Halmos–von Neumann theorem

Remarks

Here isomorphic means conjugate by a measure-preserving bijection defined up to null sets. For part 1, ergodicity makes every eigenvalue simple and every eigenfunction of constant modulus, so a product of eigenfunctions is again an eigenfunction and $\Lambda(T)$ is a group. Choose for each $\lambda \in \Lambda(T)$ an eigenfunction $f_\lambda$ of modulus $1$; the products $f_\lambda f_{\lambda'}$ and $f_{\lambda \lambda'}$ differ by constants. The proof adjusts these constants, which is possible because the circle group is divisible. It then matches the eigenfunctions of $T$ and $S$ to build a unitary map that preserves products, and such a map comes from an isomorphism of the spaces. For part 2, $K$ is the Pontryagin dual of $\Lambda(T)$ with the discrete topology, and $a$ is the inclusion character. Example: for an irrational $\alpha$, the rotation $x \mapsto x + \alpha \bmod 1$ has $\Lambda = \set{e^{2\pi i k \alpha} : k \in \Z}$. So two irrational rotations by $\alpha$ and $\beta$ are isomorphic exactly when $\beta = \pm \alpha \bmod 1$. Without the discrete spectrum hypothesis the eigenvalues do not determine the system: all weakly mixing systems have $\Lambda = \set{1}$.

Going deeper

For a student

This is the first complete classification theorem in ergodic theory, obtained by Halmos and von Neumann in 1942. It shows the power of spectral methods in the class where the spectrum is discrete, and it is the model case for the spectral theory of dynamical systems. For systems with continuous spectrum spectral invariants are far from complete, which motivated the search for other invariants such as the Kolmogorov–Sinai entropy.