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

 

Isometric isomorphisms between Banach algebras related to locally compact groups


Authors: F. Ghahramani, A. T. Lau and V. Losert
Journal: Trans. Amer. Math. Soc. 321 (1990), 273-283
MSC: Primary 43A20; Secondary 43A10, 46J99
MathSciNet review: 1005079
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Abstract: Let $ {G_1}$, $ {G_2}$ be locally compact groups. We prove in this paper that if $ T$ is an isometric isomorphism from the Banach algebra $ {\text{LUC}}{({G_1})^*}$ (the continuous dual of the Banach space of left uniformly continuous functions on $ {G_1}$, equipped with Arens multiplication) onto $ {\text{LUC}}{({G_2})^*}$, then $ T$ maps $ M({G_1})$ onto $ M({G_2})$ and $ {L^1}({G_1})$ onto $ {L^1}({G_2})$. We also prove that any isometric isomorphism from $ {L^1}{({G_1})^{**}}$ (second conjugate algebra of $ {L^1}({G_1})$) onto $ {L^1}{({G_2})^{**}}$ maps $ {L^1}({G_1})$ onto $ {L^1}({G_2})$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-1005079-2
PII: S 0002-9947(1990)1005079-2
Keywords: Locally compact groups, uniformly continuous functions, group algebra, Arens multiplication, measure algebra, isometric isomorphisms, right identities
Article copyright: © Copyright 1990 American Mathematical Society