Let $G$ be a locally compact group and let $\omega$ be a continuous weight on $G$ . We show that for each of the Banach algebras ${{L}^{1}}\left( G,\,\omega\right),\,M\left( G,\,\omega\right),\,LUC{{\left( G,\,{{\omega }^{-1}} \right)}^{*}}$ , and ${{L}^{1}}{{\left( G,\,\omega\right)}^{**}}$ , the order structure combined with the algebra structure determines the weighted group.

We show that every one-codimensional closed two-sided ideal in a boundedly approximately contractible Banach algebra has a bounded approximate identity. We use this to give a complete characterization of bounded approximate contractibility of Beurling algebras associated to symmetric weights. We give a slight modification of a criterion for bounded approximate contractibility. We use our criterion to show that, for the quasi-SIN groups, in the presence of a certain growth condition on a weight, the associated Beurling algebra is boundedly approximately amenable if and only if it is boundedly approximately contractible. We show that approximate amenability of a Beurling algebra on an IN group necessitates the amenability of the group. Finally, we show that, for every locally compact abelian group, in the presence of a growth condition on the weight, 2n-weak amenability of the associated Beurling algebra is equivalent to every point-derivation vanishing at the augmentation character.

We show that the global dimension, dgA, of every commutative Banach algebra A whose radical is a weighted convolution algebra is strictly greater than one. As an application, we see that in this case H2(A, X) ≠ 0 for some Banach A-bimodule X and thus there exists an unsplittable singular extension of the algebra A.

In [4] we have shown that any two semi-simple weighted convolution algebras L1(ω1) and L1(ω2) are isomorphic. In this paper, given any two radical weighted convolution algebras L1(ω1) and L1(ω2) we find necessary and sufficient conditions, in terms of ω1 and ω2, for L1(ω1) and L1(ω2) to be isomorphic.

We show that if a locally compact group $G$ is non-discrete or has an infinite amenable subgroup, then the second dual algebra $L^1(G)^{**}$ does not admit an involution extending the natural involution of $L^1(G)$. Thus, for the above classes of groups we answer in the negative a question raised by Duncan and Hosseiniun in 1979. We also find necessary and sufficient conditions for the dual of certain left-introverted subspaces of the space $C_b(G)$ (of bounded continuous functions on $G$) to admit involutions. We show that the involution problem is related to a multiplier problem. Finally, we show that certain non-trivial quotients of $L^1(G)^{**}$ admit involutions.

We introduce and study two new notions of amenability for Banach algebras. In particular we compare these notions with some of those studied earlier. We show that several classes of Banach algebras, including certain Banach algebras related to locally compact groups, are responsive to these notions.

In a recent paper [3] Dales and Pandey have shown that the class Sp of Segal algebras is weakly amenable. In this paper, for various classes of Segal algebras, we characterize derivations and multipliers from a Segal algebra into itself and into its dual module. In particular, we prove that every Segal algebra on a locally compact abelian group is weakly amenable and an abstract Segal subalgebra of a commutative weakly amenable Banach algebra is weakly amenable. We also introduce the Lebesgue–Fourier algebra of a locally compact group G and study its Arens regularity when G is discrete or compact.

In this paper we shall prove that the measure algebra $M(G)$ of a locally compact group $G$ is amenable as a Banach algebra if and only if $G$ is discrete and amenable as a group. Our contribution is to resolve a conjecture by proving that $M(G)$ is not amenable in the case where the group $G$ is not discrete. Indeed, we shall prove a much stronger result: the measure algebra of a non-discrete, locally compact group has a non-zero, continuous point derivation at a certain character on the algebra.

Let  be a Banach algebra and let ** be the second dual algebra of  endowed with the first or the second Arens product. We investigate relations between amenability of ** and Arens regularity of  and the rôle topological centres in amenability of **. We also find conditions under which weak amenability of ** implies weak amenability of .

It is shown that for every compact group G, L1(G)ˆ is unique and minimal among all the closed subsets I of M(G)** such that I is a proper (≠0, ≠M(G)**) algebraic ideal, and such that I is solid with respect to absolute continuity; that is, n∈L1(G)ˆ whenever n∈M(G)** and n[Lt]μ∈L1(G)ˆ.

A new proof is given for the weak amenability of the group algebras of locally compact groups.

Suppose that A is either the group algebra L1 (G) of a locally compact group G, or the Volterra algebra or a weighted convolution algebra with a regulated weight. We characterize: a) Module homomorphisms of A*, when A* is regarded an A** left Banach module with the Arens product, b) all the weak*-weak* continuous left multipliers of A**.

In [7] we gave a description of compact multipliers of Lω(X), and left open the question of whether weakly compact multipliers on Lω(X) are compact. This had already been answered for some examples of hypergroup algebras ([1], [2], [6] and [10]). Here we give a positive answer to this question in the general setting of a weighted hypergroup algebra.

Let Mω(X) be a weighted hypergroup algebra, and Lω(X) be the Banach algebra of measures μ ε Mω(X) such that the function x ↦ (1/ω(x))δx* |μ| is norm continuous. We characterize compact multipliers on Lω(X). This extends the characterization of compact multipliers on weighted group algebras and some classes of weighted semigroup algebras.

In a recent paper, E. Størmer, among other things, proves the existence of an isometric isomorphism from the measure algebra M(G) of a locally compact abelian group G into BB(L2(G)), ([6], Proposition 4.6). Here we give another proof for this result which works for non-commutative G as well as commutative G. We also prove that the algebra L1(G, λ), with λ the left (or right) Haar measure, is not isometrically isomorphic with an algebra of operators on a Hilbert space. The proofs of these two results are taken from the author's Ph.D. thesis [4], submitted to the University of Edinburgh before Størmer's paper. The author wishes to thank Dr. A. M. Sinclair for his help and encouragement.