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For each $\lambda\in\left]0,1\right]$ we exhibit an uncountable family ofcompact 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 uncountablymany 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 invariantsrelated to the scaling group modeled on the Connes invariant $T$ for vonNeumann algebras and investigate the connection between so obtained invariantsof $\mathbb{G}$ 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 beobtained as $\mathsf{L}^{\!\infty}(\mathbb{G})$ for a non-trivial compactquantum group $\mathbb{G}$.

Examples of compact quantum groups with $\operatorname{\mathsf{L}^{\!\infty}}(\mathbb{G})$ a factor