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

TL;DR

This paper constructs uncountably many examples of compact quantum groups whose associated von Neumann algebras are injective factors of various type III, and explores invariants distinguishing these examples, showing type I factors cannot arise from non-trivial compact quantum groups.

Contribution

It provides explicit constructions of compact quantum groups with von Neumann algebras of type III and introduces invariants to distinguish them, advancing understanding of quantum group von Neumann algebra types.

Findings

Uncountably many quantum groups with type III factors

Existence of quantum groups with type III_0 factors

Type I factors cannot be obtained from non-trivial quantum groups

Abstract

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 $\mathbb{G}$.