Examples in the symbolic calculus for measures

Abstract

The work is a contribution to attempts to frame converses to the generalized Wiener-Levy theorem, that (essentially) only real-anafytic functions operate on the Gelfand transforms of measures. Methods have been developed, by William Moran to exploit analytic structure in the maximal ideal space of the measure algebra of a locally compact abelian group to establish results of the kind wanted. These methods are employed to find measures on any Iocally compact abelian group on which only analytic functions operate. These measures arise from Bernoullii convolutions. The extensive machinery necessary is first developed in part 2, after a detailed. description of the problem and its context in part 1; in part 3 the three special cases of the circle group, the groups of p-adic integers, and infinite products of finite abelian groups are treated in detail. In the third case it is necessary to distinguish the case where the orders of the groups in the product are bounded. Finally a general statement for all locally compact abelian groups is deduced.