Analysis of the Plancherel weight and factoriality of the group von Neumann algebras of non-unimodular almost unimodular groups

Let $G$ be a locally compact group, $L(G)$ be its group von Neumann algebra equipped with the Plancherel weight ${\phi }_G$. In this paper, we consider the following two questions. (1) When is the restriction of ${\phi }_G$ to the subalgebra generated by a closed subgroup $H$ semifinite? If so, is it equal (up to a constant) to ${\phi }_H$? (2) When is $L(G)$ a factor? We give a complete answer to (1), and when $G$ is second countable, $G_1:=\text{ker}{\mathrm{\Delta }}_G$ is open in $G$ (called almost unimodular) and admits a sufficiently large non-unimodular part, we provide an answer to (2). When $L(G)$ is a factor, we also provide the formula of the S-invariant of $L(G)$.