Cartan subalgebras in the group algebras of virtually abelian groups

Question

Let $G$ be a virtually abelian group. Are there any general results on the existence or non-existence of Cartan subalgebras in the generated group $C^*$-algebra or group von Neumann algebra?

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Answer 1

For von Neumann algebras in the discrete case Cartan subalgebras always exist. Indeed, if we only assume that the group is amenable then its von Neumann algebra is finite and injective, so it is of the form $\bigoplus_{n \in \mathbb{N}} A_n \otimes R_n$, where $R_1=\mathcal{R}$ is the unique hyperfinite $\mathrm{II}_1$ factor, $R_n=M_n$ for $n\geqslant 2$ and $A_n$'s are abelian. Each $R_n$ is known to admit a Cartan subalgebra; for matrix algebras you just use the diagonal matrices and $\mathcal{R}$ can be presented as a crossed product, e.g using an irrational rotation. It suffices now to tensor these Cartan subalgebras with $A_n$'s and the direct sum will give a Cartan subalgebra.

EDIT: In the locally compact case (for von Neumann akgebras) the answer is also affirmative. Indeed, if a locally compact group is amenable then its von Neumann algebra is injective (but the reverse implication does not hold in this generality). What is more, if a locally compact group admits a finite index unimodular subgroup, then it is unimodular, so our group is unimodular. It follows that the Plancherel weight is tracial, hence the algebra is semifinite. Therefore the only difference with the discrete case is that there might be a type $\mathrm{II}_{\infty}$ summand. But a type $\mathrm{II}_{\infty}$ injective von Neumann algebra is of the form $A\otimes \mathcal{R} \otimes B(\ell_2)$, so it also admits a Cartan subalgebra.