Abstract
In this paper we show that for a locally compact group G, the group algebra ${L_1}(G)$ has nontrivial center if and only if G possesses a compact neighborhood of 1, invariant under inner automorphisms. Moreover, G has a basis of such neighborhoods at 1 if and only if ${L_1}(G)$ has an approximate identity consisting of functions in the center of ${L_1}$. This constitutes part of a program of finding conditions on the group algebra which characterize groups satisfying various compactness conditions (see e.g., [3]).
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