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Approximate and pseudo-amenability of various classes of Banach algebras


Y. Choi, F. Ghahramani*, Y. Zhang

* Corresponding author


Amenable Banach algebra, amenable group, approximately amenable Banach algebra, approximate diagonal, approximate identity, Fourier algebra, Segal algebra, semigroup algebra, reduced $C^*$-algebra


Appeared as J. Funct. Anal. 256 (2009), no. 10, 3158--3191.

Preprint version available at arXiv math.0801.3415 (last updated Feb. 2009). Small differences from the published version.


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We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors together with R. J. Loy. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity. Among our other results, it is shown that the Fourier algebra of the free group on two generators is not operator approximately amenable.

Further examples are obtained of l1-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate contractibility need not imply sequential approximate amenability. Results are also given for Segal algebras on locally compact groups, and algebras of p-pseudofunctions on discrete groups.


There is a mis-statement in the remark at the end of Section 3 (which does not affect the validity of the rest of the paper). See the list of corrections for more details.

This is based on joint work from 2007-2008, when I was a postdoc at the University of Manitoba. The position was supported by research grants from F. Ghahramani and Y. Zhang (U. Manitoba) and from R. Stokke (U. Winnipeg).

Subsequent work by the first two authors (Q. J. Math. (Oxford), 2011) has shown that some of the examples listed at the end of Section 3, which were shown there to not be boundedly approximately amenable, are in fact not even approximately amenable.