Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1981-0628454-9
MathSciNet review: 628454
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] F. P. Greenleaf, Norm-decreasing homomorphisms of group algebras, Pacific J. Math. 15 (1965), 1187-1219. MR 0194911 (33:3117)
  • [2] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, Springer-Verlag, Berlin and New York, 1963. MR 551496 (81k:43001)
  • [3] N. J. Kalton and G. V. Wood, Homomorphisms of group algebras with norm less than $ \surd 2$, Pacific J. Math. 62 (1976), 439-460. MR 0415205 (54:3296)
  • [4] W. Rudin, Fourier analysis on groups, Interscience, New York, 1960. MR 0152834 (27:2808)
  • [5] B. M. Schreiber, Measures with bounded convolution powers, Trans. Amer. Math. Soc. 151 (1970), 405-431. MR 0264335 (41:8931)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 43A10

Retrieve articles in all journals with MSC: 43A10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0628454-9
Keywords: Measures, bounded powers, locally compact abelian groups, algebra, homomorphisms, algebra isomorphisms
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society