Let a locally compact group $G$ act on a probability space $(X,\mu)$. Define the centralizer by $C(G)=\{\Delta\in Aut(X,\mu)\mid \Delta(gx)=g\Delta(x)\text{ almost everywhere}\}$. $Aut(X,\mu)$ denotes the set of $\mu$-preserving transformations of $X$. I am searching for actions $G\curvearrowright X$ such that the second centralizer $C(C(G))=G$. In the following, i tacitly assume that all actions are faithful and i identify the acting group $G$ with the corresponding subgroup of $Aut(X,\mu)$.

Of course, for compact groups $G$ such actions exist: let $G$ act on itself (with normalized Haar measure) by left translation. The centralizer is the right translation action of $G$, and the second centralizer is again the left translation action of $G$.

I found a paper by Daniel Rudolph that asserts that the second centralizer of the Bernoulli action of $\mathbb{Z}$ is just $\mathbb{Z}$ itself.

Now I was wondering if the similar result holds for the Gaussian action associated to the left-regular representation of an arbitrary second countable locally compact group. Has this been studied?