Weak amenability of group algebras of connected complex semisimple Lie groups

Abstract:

We consider the problem of whether every continuous derivation from a group algebra ${L^1}(G)$ into its dual ${L^\infty }(G)$ (where the ${L^1}(G)$ actions on ${L^\infty }(G)$ are the adjoint of multiplication in ${L^1}(G)$ is inner, that is, of the form $D(a) = aF - Fa$ for some $F \in {L^\infty }(G)$. This had previously been established to hold for discrete and amenable groups and is now established for $G = {\text {Gl(}}n,{\mathbf {C}})$ and for all connected semisimple complex Lie groups.