Lemma 1

Let $\varrho$ be a unitary representation of G in a Hilbert space E. The mapping $\nu \mapsto \varrho (\nu )$ from $\mathscr{M}^1(G)$ into $\mathscr{L}(E)$ is a unital morphism of involutive Banach algebras.

From what precedes, it suffices to prove that the morphism $\nu \mapsto \varrho (\nu )$ is involutive. Let $\nu \in \mathscr{M}^1(G)$. The measure $\nu^*$ is the conjugate measure of the measure $\check{\nu}($I, p. 99, example 4). According to the definition of the conjugate measure (INT, III, p. 52, § 1, n$^o5$), one calculates

for all $x$ and $y$ in E, whence $\varrho (\nu )^*=\varrho (\nu^*)$.

Let $\varrho$ be a continuous linear representation (resp. continuous and bounded) of G in a quasi-complete locally convex space E (resp. a Banach space E). If F is a closed subspace of E defining a subrepresentation $\varrho_F$ of $\varrho$, then for every measure $\nu \in \mathscr{M}_c(G)$ (resp. $\nu \in \mathscr{M}^1(G)$) the linear mapping $\varrho (\nu )$ leaves the subspace F stable and coincides with $\varrho_F(\nu )$ on F.

Conversely, let $F\subset E$ be a closed subspace, stable under the linear mappings $\varrho (f)$ for every function $f\in \mathscr{K}(G)$. Then F defines a subrepresentation of $\varrho$ (INT, VIII, p. 139, § 2, n$^o7$, prop. 10).

Recall (A, VIII, p. 388) that a function $f: G\rightarrow \mathbf{C}$ is said to be central if, for all $g$ and $h$ in G, one has $f(gh) =f(hg)$. This amounts to saying that $f$ is invariant under conjugation.