When is a quantum space not a group?

Abstract

We give a survey of techniques from quantum group theory which can be used to show that some quantum spaces (objects of the category dual to the category of $\mathrm{C}^*$-algebras) do not admit any quantum group structure. We also provide a number of examples which include some very well known quantum spaces. Our tools include several purely quantum group theoretical results as well as study of existence of characters and traces on $\mathrm{C}^*$-algebras describing the considered quantum spaces as well as properties such as nuclearity.