Mean ergodic theorem for a finite tracial system

Statement

Let $M$ be a von Neumann algebra with faithful normal tracial state $\tau$ , and let $\alpha$ be a normal $\tau$ -preserving automorphism. Write $E:M\to M^\alpha$ for the $\tau$ -preserving normal conditional expectation. Then

\left\|\frac1N\sum_{n=0}^{N-1}\alpha^n(a)-E(a)\right\|_{2,\tau}\longrightarrow0 \qquad(a\in M), \quad\|a\|_{2,\tau}=\tau(a^*a)^{1/2}.

If $M^\alpha=\mathbb C1$ , the limit is $\tau(a)1$ .

Hilbert-space mechanism

The GNS implementation is unitary on $L^2(M,\tau)$ . Its fixed subspace is $L^2(M^\alpha,\tau)$ , and the extension of $E$ is the orthogonal projection onto that subspace. The Hilbert-space mean ergodic theorem therefore gives the conclusion.

Mode of convergence

This statement is convergence in the trace $L^2$ norm. There is no underlying point $x$ at which to evaluate a general noncommutative observable. Noncommutative pointwise theorems require separately defined replacements for almost-everywhere convergence.