Classical dynamics as a state-preserving function-algebra system

Core idea

An invertible probability-preserving system $(X,\Sigma,\mu,T)$ determines

A=L^\infty(X,\mu),\qquad\varphi(f)=\int_X f\,d\mu, \qquad\alpha(f)=f\circ T^{-1}.

Here $A$ is a commutative von Neumann algebra acting by multiplication on $L^2(X,\mu)$ , $\varphi$ is a faithful normal state, and $\alpha$ is a normal state-preserving $*$ -automorphism.

Events and algebra

Pointwise multiplication and complex conjugation give $f^*f=|f|^2$ . An event $E$ , modulo null sets, corresponds to a projection $1_E=1_E^*=1_E^2$ , and $\varphi(1_E)=\mu(E)$ . Every projection of $A$ arises this way. Ergodicity becomes $A^\alpha=\mathbb C1$ .

Recovering a measurable model

On standard probability spaces, normal state-preserving isomorphisms of these function algebras correspond to measure-space isomorphisms modulo null sets. This recovers the measured system, not a preferred topology or individual representatives on null sets.

Koopman covariance

If $Kf=f\circ T^{-1}$ and $M_f$ is multiplication by $f$ , then

K M_f K^*=M_{\alpha(f)}.

The crossed product uses a regular representation on a larger Hilbert space; the covariance identity alone does not identify the algebra generated by $M_f,K$ with that crossed product.

References

Sorin Popa, Ergodic theory of group actions. §§1.1–1.2 and 4.1.