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Trace functions on inverse semigroup algebras

  • D. Easdown (a1) and W.D. Munn (a2)

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.

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[1]Barnes, B.A., ‘Representations of the ℓ1-algebra of an inverse semigroup’, Trans. Amer. Math. Soc. 218 (1976), 361396.
[2]Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups, I and II (Amer. Math. Soc, Providence, 1961 and 1967).
[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).
[4]Easdown, D. and Munn, W.D., ‘On semigroups with involution’, Bull. Austral. Math. Soc. 48 (1993), 93100.
[5]Munn, W.D., ‘Free inverse semigroups’, Proc. London Math. Soc. (3) 29 (1974), 385404.
[6]Munn, W.D., ‘A class of contracted inverse semigroup rings’, Proc. Roy. Soc. Edinburgh Ser. A 107 (1987), 175196.
[7]Reilly, N.R., ‘Free generators in free inverse semigroups’, Bull. Austral. Math. Soc. 7 (1972), 407424.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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