Continuous unitary character - Imported Knowlpedia Content

Definition

Let $G$ be a topological group and let $\mathbb T=\{z\in\mathbb C:|z|=1\}$ be the circle group. A continuous unitary character of $G$ is a continuous group homomorphism

\chi:G\longrightarrow\mathbb T.

Equivalently, it is a one-dimensional strongly continuous unitary representation of $G$ on $\mathbb C$ , with $\pi(g)z=\chi(g)z$ . No commutativity hypothesis on $G$ is needed, although every such character is trivial on commutators and therefore factors through the abelianization of $G$ .

Relation to Pontryagin duality

When $G$ is locally compact and abelian, all continuous unitary characters form the Pontryagin dual $\widehat G$ , equipped with pointwise multiplication and the compact-open topology. For a nonabelian group, unitary characters still form an abelian group, but they see only the abelianized quotient and do not describe the full unitary dual.

Examples

Every continuous unitary character of $(\mathbb R,+)$ has the form

\chi_t(x)=e^{2\pi i tx}

for a unique $t\in\mathbb R$ . The characters of $\mathbb Z$ are $n\mapsto z^n$ , parametrized by $z\in\mathbb T$ . The determinant $\det:U(n)\to\mathbb T$ is a unitary character of the finite-dimensional unitary group.

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References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapter 4, §4.1 on dual groups and continuous characters.
Walter Rudin, Fourier Analysis on Groups, Wiley-Interscience, 1962. Wiley DOI record. Relevant: Chapter 1, §1.2 on character groups.