Fixed-function criterion for ergodicity

Statement

For a probability-preserving action $G\curvearrowright(X,\mu)$ , the following are equivalent:

The action is ergodic.
$L^2(X,\mu)^G=\mathbb C1$ , where the superscript means vectors fixed by every Koopman operator.
$L^\infty(X,\mu)^G=\mathbb C1$ .

For a single probability-preserving transformation, including a noninvertible one, the corresponding criterion is $\ker(U_T-I)=\mathbb C1$ .

Proof

An invariant event $E$ gives the fixed indicator $1_E$ ; it is constant almost everywhere exactly when $\mu(E)\in\{0,1\}$ . Conversely, a nonconstant invariant complex measurable function has a nonconstant real or imaginary part. Some level set of that part has intermediate probability and is invariant modulo null sets. This contradicts ergodicity.

Algebraic form

Under inverse pullback on the function algebra, this reads $A^\alpha=\mathbb C1$ . It is a fixed-point statement, not irreducibility of a Hilbert-space representation.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. Lemma 2.2.5.