Growth rates of dimensional invariants of compact quantum groups and a theorem of Hoegh-Krohn, Landstad and Stormer

TL;DR

This paper establishes bounds on eigenvalues related to quantum group actions on C*-algebras, linking growth rates of quantum dimensions to the type of associated von Neumann algebras, extending classical finiteness results.

Contribution

It introduces new bounds for eigenvalues of the modular operator in quantum group actions, generalizing the Hoegh-Krohn, Landstad, and Stormer theorem to quantum settings.

Findings

Eigenvalue bounds depend on quantum dimension growth rates.

Quantum groups of Kac type with non-tracial states have exponentially growing representations.

Constraints on type III factors arising from quantum group actions.

Abstract

We give local upper and lower bounds for the eigenvalues of the modular operator associated to an ergodic action of a compact quantum group on a unital C*-algebra. They involve the modular theory of the quantum group and the growth rate of quantum dimensions of its representations and they become sharp if other integral invariants grow subexponentially. For compact groups, this reduces to the finiteness theorem of Hoegh-Krohn, Landstad and Stormer. Consequently, compact quantum groups of Kac type admitting an ergodic action with a non-tracial invariant state must have representations whose dimensions grow exponentially. In particular, S_{-1}U(d) acts ergodically only on tracial C*-algebras. For quantum groups with non-involutive coinverse, we derive a lower bound for the parameters 0<\lambda<1 of factors of type III_\lambda that can possibly arise from the GNS representation of the invariant state of an ergodic action with a factorial centralizer.