A characterization of SIN-groups (Q2266070)

Let G be a locally compact group. The following theorem is proved: G has small invariant neighbourhoods (i.e. the left and right uniform structures coincide) if and only if each closed proper left-ideal is annihilated by a non-zero bounded uniformly continuous function. This generalizes results by \textit{R. W. Henrichs} and \textit{H. L. Skudlarek} [Arch. Math. 39, 246-249 (1982; Zbl 0497.43002)] and the author [J. Reine Angew. Math. 99, 387-397 (1982; Zbl 0472.43004)]. Furthermore, a characterization of groups with compact invariant neighbourhoods is given, using the behaviour of the \(L^{\infty}\)-norm of convolution products.