Exercise 4, § 2

Let $H$ be a Hilbert space. A continuous representation $U$ of $G$ in $H$ is said to be unitary if the endomorphisms $U(s)$ are unitary for all $s \in G$. For every $\mu \in \mathscr{M}(G)$, let $\mu^*$ be the conjugate measure of $\check{\mu}$. Show that if $\mu \in \mathscr{M}^1(G)$ then $U(\mu^*) = U(\mu)^*$.