The unitary dual of a compact group

Assume the Axiom of Choice (The Axiom of Choice). Let $K$ be a compact Hausdorff topological group. Two strongly continuous unitary representations $\pi$ on $H$ and $\sigma$ on $H^{\mathrm{\prime }}$ of $K$ are unitarily equivalent when there is a unitary intertwiner between them, that is, a bijective linear isometry $U:H\to H^{\mathrm{\prime }}$ with $U\pi (k)=\sigma (k)U$ for every $k\in K$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners). The unitary dual (dual object) $\widehat{K}$ is the set of unitary equivalence classes of irreducible strongly continuous unitary representations of $K$.

By Irreducible unitary representations of compact groups are finite dimensional, under the Axiom of Choice (The Axiom of Choice) every irreducible strongly continuous unitary representation of $K$ has finite-dimensional carrier: the carrier admits an ordered basis of finite length $d$, its dimension $d_{\pi }:={\dim }_{\mathbb{C}}{H}_{\pi }\ge 1$ (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis), and every class contains a representative whose carrier is ${\mathbb{C}}^d$ (Every finite-dimensional real or complex inner product space has an orthonormal basis realizes the carrier as ${\mathbb{C}}^d$ through an orthonormal basis). Consequently $\widehat{K}$ is a set: it is the union over $d\ge 1$ of the set of unitary equivalence classes of irreducible representations on the fixed carrier ${\mathbb{C}}^d$, and equivalence on a fixed carrier is a relation on the set of group homomorphisms $K\to U(d)$.

Standing conventions. When a statement uses a representative $\pi$ of a class in $\widehat{K}$, or an orthonormal basis of its carrier, such choices are licensed by the Axiom of Choice (The Axiom of Choice) and every assertion made this way must be invariant under unitary equivalence; the normalized coefficient family of The normalized irreducible matrix coefficient family is the first instance. Irreducibility is understood in the sense of Strongly continuous unitary representations, invariant linear subspaces and intertwiners, so every class in $\widehat{K}$ has nonzero carrier and $d_{\pi }\ge 1$. No topology is placed on $\widehat{K}$ and no countability of $\widehat{K}$ is asserted.