A commutative hypergroup
K is, roughly speaking, a space in which the product of two elements is a
probability measure. Such spaces have been studied by Dunkl, Jewett, and
Spector. Examples include locally compact abelian groups and double-coset
spaces. K has a Haar measure m (Spector). It is shown that for several
important classes of hypergroups lhe structure space of L1(m) is a hypergroup K.
For such spaces, L1(m) is shown to be regular, in fact, super-regular, and
to have good approximate units. A Wiener-Tauberian theorem is given.
Points in the center of K are shown to be strong Ditkin sets. Examples (due
essentially to Reiter and Naimark) show that not all points in K need be spectral
sets.
