Approximate amenability of Schatten classes, Lipschitz algebras and second duals of Fourier algebras
arXiv:0906.2253 · doi:10.1093/qmath/hap034
Abstract
Amenability of any of the algebras described in the title is known to force them to be finite-dimensional. The analogous problems for \emph{approximate} amenability have been open for some years now. In this article we give a complete solution for the first two classes, using a new criterion for showing that certain Banach algebras without bounded approximate identities cannot be approximately amenable. The method also provides a unified approach to existing non-approximate amenability results, and is applied to the study of certain commutative Segal algebras. Using different techniques, we prove that \emph{bounded} approximate amenability of the second dual of a Fourier algebra implies that it is finite-dimensional. Some other results for related algebras are obtained.
22 pages. v2: minor corrections, updated references. To appear in Q. J. Math. (Oxford)