Proposition 6 (Schur’s Lemma)

Let $\varrho$ be a unitary representation of a topological group G in a nonzero Hilbert space E. Then $\varrho$ is irreducible if and only if Hom$_G(\varrho , \varrho )$ is equal to $\mathbf{C}\cdot 1_E$.

The space Hom$_G(\varrho , \varrho )$ is a unital stellar subalgebra of $\mathscr{L}(E)$ (lemma 5 of V, p. 380). For every subrepresentation F of E, the orthogonal projector $p$ with image F is an idempotent element of the stellar algebra Hom$_G(\varrho , \varrho )$ (prop. 4 of V, p. 383). If the latter is equal to $\mathbf{C}\cdot 1_E$, this means that $p= 0$ or $p= 1_E$, which means that F = 0 or F = E. Hence $\varrho$ is irreducible.

Conversely, suppose $\varrho$ irreducible. Let $u$ and $v$ be permutable elements of the stellar algebra Hom$_G(\varrho , \varrho )$ such that $uv= 0$. Suppose $u$ nonzero. Then the kernel F of $u$ defines a subrepresentation of $\varrho$ different from E, hence F is reduced to 0; since F contains the image of $v$, it follows that $v= 0$. By proposition 10 of I, p. 113, we therefore have Hom$_G(\varrho , \varrho ) =\mathbf{C}\cdot 1_E$.