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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Measures with bounded powers on locally compact abelian groups
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by G. V. Wood PDF
Trans. Amer. Math. Soc. 268 (1981), 187-210 Request permission

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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Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 268 (1981), 187-210
  • MSC: Primary 43A10
  • DOI: https://doi.org/10.1090/S0002-9947-1981-0628454-9
  • MathSciNet review: 628454