On a spectral property of one-dimensional representations of compact quantum groups

TL;DR

This paper proves that the spectrum of group-like elements in compact quantum groups forms a closed subgroup of the circle, revealing structural properties and connections to the Kadison-Kaplansky conjecture.

Contribution

It establishes a spectral property of group-like elements in compact quantum groups within the $C^*$-algebra framework, linking spectral theory to quantum group structure.

Findings

Spectrum of group-like elements is a closed subgroup of the circle

Implications for the structure of compact quantum groups

Connections to the Kadison-Kaplansky conjecture

Abstract

In the $C^*$-algebraic setting the spectrum of any group-like element of a compact quantum group is shown to be a closed subgroup of the one-dimensional torus. A number of consequences of this fact are then illustrated, along with a loose connection with the so-called Kadison-Kaplansky conjecture.