Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-12T19:56:57.468Z Has data issue: false hasContentIssue false
  • English
  • Français

Trace functions on inverse semigroup algebras

Published online by Cambridge University Press:  17 April 2009

D. Easdown  and
W.D. Munn
D. Easdown
Affiliation:
School of Mathematics and StatisticsUniversity of SydneySydneyNew South Wales 2006Australia
W.D. Munn
Affiliation:
Department of MathematicsUniversity of GlasgowGlasgow G12 8QWScotlandUnited Kingdom
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 52 , Issue 3 , December 1995 , pp. 359 - 372
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Barnes, B.A., ‘Representations of the ℓ1-algebra of an inverse semigroup’, Trans. Amer. Math. Soc. 218 (1976), 361–396.Google Scholar
[2]Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups, I and II (Amer. Math. Soc, Providence, 1961 and 1967).Google Scholar
[3]Crabb, M.J. and Munn, W.D., ‘Trace functions on the algebras of certain E-unitary inverse semigroups’, Proc. Roy. Soc. Edinburgh Ser. A (to appear).Google Scholar
[4]Easdown, D. and Munn, W.D., ‘On semigroups with involution’, Bull. Austral. Math. Soc. 48 (1993), 93–100.CrossRefGoogle Scholar
[5]Munn, W.D., ‘Free inverse semigroups’, Proc. London Math. Soc. (3) 29 (1974), 385–404.CrossRefGoogle Scholar
[6]Munn, W.D., ‘A class of contracted inverse semigroup rings’, Proc. Roy. Soc. Edinburgh Ser. A 107 (1987), 175–196.Google Scholar
[7]Reilly, N.R., ‘Free generators in free inverse semigroups’, Bull. Austral. Math. Soc. 7 (1972), 407–424.CrossRefGoogle Scholar