We characterise the existence of certain (weakly) compact multipliers of the second dual of symmetric abstract Segal algebras in both the group algebra $L^{1}(G)$ and the Fourier algebra $A(G)$ of a locally compact group G.

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 $2\times 2$ special unitary group, $\mathrm{SU} (2)$, are not approximately amenable.

In this paper we consider some notions of amenability such as ideal amenability, n-ideal amenability and approximate n-ideal amenability. The first two were introduced and studied by Gordji, Yazdanpanah and Memarbashi. We investigate some properties of certain Banach algebras in each of these classes. Results are also given for Segal algebras on locally compact groups.

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.

A number of well-known results of Ghahramani and Loy on the essential amenability of Banach algebras are generalized. It is proved that a symmetric abstract Segal algebra with respect to an amenable Banach algebra is essentially amenable. Applications to locally compact groups are given.