The context is unitary representations of locally compact topological groups.
Theorem (Gelfand–Raikov). Let $G$ be a locally compact topological group. Then $G$ is separated by its irreducible representations, i.e., for any $x,y\in G$, $x\neq y$, there is an irreducible representation $\rho:G\to U(\mathcal{H})$ such that $\rho(x)\neq\rho(y)$.
I learned this theorem and its proof from A Course in Abstract Harmonic Analysis by G. B. Folland. Then I read the Wikipedia page and saw the following remark concerning this theorem:
It states that a locally compact group is completely determined by its (possibly infinite dimensional) unitary representations.
Questions
- How do I understand this remark? What is meant by "completely determined"? In what sense?
- More generally, how do I understand the role of this theorem in the theory? Why do we care about whether the irreducible representations separate points?
Thanks in advance!
As explained in the article, for an irreducible representation $\pi: G \to U(H)$ (necessarily finite dimensional), we can define matrix elements with respect to some orthonormal basis of $H$ as $$\pi_{ij}(g) = \langle e_i, \pi(g) e_j \rangle.$$ Let $\mathcal{E}_\pi$ be the linear span of the matrix elements and let $\mathcal{E}$ be the linear span of $\cup_{[\pi]} \mathcal{E}_\pi$ where in the union we choose a representative from each unitary equivalence class of irreducible representations of $G$. It is a big part of the Peter-Weyl theorem that $\mathcal{E}$ is uniformly dense in $\mathcal{C}(G)$, the algebra of continuous functions on $G$. This follows from the Gelfand-Raikov theorem and the Stone-Weierstrass theorem. In that sense, Gelfand-Raikov determines $\mathcal{C}(G)$ and hence $G$.
In terms of the theory, Peter-Weyl then gives you a decomposition of $L^2(G)$ as $$L^2(G) = \bigoplus_{[\pi]} \mathcal{E}_\pi$$ which is the start of much of Fourier analysis on groups.