The existence of inner invariant means on \(L^{\infty}(G)\)
Let G be a locally compact group admitting identity e, and X a subspace of \(L^{\infty}(G)\) with respect to left Haar measure, containing the constant function 1 and closed under complex conjugation. The mean m on X is called inner invariant if \(m(_ xf_ x)=m(f)\) for all \(f\in X\), \(x\in G\), with \({}_ xf_ x(y)=f(x^{-1}yx)\), \(y\in G\). The group is termed [IA] group if it admits an inner invariant mean on \(L^{\infty}(G)\). This definition extends the concept of amenable group; all amenable groups and all discrete groups belong to the class [IA]. The author succeeds in obtaining for [IA] groups generalizations of characterizations of amenable groups due to Godement, Reiter, Day. Some equivalences have already been observed independently by \textit{V. Losert} and \textit{H. Rindler} [Colloq. Math. 51, 221-225 (1987; Zbl 0629.43005)]. The Dirac measure \(\delta_ e\) constitutes a trivial inner invariant mean on the space CB(G) of continuous bounded functions defined on G. The author discusses the possibility of extending \(\delta_ e\) to an inner invariant mean on \(L^{\infty}(G)\); this problem has also been studied by \textit{V. Losert} and \textit{H. Rindler} [Lect. Notes Math. 1064, 368-378 (1984; Zbl 0573.43003)].
scientific article; zbMATH DE number 4042527
scientific article