Central approximate units in a certain ideal of 𝐿¹(𝐺)

Kotzmann, Ernst; Rindler, Harald

doi:10.1090/S0002-9939-1976-0404988-0

Central approximate units in a certain ideal of $L^{1}(G)$
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by Ernst Kotzmann and Harald Rindler PDF

Proc. Amer. Math. Soc. 57 (1976), 155-158 Request permission

Abstract:

In this paper we show that for a locally compact group $G$ the ideal ${L^0}(G) = \{ f|f \in {L^1}(G),\smallint f = 0\}$ of ${L^1}(G)$ has multiple approximate units belonging to the center of ${L^0}(G)$ iff $G$ has a basis of invariant neighbourhoods of 1 and if all conjugacy classes of $G$ are precompact, or, equivalently, the group of inner automorphisms is precompact in the group of all topological automorphisms. In a sense this is part of the problem to characterize certain classes of groups by properties of the group algebra.

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