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


Yemon Choi*, Fereidoun Ghahramani

* Corresponding author



Appeared as Q. J. Math (Oxford) 62 (2011), no. 1, 39--58.
Offprints available on request.

Preprint version available at arXiv math.0906.2253


[ Math Review | Zentralblatt ]


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.


The results presented in this paper date from two spells of work: one in the summer of 2008, when I was still a postdoc at the University of Manitoba; and one during a visit to Winnipeg in April 2009. While a long gap is probably not recommended in these days of "publish or* perish", it was certainly helpful to be able to look back on the older calculations with a fresh eye, and indeed this allowed us to make several improvements. This is particularly true of the results on Segal and Lipschitz algebras, which were not covered by the original framework developed in 2008.

(*) This is "or", not XOR, of course.

Yemon Choi