Corollary 1

Let $\pi$ be an irreducible unitary representation of G in a Hilbert space E. The character of $\pi$ is a hermitian element of the involutive algebra $L^1(G)$.

Let $x\in G$. Since G is unimodular, one has $\chi^*_{\pi}(x) =$ Tr($\pi (x^{-1})$), whence $\chi^*_{\pi}(x) =$ Tr($\pi (x)$) since $\pi$ is a unitary representation.