The minimal closed non-trivial ideals of Toeplitz algebras on discrete groups (Q1586105)

Let \(G\) be a discrete, not necessary abelian, group and \(G_+\) be a subset of \(G\), such that \((G,G_+)\) be an ordered group. Let \(G_g, g\in G_+\backslash\{g\}\) be a semigroup of \(G\) generated by \(G_+\) and \(g^{-1}\) and \(G_F=\bigcap_{g\in G_+\backslash\{e\}}G_g\), so that \((G,G_F)\) be the minimal quasi-group containing \((G,G_+)\). Let also \(\gamma^{G_F,G_+}\) be the natural \(C^\ast\)-algebra morphism from Toeplitz algebra \(T^{G_+} (G)\) to Toeplitz algebra \(T^{G_F} (G)\). It is supposed that the group \(G\) is amenable and \(G_F\neq G_+\). The author shows that \(ker\;\gamma^{G_F,G_+}\) is exactly the minimal closed non-trivial ideal of \(T^{G_+} (G)\). Some applications of this result are also given.