach algebras and Jordan Banach modules will be defined below.) An inner derivation in this context is a derivation of the form: m i=1
Summary
Two fundamental questions concerning derivations from a Banach algebra A into a Banach A-bimodule M are:. (Jordan Banach algebras and Jordan Banach modules will be defined below.) An inner derivation in this context is a derivation of the form:. As shown in [43], these conditions are automatically satisfied in the case that the JB∗-triple is a C∗-algebra with the triple product (xy∗z + zy∗x)/2, leading to a new proof (cf [43, Cor. 23]) of the theorem of Ringrose quoted above as well as the results of Alaminos-Bresar-Villena and Hejazian-. It is worth noting that, besides the consequences for C∗-algebras of the main result of [43] noted above, another consequence is the automatic continuity of derivations of a JB∗-triple into its dual [43, Cor. 15], leading us to the study of weak amenability for JB∗-triples, which is the main focus of this paper. Every real or complex associative Banach algebra (resp., Jordan Banach algebra) is a real Jordan-
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