Group von Neumann algebras

Question

I have a question about group von Neumann algebras structure. If $L(G)$ is a subset of $L(H)$, can we find a subgroup $G_1$ of $H$ such that $L(G_1)$ is isomorphic to $L(G)$?

I appreciate any help.

Answer 1

If $G$ is not a subset of $H$, it is not clear how $L(G)$ could be a subset of $L(H)$. I will assume "isomorphic to". In that case, the answer is "no".

Let $G=\mathbb Z_2$, $H=\mathbb F_2$. Then $L(G)=\mathbb C^2$. AS $L(H)$ is a II$_1$ factor, it has a nontrivial projection and so $L(G)\simeq\text{span}\,\{p,1-p\}$. As $H$ has no finite-subgroups, $G_1$ cannot exist.