Quantum Stochastic Convolution Cocycles II

Abstract

Schürmann's theory of quantum Lévy 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 Markov-regular, 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 Lévy 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 Lévy process. In the examples presented, connection to the algebraic theory is emphasised by a focus on full compact quantum groups.

The Haagerup property for locally compact quantum groups

“….a/ 0 j 0 /. As argued in [27,Lemma 4.2] (compare [58,Lemma 8.7]) for the algebra Pol.G/, the map always extends to a -homomorphism Pol.G/ ! B.H/, where H is the completion of H 0 .…”

“…The last lemma yields the following result, which can be interpreted as providing a method of constructing quantum Lévy processes [58] on dual free products of compact quantum groups.…”

“…In this precise formulation it can be deduced for example from [58,Section 8]. Indeed, the correspondence between semigroups of hermitian functions, and self-adjoint L with L.1/ D 0 follows easily from one-parameter semigroup theory, and bialgebra theory, see for example the sketch in [59,Section 2].…”

Quantum stochastic convolution cocycles III

“…It was first introduced by Accardi, Schürmann and von Waldenfels, in the purely algebraic framework of * -bialgebras [1], and was further developed by Schürmann and others [7,22] who, in particular, extended it to other noncommutative forms of independence (free, boolean and monotone), still in the algebraic context. Inspired by Schürmann's reconstruction theorem, which states that every quantum Lévy process on a * -bialgebra can be equivalently realised on a symmetric Fock space, we first showed how the algebraic theory of quantum Lévy processes can be extended to the natural setting of quantum stochastic convolution cocycles [14]. These are families of linear maps (l t ) t≥0 from a * -bialgebra B to operators on the symmetric Fock space F, over a Hilbert space of the form L 2 (R + ; k), satisfying the following cocycle identity with respect to the ampliated CCR flow (σ t ) t≥0 : l s+t = l s (σ s • l t ), s, t ≥ 0, together with regularity and adaptedness conditions.…”

Quantum Random Walk Approximation on Locally Compact Quantum Groups

“…In [19] we developed a theory of quantum stochastic convolution cocycles on counital multiplier C * -bialgebras, extending the algebraic theory of quantum Lévy processes created by Schürmann and co-workers (see [25] and references therein, and, for a simplified treatment [17]), and the topological theory of quantum stochastic convolution cocycles on compact quantum groups and operator space coalgebras developed by the authors [18]. Here we apply the results of [19] to introduce and analyse a straightforward scheme for the approximation of such cocycles by quantum random walks.…”