Let ${\mathcal A}$ be a Banach algebra, and let $\varphi $ be a nonzero character on ${\mathcal A}$. For a closed ideal I of ${\mathcal A}$ with $I\not \subseteq \ker \varphi $ such that I has a bounded approximate identity, we show that $\operatorname {WAP}(\mathcal {A})$, the space of weakly almost periodic functionals on ${\mathcal A}$, admits a right (left) invariant $\varphi $-mean if and only if $\operatorname {WAP}(I)$ admits a right (left) invariant $\varphi |_I$-mean. This generalizes a result due to Neufang for the group algebra $L^1(G)$ as an ideal in the measure algebra $M(G)$, for a locally compact group G. Then we apply this result to the quantum group algebra $L^1({\mathbb G})$ of a locally compact quantum group ${\mathbb G}$. Finally, we study the existence of left and right invariant $1$-means on $ \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$.

Let H be an ultraspherical hypergroup and let $A(H)$ be the Fourier algebra associated with $H.$ In this paper, we study the dual and the double dual of $A(H).$ We prove among other things that the subspace of all uniformly continuous functionals on $A(H)$ forms a $C^*$-algebra. We also prove that the double dual $A(H)^{\ast \ast }$ is neither commutative nor semisimple with respect to the Arens product, unless the underlying hypergroup H is finite. Finally, we study the unit elements of $A(H)^{\ast \ast }.$

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.

Let $ H $ be a compact subgroup of a locally compact group $ G $. We first investigate some (operator) (co)homological properties of the Fourier algebra $A(G/H)$ of the homogeneous space $G/H$ such as (operator) approximate biprojectivity and pseudo-contractibility. In particular, we show that $ A(G/H) $ is operator approximately biprojective if and only if $ G/H $ is discrete. We also show that $A(G/H)^{**}$ is boundedly approximately amenable if and only if G is compact and H is open. Finally, we consider the question of existence of weakly compact multipliers on $A(G/H)$.

Let $\mathbb{G}$ be a locally compact quantum group and let $I$ be a closed ideal of $L^{1}(\mathbb{G})$ with $y|_{I}\neq 0$ for some $y\in \text{sp}(L^{1}(\mathbb{G}))$. In this paper, we give a characterization for compactness of $\mathbb{G}$ in terms of the existence of a weakly compact left or right multiplier $T$ on $I$ with $T(f)(y|_{I})\neq 0$ for some $f\in I$. Using this, we prove that $I$ is an ideal in its second dual if and only if $\mathbb{G}$ is compact. We also study Arens regularity of $I$ whenever it has a bounded left approximate identity. Finally, we obtain some characterizations for amenability of $\mathbb{G}$ in terms of the existence of some $I$-module homomorphisms on $I^{\ast \ast }$ and on $I^{\ast }$.

We characterize two important notions of amenability and compactness of a locally compact quantum group $\mathbb{G}$ in terms of certain homological properties. For this, we show that $\mathbb{G}$ is character amenable if and only if it is both amenable and co-amenable. We finally apply our results to Arens regularity problems of the quantum group algebra ${{L}^{1}}\left( \mathbb{G} \right)$ . In particular, we improve an interesting result by Hu, Neufang, and Ruan.

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.

For a Banach algebra 𝒜 and a character ϕ on 𝒜, we introduce and study the notion of essential ϕ-amenability of 𝒜. We give some examples to show that the class of essentially ϕ-amenable Banach algebras is larger than that of ϕ-amenable Banach algebras introduced by Kaniuth et al. [‘On ϕ-amenability of Banach algebras’, Math. Proc. Cambridge Philos. Soc.144 (2008), 85–96]. Finally, we characterize the essential ϕ-amenability of various Banach algebras related to locally compact groups.

For a Banach algebra 𝒜 and a character ϕ on 𝒜, we have recently introduced and studied the notion of ϕ-pseudo-amenability of 𝒜. Here, we give some characterizations of this notion in terms of derivations from 𝒜 into various Banach 𝒜-bimodules.

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.