Measures with Bounded Powers on Locally Compact Abelian Groups

Abstract

Abstract. If fi is a measure on a locally compact abelian group with its positive and negative convolution powers bounded in norm by K <j(4 cos(w/9) + 1) -1.58626, then p. has the form p. = \(cos 9SX + i sin 9SXU) where |\| = 1 and u2 = e.Applications to isomorphism theorems are given. In particular, if G, and G2 are l.c.a. groups and T is an isomorphism of L'(G,) onto Ll(G£ with ||r|| < 1(4 cos(w/9) + 1), then either G, and G2 are isomorphic, or they both have subgroups of order 2 with isomorphic quotients.

Inequalities in l1 and lp and applications to group algebras

“…In [4] measures with bounded powers on abelian groups were characterised by the size of the bound. In particular, it was shown that if G is a locally compact abelian group and /i is a measure on G with \\n"\\ <^(l+ > /3), for all n in Z, then fi has the form aS x for a group element x, and | a | = l. In fact, this result follows from [1, Theorem 2.6].…”