We give sufficient conditions and necessary conditions for a Banach algebra, which is ℓ1-graded over a semi-lattice, to be biflat or biprojective. As an application we characterise biflat and biprojective discrete convolution algebras for commutative semi-groups.

This paper is concerned with two notions of cohomological triviality for Banach algebras, weak amenability and cyclic amenability. The first is defined within Hochschild cohomology and the latter within cyclic cohomology. Our main result is that where ℱ is a Banach algebraic free product of two Banach algebras and ℬ. It follows that cyclic amenability is preserved under the formation of free products.

Let T: A → B be a linear operator between two Banach algebras A and B. The basic problem in the theory of automatic continuity is to find algebraic conditions on T, A, and B which ensure that T is continuous. As a means to study continuity properties of T the separating space of T has played a crucial role. It is defined as

In this paper we study the cohomology groups Hn(I, I*) and Hn([Uscr], [Uscr]*) where [Uscr] is a Banach algebra with a bounded approximate identity and I is a codimension one closed two-sided ideal of [Uscr]. This is applied to the case when [Uscr] is the group algebra L1(G) of a locally compact group G and I={f∈L1(G)[mid ]∫Gf=0}, the augmentation ideal of G. We show that if G is inner amenable, then I is always weakly amenable, i.e. [Hscr]1(I, I*)={0}.

Let X be a Banach space and let A be a uniformly closed algebra of compact operators on X, containing the finite rank operators. We set up a general framework to discuss the equivalence between Banach space approximation properties and the existence of right approximate identities in A. The appropriate properties require approximation in the dual X* by operators which are adjoints of operators on X. We show that the existence of a bounded right approximate identity implies that of a bounded left approximate identity. We give examples to show that these properties are not equivalent, however. Finally, we discuss the well known result that, if X* has a basis, then X has a shrinking basis. We make some attempts to generalize this to various bounded approximation properties.

Let E and F constitute a Banach pairing. We prove that the algebra of F-nuclear operators on E, Nf (E), is amenable if and only if E is finite dimensional and is weakly amenable if and only if dim KF ≦ 1, and the trace on E⊗F is injective on KF. Here KF is the kernel of the canonical map E⊗^F →NF(E). On the route we find the corresponding statements for the associated tensor algebra, E⊗^F.

In this paper we apply a theorem of Khelemskiĭ and Sheĭnberg, characterising amenability by means of bounded approximate identities, to weighted discrete convolution algebras. In doing this we give a condition for a weighted discrete convolution algebra to have a bounded approximate identity. Under the condition that the semigroup (S,.) is one-sided cancellative, we prove that, if some weighted discrete convolution algebra on S is amenable, then (S,.) is actually a group. We further characterise all amenable weighted discrete convolution algebras on groups, thus extending a well-known theorem of B. E. Johnson [9].