Trace functions on inverse semigroup algebras

Abstract

Let S be an inverse semigroup and let F be a subring of the complex field containing 1 and closed under complex conjugation. This paper concerns the existence of trace functions on F [ S ], the semigroup algebra of S over F . Necessary and sufficient conditions on S are found for the existence of a trace function on F [ S ] that takes positive integral values on the idempotents of S . Although F [ S ] does not always admit a trace function, a weaker form of linear functional is shown to exist for all choices of S . This is used to show that the natural involution on F [ S ] is special. It also leads to the construction of a trace function on F [ S ] for the case in which F is the real or complex field and S is completely semisimple of a type that includes countable free inverse semigroups.