Central functions in group algebras

Abstract

In this paper we show that for a locally compact group G, the group algebra-Li(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 Z-i(G) has an approximate identity consisting of functions in the center of L\. This constitutes part of a program of finding conditions on the group algebra which characterize groups satisfying various compactness conditions (see e.g., [3]).

Product-Convolution Operators and Mixed-Norm Spaces

“…To prevent this, we assume the group is (IN), i.e., has a compact invariant neighborhood of the identity. This classical condition [6] is equivalent to existence of a nontrivial center in the group algebra [13]. (It seems unlikely (IN) is the weakest useful condition, but attempting to use unimodularity alone proved intractable.…”

Segal algebras on non-abelian groups

“…Analogously to the paper [6] of Mosak we characterize the class [SIN] by the existence of central approximate units in a Segal algebra. In contrast to the case L\G) it is not trivial that there must always exist central approximate units in any given Segal algebra.…”

“…Moreover, we may assume that both V and K are F-invariant [1, (1.1) Theorem]. For a finite subset F c SX(G) and e > 0, we can take u E ZX(G), supp u C H, u > 0, \\u\\x = 1, such that 0) \\u*f-f\\s<e/2, fEF, holds; this follows from (S3) and [6]. If we can find a v £ ZS(G) with (2) \\u-v\\x<e/2M, M = max{||./l|J/£F}, \\v\\x= 1.…”

“…Since ZS(G) = ZX(G) n SX(G) holds, we may assume that Z\G) ¥= {0}-otherwise there is nothing to prove. But then G belongs to [77V] [6].…”

Segal Algebras on Non-Abelian Groups

“…For x # e, the operator Lx cannot reduce to identity on (ua)aeA because SX(G) is dense in LX(G). We repeat the arguments of [6] to prove G £ [SIN].…”