J Austral Math Soc Ser A 46 pp319--342, 1989.

The Structure of Hypergroup Measure Algebras

W. Christopher Lang

Abstract

A close analogue for some hypergroup measure algebras of the structure semigroup theorem of J. L. Taylor for convolution measure algebras is constructed: a structure semihypergroup representation is made for the hypergroup measure and its spectrum. This is done for those hypergroup measure algebras that satisfy a condition known as the structure-strong condition. This condition is that the norm-closure of the linear span of the spectrum of the hypergroup measure algebra is a commutative B*-algebra. Then examples of hypergroups whose measure algebras satisfy this condition are given. They include the space of B-orbits of G, where B is a finite solvable group of automorphisms on a locally compact abelian group G. (the hypergroup measure algebra may be identified with the algebra of B-invariant measures on G.) Other examples are the algebra of central measures on a compact, connected, semisimple Lie group, and the algebra of rotation invariant measures on the plane.

1980 AMS Subject Classification (1985 Revision): 43A10

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Authors

W. Christopher Lang

Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Mississippi 39762, U.S.A.