  For each $\lambda\in\left]0,1\right]$ we exhibit an uncountable family of
compact quantum groups $\mathbb{G}$ such that the von Neumann algebra
$\mathsf{L}^{\!\infty}(\mathbb{G})$ is the injective factor of type
$\mathrm{III}_\lambda$ with separable predual. We also show that uncountably
many injective factors of type $\mathrm{III}_0$ arise as
$\mathsf{L}^{\!\infty}(\mathbb{G})$ for some compact quantum group
$\mathbb{G}$. To distinguish between our examples we introduce invariants
related to the scaling group modeled on the Connes invariant $T$ for von
Neumann algebras and investigate the connection between so obtained invariants
of $\mathbb{G}$ and the Connes invariants
$T(\mathsf{L}^{\!\infty}(\mathbb{G}))$, $S(\mathsf{L}^{\!\infty}(\mathbb{G}))$.
In the final section we show that factors of type $\mathrm{I}$ cannot be
obtained as $\mathsf{L}^{\!\infty}(\mathbb{G})$ for a non-trivial compact
quantum group $\mathbb{G}$.
