Much work has been done on the ℓ¹-algebras of groups, but much less on ℓ¹-algebras of semigroups. This thesis studies those of inverse semigroups, also known as generalised groups, with emphasis on the involutive structure. Where results extend to the semigroup ring, I extend them.
I determine the characters of a semilattice in terms of its order structure. The simplest suffice to separate its ℓ¹-algebra. I also determine the algebra’s minimal idempotents.
I introduce a generalisation of Banach *-algebras which has good hereditary properties and includes the inverse semi groups rings. These latter have an ultimate identity which can be used to test for representability. Involutive semigroups with s*s an idempotent yield inverse semi groups when quotiented by the congruence
induced by their algebras’ *-radical.
The left regular *-representation of inverse seroigroups is faithful and acts like that of groups. The corresponding idea of amenability coincides with the traditional one. Brandt semi groups have the weak containment property iff the associated group does. The relationship of ideals to weak containment is studied, and inverse semigroups with well ordered semilattices are shown to have the property if all their subgroups do. The converse is extended for Clifford semigroups.
Symmetry and related ideas are considered, and basic results proved for the above mentioned generalisation, and a better version for a possibly more restricted generalisation. The symmetry of an ℓ¹-algebra of an E-unitary inverse semi group is shown to depend on the symmetry of the ℓ¹-algebra of its maximal group homomorphic image if the semilattice has a certain structure or the semigroup is a Clifford semigroup. Inverse semi groups with well ordered semilattices are shown to have symmetric ℓ¹-algebra if all the subgroups do.
Finally, some topologically simple ℓ¹-algebras and simple semigroup rings are constructed, extending results on simple inverse semigroup rings.