Approximate and pseudo-amenability of various classes of Banach algebras
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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 $PF_p(Γ)$ of $p$-pseudofunctions on a discrete group $Γ$ (of which the reduced $C^*$-algebra is a special case).
35 pages, revision of Jan '08 preprint. Abstract and MSC added; bibliograpy updated; slight tweaks to Section 4; and correction of a few typos. The final version is to appear in J. Funct. Anal
35 pages, revision of Jan '08 preprint. Abstract and MSC added; bibliograpy updated; slight tweaks to Section 4; and correction of a few typos. The final version is to appear in J. Funct. Anal