Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 050, 43 pages      arXiv:2201.10931      https://doi.org/10.3842/SIGMA.2022.050

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

Yoshimichi Ueda
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan

Received February 07, 2022, in final form June 26, 2022; Published online July 05, 2022

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.

Key words: spherical representation; KMS state; ordered $*$-vector space; operator system; inductive limit; quantum group; $\sigma$-$C^*$-algebra.

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