It it well-known in the von Neumann algebra theory that for $\Gamma$ a non-trivial countable group, the von Neumann algebra $L(\Gamma)$ generating by $\Gamma$ acting by left multiplication on $l^2(\Gamma)$, is a ${\rm II}_1$ factor iff $\Gamma$ is an ICC group.

Now let $(G \subset \Gamma)$ be an inclusion of a *finite* group $G$ in a countable group $\Gamma$, Let $\mathbb{C}(G \backslash \Gamma / G) $ be the (Hecke) double coset algebra: the subalgebra of $\mathbb{C}G $, generated by the elements $a_{\gamma} = \sum_{\alpha \in G \gamma G} \alpha$ (well-defined because $G$ finite) with $\gamma \in \Gamma$. Let $L(\Gamma,G)$ be the von Neumann algebra generated by $\mathbb{C}(G \backslash \Gamma / G) $ acting by left multiplication on $l^2(G \backslash \Gamma)$.

*Question*: What's the necessary and sufficient condition on $(G \subset \Gamma)$ for $L(\Gamma,G)$ to be a ${\rm II}_1$ factor?

In the case that there are inclusions $(G \subset \Gamma)$ with $\Gamma$ ICC, $G \neq \{ e \}$ and $L(\Gamma,G)$ ${\rm II}_1$ factor:

*Optional question 1*: Is it true that $L(\Gamma,G) \simeq L(\Gamma)$?

*Optional question 2*: How to generalize the construction above for $G$ infinite?