Example 7

Let G be a group (with unit element 1) and $\pi$ a homomorphism from G into the group of automorphisms of a complex hilbertian space E ; we say that $\pi$ is a unitary representation of G in E. Let $a \in E$ ; we put

for all $x \in G$. Then $\phi : G \to \mathbf{C}$ is positive definite, in other words satisfies the relation :

(PD) For every $\lambda_1, ..., \lambda_n$ in $\mathbf{C}$ and $x_1, ..., x_n$ in $G$, we have

In fact, the first member of (11) is precisely $\| \sum_{i=1}^n \lambda_i \pi(x_i) . a \| ^2$.

Conversely, let $\phi$ be a positive definite function on $G$. Let $C^{(G)}$ be the vector space of all functions with finite support on $G$. We define a hermitian form $\Phi$ on $G$ by

and the relation (PD) expresses the fact that $\Phi$ is positive. By the corollary of prop. 4, there exists a hilbertian space $E$ and a linear mapping $\rho : C^{(G)} \to E$, with a dense image, such that

For every $x \in G$, let $\gamma_x$ be the left translation by $x$ in $C^{(G)}$ defined by $\gamma_x u(y) = u(x^{-1} y)$ for $u \in C^{(G)}$ and $y \in G$. We have $\Phi(\gamma_x u, \gamma_x v) = \Phi(u, v)$. Now apply assertion b) of the corollary of prop. 4 to $\rho$ and $\rho \circ \gamma_x$: there exists a unique automorphism $\pi(x)$ of the hilbertian space $E$ such that $\rho \circ \gamma_x = \pi(x) \circ \rho$. We see immediately that $\pi$ is a homomorphism from $G$ into the group of automorphisms of $E$.

Let $\delta$ be the element of $C^{(G)}$ defined by $\delta(1) = 1, \delta(x) = 0$ for $x \neq 1$ in $G$. We have $u = \sum_{x \in G} u(x) . \gamma_x \delta$ for all $u \in C^{(G)}$, and so $\rho(u) = \sum_{x \in G} u(x) \ \pi(x) . a$ by putting $a = \rho(\delta)$.

Formulas (12) and (13) imply that $\phi(x) = \langle a|\pi(x).a \rangle$ for all $x \in G$. We remark that the set of vectors $\pi(x).a$ for all $x \in G$, is total in $E$.