Lemma: Two ordinary ergodic limits

Under (eq:scalarcentralizer), for every $a\in M$ and $\psi \in M_{\ast }$,

Proof. The unitary mean ergodic theorem gives a Hilbert-norm limit in (eq:Hilbertmean). The operator averages are bounded by $\Vert a\Vert$. Translation by fixed $u$ changes them in norm by at most $|u|\Vert a\Vert /k$ for large $k$. Every weak-star cluster point is therefore fixed by $\sigma$ and hence scalar. Evaluation by $\phi$ identifies it as $\phi (a)1$, which also identifies the vector limit.

The modular action on $M_{\ast }$ is norm continuous on each functional: positive functionals are standard-form vector functionals, and general functionals are linear combinations of positive ones. Thus the averages are Bochner integrals and contractions.

Let $N$ be the norm-closed span of $\chi \circ {\sigma }_u-\chi$, $\chi \in M_{\ast },u\in \mathbb{R}$. Its annihilator in $M$ is the scalar centralizer. Hahn–Banach consequently identifies $N=\{\psi :\psi (1)=0\}$. Interval boundary estimates make the averages tend to zero on the generators of $N$, hence on $N$ by contractivity. They fix $\phi$, proving (eq:predualmean).