|
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.
|
