Approximate and pseudo-amenability of various classes of Banach algebras

Details Approximate and pseudo-amenability of various classes of Banach algebras Approximate and pseudo-amenability of various classes of Banach algebras

2008-01-22

arxiv.org/abs/0801.3415v2

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

Mathematics

Functional Analysis

35 pages, revision of Jan '08 preprint. Abstract and MSC added; bibliograpy updated; slight tweaks to Section 4; and correctio

Scientific paper

10.1016/j.jfa.2009.02.012

We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors. 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 approximately amenable. Further examples are obtained of ${\ell}^1$-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate amenability need not imply sequential approximate amenability. Results are also given for Segal subalgebras of $L^1(G)$, where $G$ is a locally compact group, and the algebras $PF_p(\Gamma)$ of $p$-pseudofunctions on a discrete group $\Gamma$ (of which the reduced $C^*$-algebra is a special case).

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