Ergodicity and factoriality of a free crossed product

Statement

Let a countable discrete group $G$ act essentially freely by probability-preserving transformations on a standard probability space. For $M=L^\infty(X,\mu)\rtimes G$ ,

Z(M)=L^\infty(X,\mu)^G.

Consequently, $M$ is a factor if and only if the action is ergodic. If also $X$ is atomless, the factor is of type $\mathrm{II}_1$ .

Reason

Essential freeness makes the included $A=L^\infty(X)$ maximal abelian. A central element must therefore lie in $A$ . It commutes with the implementing unitaries exactly when it is fixed by every $\alpha_g$ . This gives the center formula; the fixed-function criterion gives the ergodicity equivalence.

Freeness is a hypothesis

For an $n$ -cycle viewed as a $\mathbb Z$ -action, $u^n$ is central in the regular crossed product and is not scalar. Thus the action is ergodic but the crossed product is not a factor. Omitting essential freeness would make the stated equivalence false.

References

Sorin Popa, Ergodic theory of group actions. Theorem 4.1.1.