A convolution measure algebra
is a partially ordered Banach algebra in which the norm, order, and algebraic
operations are related in special ways. Examples include the group algebra
L1(G) and measure algebra M(G) on a locally compact group G and, more
generally, the measure algebra M(S) on any locally compact semigroup
S.
This paper demonstrates several ways in which a convolution measure algebra can
be realized as an algebra of measures on a compact semigroup. A relationship is
established between such realizations and certain classes of Banach space
representations of the algebra. These results give a partial extension to the
noncommutative case of the structure theory of commutative semi-simple convolution
measure algebras.
