Lemma 5

Let G be a topological group, and let $\varrho$ be a unitary representation of G in a complex Hilbert space E. The space Hom$_G(\varrho , \varrho )$ is a unital involutive subalgebra of $\mathscr{L}(E)$.

What precedes shows that Hom$_G(\varrho , \varrho )$ is a unital self-adjoint subalgebra of $\mathscr{L}(E)$. Since it is closed in $\mathscr{L}$ (E), it is an involutive subalgebra of $\mathscr{L}(E)$.