Chapter V

Unitary representations

For every vector space E, we denote by $1_E$ the identity mapping of E. If E, F and G are vector spaces and if $u: F\rightarrow E$ and $v: G\rightarrow F$ are linear mappings, we sometimes denote by $uv$ the linear mapping $u\circ v$ of G into E.

For a Hilbert space E, we denote by $\langle x|y\rangle_E$, or simply $\langle x|y\rangle$, the scalar product on E.

If X is a locally compact topological space and $\mu$ a measure on X, we shall denote by $\mathscr{L}^p(X, \mu) =\mathscr{L}_{\mathbf{C}}^p(X, \mu)$ and $L^p(X, \mu) = L^p_{\mathbf{C}}(X, \mu)$ for $p\in [1,+\infty ]$.

We denote by $e$ the unit element of a group G.

Let A be an associative algebra, E a topological vector space and $\pi : A\rightarrow \mathscr{L}(E)$ a representation of A in E. We say that $\pi$ is non-degenerate if the set of vectors $\pi (a)x$ for $a\in A$ and $x\in E$ is total in E.