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

 

Maximal ideals in the group algebra of an extension, with applications


Author: Arnold J. Insel
Journal: Trans. Amer. Math. Soc. 172 (1972), 195-206
MSC: Primary 43A20; Secondary 43A35
MathSciNet review: 0312154
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Abstract: Let $ A$ be a closed central subgroup of $ E$ (all groups are locally compact and second countable). Let $ G = E/A$. For each $ a\in \hat A$, the dual of $ A$, a multiplication is introduced with respect to which the Banach space $ {L^1}(G)$ is a Banach algebra, denoted by $ {L^1}(G,a(\sigma ))$. A one-to-one correspondence is established between the maximal closed (right, left, $ 2$-sided) ideals of the group algebra $ {L^1}(E)$ and the totality of maximal closed (right, left, $ 2$-sided) ideals of $ {L^1}(G,a(\sigma ))$, where $ a$ varies over $ \hat A$. Applications include a bound for the spectral norm of an element of $ {L^1}(E)$ and the representation of a continuous positive definite function on $ E$ as an integral (a 'Bochner' theorem).


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0312154-3
PII: S 0002-9947(1972)0312154-3
Keywords: Topological group extension, group algebra, maximal closed ideal, spectral norm, positive definite function, Bochner theorem
Article copyright: © Copyright 1972 American Mathematical Society