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Transactions of the American Mathematical Society

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



Harmonic analysis and centers of group algebras

Authors: J. Liukkonen and R. Mosak
Journal: Trans. Amer. Math. Soc. 195 (1974), 147-163
MSC: Primary 43A20; Secondary 43A45
MathSciNet review: 0350322
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Abstract: The purpose of this paper is to present some results of harmonic analysis on the center of the group algebra $ Z({L^1}(G))$ where G is a locally compact group. We prove that $ Z({L^1}(G))$ is a regular, Tauberian, symmetric Banach $ ^\ast$-algebra and contains a bounded approximate identity. Wiener's generalized Tauberian theorem is therefore applicable to $ Z({L^1}(G))$. These results complement those of I. E. Segal relating to the group algebra of locally compact abelian and compact groups. We also prove that if G contains a compact normal subgroup K such that G/K is abelian, then $ Z({L^1}(G))$ satisfies the condition of Wiener-Ditkin, so that any closed set in its maximal ideal space whose boundary contains no perfect subset is a set of spectral synthesis. We give an example of a general locally compact group for which $ Z({L^1}(G))$ does not satisfy the condition of Wiener-Ditkin.

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Keywords: Centers of group algebras, regular algebras, Tauberian algebras, symmetric $ ^\ast$-algebras, approximate identities, condition of Wiener-Ditkin, [IN] groups, $ {[FC]^ - }$ groups, $ {[FIA]^ - }$ groups, spherical functions
Article copyright: © Copyright 1974 American Mathematical Society