Spherical Representations of $C^*$-Flows II: Representation System and Quantum Group Setup

TL;DR

This paper extends the theory of spherical representations to quantum groups, introducing analogs of dimension groups and applying them to inductive limits of compact quantum groups, supporting asymptotic representation theory.

Contribution

It develops a framework for spherical representations in quantum groups, including analogs of dimension groups and applications to inductive limits, advancing quantum asymptotic representation theory.

Findings

Established analogs of dimension groups using operator systems.

Applied the theory to inductive limits of compact quantum groups.

Justified Sato's approach to asymptotic representation theory for quantum groups.

Abstract

This paper is a sequel to our previous study of spherical representations in the operator algebra setup. We first introduce possible analogs of dimension groups in the present context by utilizing the notion of operator systems and their relatives. We then apply our study to inductive limits of compact quantum groups, and establish an analogue of Olshanski's notion of spherical unitary representations of infinite-dimensional Gelfand pairs of the form $G < G\times G$ (via the diagonal embedding) in the quantum group setup. This, in particular, justifies Ryosuke Sato's approach to asymptotic representation theory for quantum groups.