Approximate and pseudo-amenability of various classes of Banach algebras

Abstract

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 $P{F}_p(\Gamma )$ of $p$-pseudofunctions on a discrete group $\Gamma$ (of which the reduced $C^{\ast }$-algebra is a special case).

arXiv comment35 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

References in corpus (1)

• Approximate characterizations of projectivity and injectivity for Banach modules

Cited by in corpus (8)

• Approximate amenability of Schatten classes, Lipschitz algebras and second duals of Fourier algebras
• Approximate amenability of Segal algebras II
• Amenability properties of Banach algebra valued continuous functions
• 2n-Weak module amenability of semigroup algebras
• Solved and unsolved problems in generalized notions of amenability for Banach algebras
• Singly generated operator algebras satisfying weakened versions of amenability
• Generalized notion of amenability for a class of matrix algebras
• Weak amenability of commutative Beurling algebras