Examples of compact quantum groups with a factor

For each $\lambda \in ]0,1]$ we exhibit an uncountable family of compact quantum groups $\mathbb{G}$ such that the von Neumann algebra ${\mathsf{L}}^{\text{\hspace{0.17em}⁣}\mathrm{\infty }}(\mathbb{G})$ is the injective factor of type ${\mathrm{I}\mathrm{I}\mathrm{I}}_{\lambda }$ with separable predual. We also show that uncountably many injective factors of type ${\mathrm{I}\mathrm{I}\mathrm{I}}_0$ arise as ${\mathsf{L}}^{\text{\hspace{0.17em}⁣}\mathrm{\infty }}(\mathbb{G})$ 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 $T({\mathsf{L}}^{\text{\hspace{0.17em}⁣}\mathrm{\infty }}(\mathbb{G}))$, $S({\mathsf{L}}^{\text{\hspace{0.17em}⁣}\mathrm{\infty }}(\mathbb{G}))$. In the final section we show that factors of type $\mathrm{I}$ cannot be obtained as ${\mathsf{L}}^{\text{\hspace{0.17em}⁣}\mathrm{\infty }}(\mathbb{G})$ for a non-trivial compact quantum group $\mathbb{G}$.