Ergodic action on an operator algebra

A state-preserving action $\alpha:G\to\operatorname{Aut}(A)$ on a unital operator algebra is ergodic in the fixed-algebra sense if its fixed-point algebra is

A^G=\{a\in A:\alpha_g(a)=a\text{ for every }g\in G\}=\mathbb C1.

For one automorphism, take the action of $\mathbb Z$ that it generates.

Classical comparison

For $A=L^\infty(X,\mu)$ , this is precisely measurable ergodicity. For $A=C(X)$ , it concerns only continuous invariant functions; by itself it neither determines a unique invariant probability measure nor implies ergodicity for every invariant measure. The algebra and the notion of invariance must therefore be specified.

Noncommutative example

The full unitary group acts ergodically on matrices by unitary conjugation, preserving normalized trace.