= Continuous unitary character
{title2=$\chi:G\longrightarrow\mathbb T$}
= Group character
{synonym}
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> $\rho(z)=\operatorname{diag}(z,z^{-1})$ has <character of a representation> $\operatorname{tr}\rho(e^{i\theta})=2\cos\theta$. Its value at the <identity element> is $2$, so this <trace> is not a <continuous unitary character>.
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