Some remarks on a paper by Liu and van Rooij
In the paper [\textit{T.-S. Liu} and \textit{A. C. M. van Rooij}, Nederl. Akad. Wet., Proc., Ser. A 77, 306--316 (1974; Zbl 0305.43005)], the authors proved that if a discrete group \(G\) is amenable, then there exist non-zero left invariant operators on \(L^\infty(G)\) with values either in \(L^p(G)\), \(2\leq p<\infty\), or in \(C^0(G)\). This result raises the following question: Does the existence of a non-zero left invariant operator from \(L^\infty(G)\) into \(L^p(G)\), \(2\leq p<\infty\), or in \(C^0(G)\) imply the amenability of \(G\)? In the paper under review, the author gives several affirmative answers for locally compact groups. For instance, the author obtains the following results: Let \(G\) be locally compact and \(2\leq p<\infty\). (a) If there is a non-zero left invariant operator from \(L^\infty(G)\) into \(L^p(G)\), then \(G\) is amenable. (b) \(G\) is amenable if and only if there is a non-zero weakly compact left invariant operator from \(L^\infty(G)\) into itself. (c) \(G\) is inner amenable if and only if there is a non-zero weakly compact conjugation invariant operator from \(L^\infty(G)\) into itself. The author also obtains several characterizations for compact groups.
scientific article; zbMATH DE number 5311053
scientific article