Bulletin of the Australian Mathematical Society
- Bulletin of the Australian Mathematical Society / Volume 52 / Issue 03 / December 1995, pp 359-372
- Copyright © Australian Mathematical Society 1995
- DOI: http://dx.doi.org/10.1017/S0004972700014854 (About DOI), Published online: 17 April 2009
Research Article
Trace functions on inverse semigroup algebras
D. Easdowna1 and W.D. Munna2
a1 School of Mathematics and Statistics University of Sydney Sydney New South Wales 2006 Australia
a2 Department of Mathematics University of Glasgow Glasgow G12 8QW Scotland United Kingdom
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.
(Received January 17 1995)
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