Harmonic analysis and centers of group algebras

Liukkonen, J.; Mosak, R.

doi:10.1090/S0002-9947-1974-0350322-7

Harmonic analysis and centers of group algebras
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Trans. Amer. Math. Soc. 195 (1974), 147-163 Request permission

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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