Non existence of group structure on some quantum spaces

TL;DR

This paper demonstrates that certain well-known compact quantum spaces, such as quantum tori and quantum two-spheres, cannot be endowed with a compact quantum group structure due to algebraic and analytical obstructions.

Contribution

It provides a proof that specific quantum spaces lack a quantum group structure by analyzing traces, characters, and nuclearity of their C*-algebras.

Findings

Quantum tori do not admit a compact quantum group structure.

Quantum two-spheres lack a quantum group structure.

Obstructions are identified via traces, characters, and nuclearity analysis.

Abstract

We prove that some well known compact quantum spaces like quantum tori and some quantum two-spheres do not admit a compact quantum group structure. This is achieved by considering existence of traces, characters and nuclearity of the corresponding $\mathrm{C}^*$-algebras.