 Continuous unitary character

A continuous unitary character is a continuous group homomorphism from a topological group to the unit circle with multiplication. For abelian groups, multiplication of continuous unitary characters is pointwise. Such continuous unitary characters must be distinguished from the traces of higher-dimensional group representations, called characters of a representation, and from characters of an algebra.

For every integer $n$, the map ${\chi }_n(z)=z^n$ is a continuous unitary character of the circle group: ${\chi }_n(zw)={\chi }_n(z){\chi }_n(w)$ and $|{\chi }_n(z)|=1$. By contrast, the two-dimensional group representation $\rho (z)=\mathrm{diag}(z,z^{-1})$ has character of a representation $\mathrm{tr}\rho (e^{i\theta })=2\cos \theta$. Its value at the identity element is $2$, so this trace is not a continuous unitary character.