Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory
Abstract
Strongly continuous semigroups of unital completely positive maps (i.e. quantum Markov semigroups or quantum dynamical semigroups) on compact quantum groups are studied. We show that quantum Markov semigroups on the universal or reduced C${}^*$algebra of a compact quantum group which are translation invariant (w.r.t. to the coproduct) are in onetoone correspondence with Lévy processes on its $*$Hopf algebra. We use the theory of Lévy processes on involutive bialgebras to characterize symmetry properties of the associated quantum Markov semigroup. It turns out that the quantum Markov semigroup is GNSsymmetric (resp. KMSsymmetric) if and only if the generating functional of the Lévy process is invariant under the antipode (resp. the unitary antipode). Furthermore, we study Lévy processes whose marginal states are invariant under the adjoint action. In particular, we give a complete description of generating functionals on the free orthogonal quantum group $O_n^+$ that are invariant under the adjoint action. Finally, some aspects of the potential theory are investigated. We describe how the Dirichlet form and a derivation can be recovered from a quantum Markov semigroup and its Lévy process and we show how, under the assumption of GNSsymmetry and using the associated Schürmann triple, this gives rise to spectral triples. We discuss in details how the above results apply to compact groups, group C$^*$algebras of countable discrete groups, free orthogonal quantum groups $O_n^+$ and the twisted $SU_q (2)$ quantum group.
 Publication:

arXiv eprints
 Pub Date:
 October 2012
 arXiv:
 arXiv:1210.6768
 Bibcode:
 2012arXiv1210.6768C
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Probability;
 20G42;
 43A05;
 46L57;
 60G51;
 81R50
 EPrint:
 54 pages, thoroughly revised, to appear in the Journal of Functional Analysis