Lemma 2

The graph of the equivalence relation “ $x\in G$ and $y\in G$ and $x$ is conjugate to $y$ ” in G is closed.

In fact, this graph is the image of the continuous mapping of the compact space $G\times G$ into itself defined by $(x, y)\mapsto (x, yxy^{-1})$.

We denote by $G^{\sharp}$ the space of conjugacy classes of G endowed with the quotient topology; it is a compact space by lemma 2 and TG, I, p. 78, prop. 8. Let $\varpi : G\rightarrow G^{\sharp}$ be the canonical projection. The mapping of $\mathscr{C}(G^{\sharp})$ into $\mathscr{C}(G)$ defined by $f\mapsto f\circ \varpi$ identifies the involutive algebra $\mathscr{C}(G^{\sharp})$ with the involutive subalgebra of $\mathscr{C}(G)$ formed by the continuous central functions.

The measures on $G^{\sharp}$ are identified with the central measures on G (V, p. 402, def. 1).

The linear form on $\mathscr{C}(G^{\sharp})$ defined by $f\mapsto \int_Gf$ is then a positive measure of mass 1 on $G^{\sharp}$, denoted by $\mu^{\sharp}$. For every $p\in [1,+\infty ]$, we denote by $\mathscr{L}^p(G^{\sharp})$ (resp. $L^p(G^{\sharp})$) the space $\mathscr{L}_{\mathbf{C}}^p(G^{\sharp}, \mu^{\sharp})$ (resp. $L^p_{\mathbf{C}}(G^{\sharp}, \mu^{\sharp})$). We identify $L^p(G^{\sharp})$ with the closure of $\mathscr{C}(G^{\sharp})$ in $L^p(G)$; in particular, it is a closed subspace of $L^p(G)$.

We also denote by $\Theta (G^{\sharp}) = \Theta (G)\cap \mathscr{C}(G^{\sharp})$ the space of central matrix coefficients of G. It is a unital involutive subalgebra of $\mathscr{C}(G^{\sharp})$.

When G is a Lie group, the space $\Theta (G^{\sharp})$ is also denoted by ZΘ(G) in LIE, IX, p. 71.