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

Abstract

We prove a number of results having to do with equipping type-I ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-algebra in question is an extension of a non-zero finite direct sum of elementary ${\mathrm{C}}^{\ast }$-algebras by a commutative unital ${\mathrm{C}}^{\ast }$-algebra then it must be finite-dimensional.