Corollary 7

Let $\pi$ be an irreducible unitary representation of a topological group G in a Hilbert space E. There exists a continuous homomorphism $\chi$ of the center C of G into $\mathbf{U}$ such that $\pi (z) =\chi (z)\cdot 1_E$ for all $z\in C$. In particular, if G is commutative, one has dim(E) = 1.

For $z\in C$, the mapping $\pi (z)$ belong to Hom$_G(\pi , \pi )$; it is therefore of the form $\chi (z)\cdot 1_E$ for a certain complex number $\chi (z)$ (prop. 6). Since $\pi (z)$ is a unitary mapping, one has $|\chi (z)|= 1$. Moreover, since $\pi$ is a homomorphism, the mapping $z\mapsto \chi (z)$ is a homomorphism. Let us fix $v\not = 0$ in E; the mapping $z\mapsto \chi (z)v=\pi (z)v$ of C into $\mathbf{U}$ is continuous, and therefore the homomorphism $\chi$ is continuous.

Finally, if G is commutative, one has C = G, hence $\pi (g) =\chi (g)\cdot 1_E$ for all $g$ in G; every subspace of dimension 1 of E is then a subrepresentation of $\pi$, and since $\pi$ is irreducible, the space E must be of dimension 1.