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Amenability of any of the algebras described in the title is known to force them to be finite-dimensional. The analogous problems for  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  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.

APPROXIMATE AMENABILITY OF SCHATTEN CLASSES, LIPSCHITZ ALGEBRAS AND SECOND DUALS OF FOURIER ALGEBRAS