Corollary

For every elements $g\not =h$ in G, there exists an irreducible representation $\pi \in \widehat{G}$ and a matrix coefficient $f$ of $\pi$ such that $f(g)\not =f(h)$. In particular, the subalgebra Υ(G) of $\mathscr{C}_b(G)$ separates the points of G.

Let $g\not =h$ in G. There exists an irreducible representation $\pi \in \widehat{G}$ such that $\pi (g)\not =\pi (h)$ (th. 4); hence there exist $x$ and $y$ in the space of $\pi$ such that $\langle x|\pi (g)y\rangle \not =\langle x|\pi (h)y\rangle$.