TITLE: Actions of locally compact groups on the hyperfinite $II_1$ factor
QUESTION [11 upvotes]: Let $R$ be the hyperfinite $II_1$ factor, and let $G$ be a locally compact group.

(1) Does there always exist a continuous, (faithful) outer action of $G$ on $R$?
  (2) If so, how does one construct one?
  (3) Is there any hope of classifying such actions?


(1) Here, an "action" is a continuous homomorphism $\rho:G \to Aut(R)$, where the latter is equipped with the $u$-topology, i.e., the topology of pointwise convergence on the predual $R_*$. Such an action is called "outer" if $\rho(G)\cap Inn(R)=\{\mathrm{id}_R\}$.
(2) Note that for $G$ discrete, one can do a "Bernoulli shift" action on $\bigotimes_G M_2(\mathbb C)$. Is there a continuous analog of the above construction?
(3) In his PhD thesis, Vaughan Jones provided a complete answer to that question when $G$ is finite.
Injective homomorphisms $\bar \rho:G\to Out(R)$ are classified, up to conjugation in $Out(R)$, by a cohomology class $\omega_{\bar \rho}\in H^3(G,U(1))$. And $\bar\rho$ comes from a homomorphism $\rho:G\to Aut(R)$ iff $\omega_{\bar \rho}=0$. It is perhaps reasonable to expect this result to extend to all ameanable groups. Has this question been investigated?

REPLY [5 votes]: A positive answer to my question (2) is provided in the last section of Popa Takesaki "Topological structure of unitary and automorphism groups".
They show that for any locally compact group $G$, the action of $G$ on the $II_1$ factor $\{\mathit{CAR}(L^2G)\}''$ is appropriately continuous. Here, the double commutant is taken on $L^2(\mathit{CAR}(L^2G),tr)$ with respect to the unique trace on $\mathit{CAR}(L^2G)$, and $\mathit{CAR}$ is the algebra of canonical anticommutation relations (a.k.a. Clifford algebra).
Answer courtesy of Daniel Bruegmann.