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Amenability of any of the algebras described in the title is known to forcethem to be finite-dimensional. The analogous problems for \emph{approximate}amenability have been open for some years now. In this article we give acomplete solution for the first two classes, using a new criterion for showingthat certain Banach algebras without bounded approximate identities cannot beapproximately amenable. The method also provides a unified approach to existingnon-approximate amenability results, and is applied to the study of certaincommutative Segal algebras.  Using different techniques, we prove that \emph{bounded} approximateamenability of the second dual of a Fourier algebra implies that it isfinite-dimensional. Some other results for related algebras are obtained.

Approximate amenability of Schatten classes, Lipschitz algebras and second duals of Fourier algebras