I asked a question over on Math.stackexchange a few days ago, but it didn't get much activity. Hopefully this question isn't considered too elementary by the standards of Mathoverflow. Here is what I am wondering, but first some setup.

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 $\sigma : G \to \mathrm{Aut}(N)$ and $v : G \times G \to \mathcal{U}(M)$ satisfying the following three conditions for all $k,l,m \in G$:

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

(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 $\ell^2(G,\mathcal{H})$ 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?

  • $\begingroup$ Far from an answer, but you might start with looking at Kleppner's condition on the cocycle (at least when $M = \mathbb C$). See for example Proposition 1.3 in "Twisted group $C^*$-algebras corresponding to nilpotent discrete groups" by J. A. Packer. $\endgroup$
    – pitariver
    Jan 7 at 8:21


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