Ergodic actions of compact quantum groups from solutions of the conjugate equations

Abstract

We use a tensor $C^{\ast }$-category with conjugates and two quasitensor functors into the category of Hilbert spaces to define a ${}^{\ast }$-algebra depending functorially on this data. If one of them is tensorial, we can complete in the maximal $C^{\ast }$-norm. A particular case of this construction allows us to begin with solutions of the conjugate equations and associate ergodic actions of quantum groups on the $C^{\ast }$-algebra in question. The quantum groups involved are $A_u\left(Q\right)$ and $B_u\left(Q\right)$.