Vol. 47, No. 2, 1973

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ISSN: 0030-8730
Structure hypergroups for measure algebras

Charles F. Dunkl

Vol. 47 (1973), No. 2, 413–425
Abstract

An abstract measure algebra A is a Banach algebra of measures on a locally compact Hausdorff space X such that the set of probability measures in A is mapped into itself under multiplication, and if μ is a finite regular Borel measure on X and μ << ν A then μ A. If A is commutative then the spectrum of A,ΔA, is a subset of the dual of A,A, which is a commutative W-algebra. In this paper conditions are given which insure that the weak-* closed convex hull of ΔA, or of some subset of ΔA, is a subsemigroup of the unit ball of A. This statement implies the existence of certain bypergroup structures. An example is given for which the conditions fail.

The theory is then applied to the measure algebra of a compact p-hypergroup, for example, the algebra of central measures on a compact group, or the algebra of measures on certain homogeneous spaces. A further hypothesis, which is satisfied by the algebra of measures given by ultraspherical series, is given and it is used to give a complete description of the spectrum and the idempotents in this case.

Mathematical Subject Classification 2000
Primary: 43A10
Milestones
Received: 10 May 1972
Published: 1 August 1973
Authors
Charles F. Dunkl