Quantum actions on discrete quantum spaces and a generalization of Clifford’s theory of representations

Abstract

To any action of a compact quantum group on a von Neumann algebra which is a direct sum of factors we associate an equivalence relation corresponding to the partition of a space into orbits of the action. We show that in case all factors are finite-dimensional (i.e. when the action is on a discrete quantum space) the relation has finite orbits. We then apply this to generalize the classical theory of Clifford, concerning the restrictions of representations to normal subgroups, to the framework of quantum subgroups of discrete quantum groups, itself extending the context of closed normal quantum subgroups of compact quantum groups. Finally, a link is made between our equivalence relation in question and another equivalence relation defined by R. Vergnioux.

On C∗‐completions of discrete quantum group rings

“…Throughout the paper will denote a compact quantum group in the sense of Woronowicz, and will be the discrete quantum group dual to . For precise definitions and all the related terminology we refer (for example) to [, Section 2]; we will follow the conventions of that paper. We will be mainly interested in the quantum group ring (in other words the Hopf ‐algebra ), and its reduced and universal completions and (in other words and ), with the latter being the universal enveloping ‐algebra of .…”

Quantum correlations on quantum spaces

“…in the theory of quantum groups (cf. [2,6,7,29,34]) the correspondence between quantum spaces and C * -algebras is expressed by denoting C * -algebras by C(X) or C 0 (X) where X is the corresponding quantum space. The distinction between C(•) and C 0 (•) is based on whether the C * -algebra is unital (in the former case) or not (in the latter case).…”

Lattice of Idempotent States on a Locally Compact Quantum Group

“…It follows that N ω has a finite dimensional direct summand. Furthermore the natural action of G on N ω is clearly ergodic, so by[5, Theorem 3.4] we have dim N ω ă`8.Conversely, if dimpN ω q ă`8 then ηpN ω q Ă L 2 pGq is finite dimensional. Thus P ω " pid b ωqp Wq " pid b ωqpWq, which is the projection onto L 2 pN ω q, intersects only finitely many isotypical components of L 8 pGq (cf [26,.…”

“…Now, by Theorem 4.12 applied to p G, we find that ω P Idem nor pGq.Remark 4.17. Let us note that the first part of the proof of Theorem 4.16 can also be performed without using[5, Theorem 3.4]. Indeed, if G is a compact quantum group and ω P Idem nor pGq then we already know from Theorem 4.12 and its proof that…”