Let B be an abstract Segal algebra with respect to A. For a nonzero character φ on A, we study φ-amenability, and φ-contractibility of A and B. We then apply these results to abstract Segal algebras related to locally compact groups.2000 Mathematics subject classification: primary 46H05; secondary 43A07.

Abstract. We calculate the exact amenability constant of the centre of ℓ 1 (G) when G is one of the following classes of finite group: dihedral; extraspecial; or Frobenius with abelian complement and kernel. This is done using a formula which applies to all finite groups with two character degrees. In passing, we answer in the negative a question raised in work of the third author with Azimifard and Spronk (J. Funct. Anal. 2009).

In this paper we first show that for a locally compact amenable group G, every proper abstract Segal algebra of the Fourier algebra on G is not approximately amenable; consequently, every proper Segal algebra on a locally compact abelian group is not approximately amenable. Then using the hypergroup generated by the dual of a compact group, it is shown that all proper Segal algebras of a class of compact groups including the 2 × 2 special unitary group, SU(2), are not approximately amenable.2010 Mathematics subject classification: primary 46H20; secondary 43A20, 43A62, 46H10.

Let G be a restricted direct product of finite groups {G i } i∈I , and let Zℓ 1 (G) denote the centre of its group algebra. We show that Zℓ 1 (G) is amenable if and only if G i is abelian for all but finitely many i, and characterize the maximal ideals of Zℓ 1 (G) which have bounded approximate identities. We also study when an algebra character of Zℓ 1 (G) belongs to c 0 or ℓ p and provide a variety of examples.Keywords: Centre of group algebras, restricted direct product of finite groups, amenability, absolutely idempotent characters, maximal ideals.MSC 2010: 43A20 (primary); 20C15 (secondary).

We let the central Fourier algebra, ZA(G), be the subalgebra of functions u in the Fourier algebra A(G) of a compact group, for which u(xyx −1 ) = u(y) for all x, y in G.