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Proceedings of the American Mathematical Society

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An analytic characterization of groups with no finite conjugacy classes

Author: E. R. Cowie
Journal: Proc. Amer. Math. Soc. 87 (1983), 7-10
MSC: Primary 46H99; Secondary 20F38, 43A20
MathSciNet review: 677219
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Abstract: Let $ A$ be a unital Banach algebra and $ \mathcal{G}$ the group of isometries in $ A$. The norm in $ A$ is uniquely maximal if $ \mathcal{G}$ is not contained in any larger bounded group in $ A$ and there is no equivalent norm on $ A$ with the same group of isometries. We use a group theory result of B. H. Neumann to prove that the discrete measure algebra $ {l^1}(G)$ is uniquely maximal if and only if $ G$ has no finite conjugacy classes.

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

  • [1 E] R. Cowie, Isometries in Banach algebras, Ph.D. thesis, Swansea, Wales, U.K., 1981.
  • [2] B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc. 29 (1954), 236-248. MR 0062122 (15:931b)
  • [3] W. Rudin, Functional analysis, McGraw-Hill, New York, 1973. MR 0365062 (51:1315)

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Keywords: Unital Banach algebras, maximal group, equivalent norms, uniquely maximal norm, convex transitive norm, finite conjugacy classes, centralizers, finite index, discrete measure algebras
Article copyright: © Copyright 1983 American Mathematical Society

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