Tracial central states on compact quantum groups

TL;DR

The work classifies extremal central tracial states on the Hopf *-algebras of several families of compact quantum groups. By leveraging the maximal Kac quantum subgroup and a mix of root-system combinatorics, Weingarten calculus, and convolution-semigroup methods, the authors reduce the problem to tractable classical data (torus centers or finite centers) and obtain explicit finite sets of extremals. Specifically, extremals are indexed by central data for $\mathbb{G}_q$ (via $Z(G)$), and for the free quantum groups $O_N^+$, $S_N^+$, and $H_N^+$ the extremals are $h$, $\varepsilon$, and in some cases $\varepsilon_{\text{alt}}$, with all central tracial states extending to $C^u(\mathbb{G})$. These results connect central trace properties to intrinsic quantum-group structure and have implications for quantum Lévy processes and noncommutative geometry.

Abstract

Motivated by classical investigation of conjugation invariant positive-definite functions on discrete groups, we study tracial central states on universal C*-algebras associated with compact quantum groups, where centrality is understood in the sense of invariance under the adjoint action. We fully classify such states on q-deformations of compact Lie groups, on free orthogonal quantum groups, quantum permutation groups and on quantum hyperoctahedral groups.

Theorems & Definitions (61)

• Remark 2.1
• Lemma 2.2
• proof
• Proposition 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• proof