Quantum stochastic convolution cocycles II
Abstract
Schuermann's theory of quantum Levy processes, and more generally the theory of quantum stochastic convolution cocycles, is extended to the topological context of compact quantum groups and operator space coalgebras. Quantum stochastic convolution cocycles on a C*hyperbialgebra, which are Markovregular, completely positive and contractive, are shown to satisfy coalgebraic quantum stochastic differential equations with completely bounded coefficients, and the structure of their stochastic generators is obtained. Automatic complete boundedness of a class of derivations is established, leading to a characterisation of the stochastic generators of *homomorphic convolution cocycles on a C*bialgebra. Two tentative definitions of quantum Levy process on a compact quantum group are given and, with respect to both of these, it is shown that an equivalent process on Fock space may be reconstructed from the generator of the quantum Levy process. In the examples presented, connection to the algebraic theory is emphasised by a focus on full compact quantum groups.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2006
 arXiv:
 arXiv:math/0611497
 Bibcode:
 2006math.....11497L
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Probability;
 46L53;
 81S25 (Primary);
 22A30;
 47L25;
 16W30 (Secondary)
 EPrint:
 32 pages, expanded introduction and updated references. The revised version will appear in Communications in Mathematical Physics