Simplicial cohomology of totally ordered semigroup algebras

Abstract

It is known that some discrete semigroup algebras have trivial continuous simplicial cohomology, at least in high dimensions. The aim of this work is to investigate the situation for the locally compact case, which even for the important example of the positive real numbers is not clear. The initial focus of this thesis is on the continuous simplicial cohomology groups for the algebra L1(R+; _). We then adapt our methods and progress to investigating the more general case of L1(X; ; c) where (X; ) is a totally ordered semigroup with the binary operation max and which is locally compact in its order topology and c is a continuous, - nite, positive, regular Borel measure on X. The rst continuous simplicial cohomology of L1(R+; _) was already known, but we o er a new method of deriving this result and then use this method to prove the triviality of higher dimensional continuous simplicial cohomology groups for this algebra. We then modify our method to derive analogous results for the algebra L1(X; ; c) by analysing the algebra L1(R+; c).