We continue the study of operators from an Archimedean vector lattice E into a cofinal sublattice H which have the property that there is λ > 0 such that if x ∈ E, h ∈ H and |x|≤|h|, then |Tx| ≤ λ|h|. The collection Z(E|H) of all of those operators forms an algebra under composition. We investigate the relationship between the properties of having an identity, being Abelin and being semi-simple for such-algebras, culminating in a proof that they are equivalent if H is Dedekind complete. We also study various for such an operator T, showing that, apart from 0, its spectrum relative to Z(E|H) is the same as that of T|H relative to Z(H) and that of T relative to ℒ(E) (Provided E is a Banach lattice and H is closed).

This paper is concerned with the problem of automatic continuity of derivations from group algebras L1(G) is a locally compact group, and convolution algebras L1(ω), where ω is a weight function. In the case of group algebras, it is shown that either the problem reduces to the case when G is the free group on a countably infinite number of generators or there is a non-discrete group G with a discontinuous l1(G)-bimodule homomorphism from L1(G). It is also shown that every derivation from L1(G) to a commutative L1(G)-bimodule is continuous. Similar results are obtained for weighted convolution algebras.

We give a survey of some of the realisations that have been given of monogenic inverse semigroups and discuss their relation to one another. We then analyse the representations by bijections, combined under composition, of monogenic inverse semigroups, and classify these into isomorphism types. This provides a particularly easy way of classifying monogenic inverse semigroups into isomorphism types. Of interest is that we find two quite distinct representations by bijections of free monogenic inverse semigroups and show that all such representations must contain one of these two representations. We call a bijection of the form ai ↦ ai+1, i = 1,2,…, r − 1, a finite link of length r, and one of the form ai ↦ ai+1, i = 1,2…, a forward link. The inverse of a forward link we call a backward link. Two bijections u: A → B and r: C → D are said to be strongly disjoint if A ∩ C, A ∩ D, B ∩ C and B ∩ D are each empty. The two distinct representations of a free monogenic inverse semigroup, that we have just referred to, are first, such that its generator is the union of a counbtable set os finite links that are pairwise storongly disjoint part of any representation of a free monogenic inverse semigroup, the remaining part not affecting the isomorphism type. Each representation of a monogenic inverse semigroup that is not free contains a strongly disjoint part, determining it to within isomorphism, that is generated by either the strongly disjoint union of a finite link and a permutation or the strongly disjoint union of a finite and a forward link.

The history of the proof of the integration by parts formula for the Perron integral, and for the SCP-integral of Burkill, is discussed.

Given a variety K of lattice-ordered algebras, A ∈ K is catalytic if for all B ∈ K, K(A, B) is a lattice for the pointwise order. The catalytic objects are determined for various varieties of distributive-lattice-ordered algebras. The characterisations obtained do not show an overall unity and exhibit diverse behaviour. Duality is employed extensively. Its usefulness in this context depends on the existence of an order-isomorphism between K(A, B) and the corresponding dual horn-set. Criteria for the existence of such an order-isomorphism are investigated for dualities of the Davey-Werner type. The relationship between catalytic objects and colattices is also discussed.

The distance between two operator algebras acting on a Hilbert space H is defined to be the Hausdorff distance between their unit balls. We investigate the structural similarities between two close AW*-algebras A and B acting on a Hilbert space H. In particular, we prove that if A is of type I and separable, then A and B are *-isomorphic.

It is shown that if {Gn: n = 1, 2,…} is a countable family of Hausdorff kω-topological groups with a common closed subgroup A, then the topological amalgamated free product *AGn exists and is a Hausdorff kω-topological group with each Gn as a closed subgroup. A consequence is the theorem of La Martin that epimorphisms in the category of kω-topological groups have dense image.