Compact quantum group structures on type-I $\mathrm{C}^*$-algebras

TL;DR

This paper investigates the conditions under which type-I C*-algebras can have compact quantum group structures, showing co-amenability and finite-dimensionality under certain extensions.

Contribution

It establishes that such quantum groups are necessarily co-amenable and characterizes finite-dimensionality for specific algebra extensions.

Findings

Compact quantum groups on type-I C*-algebras are co-amenable.

Certain extensions of C*-algebras imply finite-dimensionality.

Results restrict possible structures of quantum groups on these algebras.

Abstract

We prove a number of results having to do with equipping type-I $\mathrm{C}^*$-algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the $\mathrm{C}^*$-algebra in question is an extension of a non-zero finite direct sum of elementary $\mathrm{C}^*$-algebras by a commutative unital $\mathrm{C}^*$-algebra then it must be finite-dimensional.