Biflatness and Pseudo-amenability of Segal Algebras

We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, L (G), and the Fourier algebra, A(G), of a locally compact group, G. Barry Johnson introduced the important concept of amenability for Banach algebras in [20], where he proved, among many other things, that a group algebra L1(G) is amenable precisely when the locally compact group, G, is amenable. For other Banach algebras, it is often useful to relax some of the conditions in the original definition of amenability and a popular theme in abstract harmonic analysis has been to find, for various classes of Banach algebras associated to locally compact groups, a “correct notion” of amenability in the sense that it singles out the amenable groups. For example, the measure algebra M(G) is amenable if and only if G is both amenable and discrete [6], however M(G) is Connes-amenable (a definition of amenability for dual Banach algebras) exactly when G is amenable [33]. As another example, the Fourier algebra, A(G), can fail to be amenable even for compact groups [22], but is operator amenable (a version of amenability that makes sense for Banach algebras with an operator space structure) if and only if G is amenable [30]. The purpose of this paper is to examine the amenability properties of Segal algebras, in both L1(G) and A(G). All of the aforementioned versions of amenability imply the existence of a bounded approximate identity (or identity in the case of Connes-amenability), however, a proper Segal algebra never has a bounded approximate identity [2]. Ghahramani, Loy and Zhang have introduced several notions of “amenablility without boundedness”, including approximate and pseudo-amenability, which do not a priori imply the existence of bounded approximate identities [14], [17]. It is thus natural to try to determine when a Segal algebra is approximately/pseudo-amenable. Indeed, this has already been considered in [14] and [17]. In particular, Date: February 2, 2008. 2000 Mathematics Subject Classification. Primary 43A20, 43A30; Secondary 46H25, 46H10, 46H20, 46L07.