Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Symmetric amenability and the nonexistence of Lie and Jordan derivations

B. E. Johnsona1

a1 The University of Newcastle, Newcastle upon Tyne, NE1 7RU

A. M. Sinclair has proved that if S0305004100075010_inline1 is a semisimple Banach algebra then every continuous Jordan derivation from S0305004100075010_inline1 into S0305004100075010_inline1 is a derivation ([12, theorem 3·3]; ‘Jordan derivation’ is denned in Section 6 below). If S0305004100075010_inline2 is a Banach S0305004100075010_inline1-bimodule one can consider Jordan derivations from S0305004100075010_inline1 into S0305004100075010_inline2 and ask whether Sinclair's theorem is still true. More recent work in this area appears in [1]. Simple examples show that it cannot hold for all modules and all semisimple algebras. However, for more restricted classes of algebras, including C*-algebras one does get a positive result and we develop two approaches. The first depends on symmetric amenability, a development of the theory of amenable Banach algebras which we present here for the first time in Sections 2, 3 and 4. A Banach algebra is symmetrically amenable if it has an approximate diagonal consisting of symmetric tensors. Most, but not all, amenable Banach algebras are symmetrically amenable and one can prove results for symmetric amenability similar to those in [8] for amenability. However, unlike amenability, symmetric amenability does not seem to have a concise homological characterisation. One of our results [Theorem 6·2] is that if S0305004100075010_inline1 is symmetrically amenable then every continuous Jordan derivation into an S0305004100075010_inline1-bimodule is a derivation. Special techniques enable this result to be extended to other algebras, for example all C*-algebras. This approach to Jordan derivations appears in Section 6.

(Received May 25 1995)