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The structure of hypergroup measure algebras

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.