Proposition 4

The space Θ(G) is dense in $\mathscr{C}(G)$ and in $L^2(G)$.

The unital subalgebra Θ(G) of $\mathscr{C}(G)$ is stable under conjugation. It coincides with the subalgebra Υ(G) by prop. 2 of V, p. 457, hence it separates the points of G (cor. of theorem 4 of V, p. 454). Consequently, it is dense in $\mathscr{C}(G)$ by TG, X, p. 40, cor. 2, and, fortiori, it is dense in the space $L^2(G) ($cf. INT, IV, p. 155, §4, n$^o7$, prop. 13).

Recall that the unitary biregular representation of G is the representation $\boldsymbol{\varrho }_G$ of $G\times G$ in $L^2(G)$ such that $(g_1, g_2)\mapsto \boldsymbol{\gamma }_G(g_1)\boldsymbol{\delta }_G(g_2)$.