M(R, X) with order convolution

Abstract

for allf in C(I). Let X be a commutative Banach algebra with identity e, Ilell = 1. Denote by M(I, X) the set of all countably additive, regular vector-valued measures defined on the a-algebra 'MJ (I) of Borel sets in I with values in X, which have finite total variation [3]. With the total variation as norm, Ilmll = Iml(I), m E M(I, X), M(I, X) is a Banach space ([6, p. 161] or [8, p. 103]). Following [3, p. 379] a linear operation U: C(I) -> X is said to be dominated if there exists a regular positive Borel measure v such that