Koopman representation

For a measure-preserving left action $g\mapsto T_g$ , its Koopman representation is

\kappa:G\longrightarrow\mathcal U(L^2(X,\mu)),\qquad (\kappa(g)f)(x)=f(T_{g^{-1}}x).

It satisfies $\kappa(e)=I$ , $\kappa(gh)=\kappa(g)\kappa(h)$ , and each $\kappa(g)$ is unitary. For topological groups we explicitly assume strong continuity when invoking results that require it.

Why the inverse is present

Applying two pullbacks yields

\kappa(g)\kappa(h)f =f\circ T_{h^{-1}}\circ T_{g^{-1}} =f\circ T_{(gh)^{-1}}.

The norm identity is $\|f\circ T_{g^{-1}}\|_2=\|f\|_2$ , and $\kappa(g^{-1})$ is the inverse. This verifies all representation axioms.

What it records

States are points of $X$ ; vectors of the representation are observables. Nonconstant fixed vectors obstruct ergodicity, and matrix coefficients are correlations. The reduced representation removes the constants that are always fixed; a nonzero finite-dimensional invariant subspace in that reduced representation obstructs weak mixing.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §2.2.