# Twisted Crossed Product von Neumann Algebra

Let $$M$$ be a (tracial) von Neumann algebra acting on a Hilbert space $$\mathcal{H}$$. A cocycle action of a discrete group $$G$$ on $$M$$ is a pair $$(\sigma, v)$$ with $$\sigma : G \to Aut(N)$$ and $$v : G \times G \to \mathcal{U}(M)$$ satisfying the following three conditions for all $$k,l,m \in G$$:

(1) $$\sigma_{k} \sigma_{l} = ad(v(k,l))\sigma_{kl}$$

(2) $$v(k,l)v(kl,m) = \sigma_{k}(v(l,m)) v(k,lm)$$

(3) $$v(1,l) = v(l,1) = 1$$

The twisted crossed product of $$M$$ by $$G$$, denoted by $$M \rtimes_{(\sigma,v)} G$$, is defined as the von Neumann algebra acting on $$\ell^2(G,\mathcal{G})$$ generated by $$\pi_{\sigma}(M)$$ and $$\lambda_{v}(G)$$, where $$\pi_{\sigma}$$ is the faithful normal representation of $$M$$ on $$\ell^2(G,\mathcal{H})$$ defined via

$$(\pi_{\sigma}(x) \xi)(l) = \sigma_{l^{-1}}(x) \xi (l)$$

and, for each $$g \in G$$, $$\lambda_{v}(g)$$ is the unitary operator defined by

$$(\lambda_{v}(k)(\xi))(l) = v(l^{-1},k) \xi (k^{-1}l)$$

for all $$x \in M$$, $$\xi \in \ell^2(G,\mathcal{H})$$, and $$l \in G$$.

My question is,

are there necessary and sufficient conditions for when the twisted crossed product $$M \rtimes_{(\sigma,v)} G$$ is a $$II_1$$ factor?