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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Measures with bounded powers on locally compact abelian groups

Author: G. V. Wood
Journal: Trans. Amer. Math. Soc. 268 (1981), 187-210
MSC: Primary 43A10
MathSciNet review: 628454
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Abstract: If $ \mu $ is a measure on a locally compact abelian group with its positive and negative convolution powers bounded in norm by $ K < \tfrac{1} {3}(4\cos (\pi /9) + 1) \sim 1.58626$ , then $ \mu $ has the form $ \mu = \lambda (\cos \theta {\delta _x} + i\,\sin \theta {\delta _{xu}})$ where $ \vert\lambda \vert = 1$ and $ {u^2} = e$. Applications to isomorphism theorems are given. In particular, if $ {G_1}$ and $ {G_2}$ are l.c.a. groups and $ T$ is an isomorphism of $ {L^1}({G_1})$ onto $ {L^1}({G_2})$ with $ \left\Vert T \right\Vert < \tfrac{1} {3}(4\,\cos (\pi /9) + 1)$, then either $ {G_1}$ and $ {G_2}$ are isomorphic, or they both have subgroups of order $ 2$ with isomorphic quotients.

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PII: S 0002-9947(1981)0628454-9
Keywords: Measures, bounded powers, locally compact abelian groups, algebra, homomorphisms, algebra isomorphisms
Article copyright: © Copyright 1981 American Mathematical Society

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