Convolution powers of singular-symmetric measures

Abstract

Let G be an infinite compact abelian group such that its dual group contains an infinite independent subset. IS(G) denotes the sum of all radicals of group algebras contained in the measure algebra on G. Then, for a positive integer k, there is a measure ,u on G such that p I is singularsymmetric for 1 k. Let G be an infinite compact abelian group and G its dual group. A subset E of G is said to be independent if for every choice of distinct points xl, ... , x, of E and integers n1, . . . , ni, either nIxI = n2x2 = * = nixi = 0 or n1x1 + n2x2 + * * * + nixi # 0. In this paper, we assume that G contains an infinite independent set. Let M(G) be the measure algebra on G and L'(G) the group algebra on G. We denote by Rad L'(G) the radical of L'(G) in M(G). We put C(G) = ITRad L'(G7), where T runs through over L.C.A. group topologies on G which are stronger than the original one. We denote by A the maximal ideal space of M(G). For 1i E M(G), we put s*(E) = E for every Borel subset E of G. We denote by TU the set of all symmetric measures of M(G), that is, T = {,u E M(G); ii*(f) = Ai(f) for everyf E A}, where ii is the Gel'fand transform of yt. Then we have e (G) c W. A measure jI 9E is called singular-symmetric if It is singular with e,(G) (y IeC(G)). In [1] and_[2], the author showed that there exists a singular-symmetric measure on R such that yt * It E e(R), where R is the Bohr compactification of the real line R. In [5], Shimizu showed that there exists a singular-symmetric measure on H = IIH0, where H, is an infinite compact abelian group (n = 1, 2, . . . ), using essentially the same method as in [1]. In the previous paper [3], the author showed that there exists a singular-symmetric measure IL on R such that ,n is singular-symmetric for every positive integer n, where An = nPl * p and pI = Au(n > 2). In this paper, we show the following. The essential idea of the proof is found in [1]. THEOREM. Let G be a compact abelian group whose dual group G contains an infinite independent set. Then, for each positive integer k, there exists a Received by the editors February 2, 1976. AMS (MOS) subject classifications (1970). Primary 43A05, 43A10, 43A32.