Definition 1

A continuous homomorphism of G into the multiplicative group $\mathbf{U}$ of complex numbers of modulus 1 is called a unitary character of G.

In other words, a unitary character is a continuous complex-valued function $\chi$ on G such that:

In this chapter, we shall often say simply "character" instead of "unitary character".

Let E be a Hilbert space of dimension 1, and let $\chi$ be a unitary character of G. The mapping which to $x\in G$ associates the homothety of ratio $\chi (x)$ in E is a continuous isometric linear representation of G in E. Conversely, every bounded continuous linear representation of G in E is obtained by this process, and in particular is unitary.

It is immediate that the product of two unitary characters, the inverse of a unitary character, and the constant function equal to 1 are unitary characters. Hence the set $\widehat{G}$ of unitary characters of G is a group for multiplication. This group is commutative. On the other hand, the mapping $(\chi_1, \chi_2)\mapsto \chi_1\chi^{-1}_2=\chi_1\overline{\chi}_2$ is continuous for the topology of compact convergence, and $\widehat{G}$ endowed with the topology of compact convergence is a topological group (TG, X, p. 6, Corollary 2 and Remark 1).