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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 $|\lambda | = 1$ 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 $\left \| T \right \| < \tfrac {1} {3}(4 \cos (\pi /9) + 1)$, then either ${G_1}$ and ${G_2}$ are isomorphic, or they both have subgroups of order $2$ with isomorphic quotients.

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Keywords: Measures, bounded powers, locally compact abelian groups, algebra, homomorphisms, algebra isomorphisms
Article copyright: © Copyright 1981 American Mathematical Society