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We describe all Gaussian generating functionals on several easy quantum groups given by non-crossing partitions. This includes in particular the free unitary, orthogonal and symplectic quantum groups. We further characterize central Gaussian generating functionals and describe a centralization procedure yielding interesting (non-Gaussian) generating functionals., Comment: 24 pages

Consider a locally compact quantum group $\mathbb{G}$ with a closed classical abelian subgroup $\Gamma$ equipped with a $2$-cocycle $\Psi:\hat{\Gamma}\times\hat{\Gamma}\to\mathbb{C}$. We study in detail the associated Rieffel deformation $\mathbb{G}^{\Psi}$ and establish a canonical correspondence between $\Gamma$-invariant convolution semigroups of states on $\mathbb{G}$ and on $\mathbb{G}^{\Psi}$., Comment: 37 pages. v3: minor changes. To appear in Letters in Mathematical Physics

We introduce the Gaussian part of a compact quantum group $\mathbb{G}$, namely the largest quantum subgroup of $\mathbb{G}$ supporting all the Gaussian functionals of $\mathbb{G}$. We prove that the Gaussian part is always contained in the Kac part, and characterise Gaussian parts of classical compact groups, duals of classical discrete groups and $q$-deformations of compact Lie groups. The notion turns out to be related to a new concept of "strong connectedness" and we exhibit several examples of both strongly connected and totally strongly disconnected compact quantum groups., Comment: 23 pages; v3 changes the title

Every symmetric generating functional of a convolution semigroup of states on a locally compact quantum group is shown to admit a dense unital $*$-subalgebra with core-like properties in its domain. On the other hand we prove that every normalised, symmetric, hermitian conditionally positive functional on a dense $*$-subalgebra of the unitisation of the universal C$^*$-algebra of a locally compact quantum group, satisfying certain technical conditions, extends in a canonical way to a generating functional. Some consequences of these results are outlined, notably those related to constructing cocycles out of convolution semigroups., Comment: 25 pages. v2: added an example and made several minor changes. To appear in International Mathematics Research Notices

The subject of this paper is the study of convolution semigroups of states on a locally compact quantum group, generalising classical families of distributions of a L\'{e}vy process on a locally compact group. In particular a definitive one-to-one correspondence between symmetric convolution semigroups of states and noncommutative Dirichlet forms satisfying the natural translation invariance property is established, extending earlier partial results and providing a powerful tool to analyse such semigroups. This is then applied to provide new characterisations of the Haagerup Property and Property (T) for locally compact quantum groups, and some examples are presented. The proofs of the main theorems require developing certain general results concerning Haagerup's $L^{p}$-spaces., Comment: 52 pages. v2: minor changes. To appear in Journal de Math\'ematiques Pures et Appliqu\'ees

We describe basic motivations behind quantum or noncommutative probability, introduce quantum L\'evy processes on compact quantum groups, and discuss several aspects of the study of the latter in the example of quantum permutation groups. The first half of this paper is a survey on quantum probability, compact quantum groups, and L\'evy processes on compact quantum groups. In the second half the theory is applied to quantum permutations groups. Explicit examples are constructed and certain classes of such L\'evy processes are classified., Comment: 60 pages

We develop fully noncommutative Feynman-Kac formulae by employing quantum stochastic processes. To this end we establish some theory for perturbing quantum stochastic flows on von Neumann algebras by multiplier cocycles. Multiplier cocycles are constructed via quantum stochastic differential equations whose coefficients are driven by the flow. The resulting class of cocycles is characterised under alternative assumptions of separability or Markov regularity. Our results generalise those obtained using classical Brownian motion on the one hand, and results for unitarily implemented flows on the other., Comment: 27 pages. Minor corrections to version 2. To appear in the Journal of the London Mathematical Society

Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given for the case of finite-dimensional initial space or, more generally, for coefficients satisfying a finite localisability condition. Necessary and sufficient conditions are obtained for a conjugate pair of quantum stochastic cocycles on a finite-dimensional operator space to strongly satisfy such a quantum stochastic differential equation. This gives an alternative approach to quantum stochastic convolution cocycles on a coalgebra., Comment: 20 pages

Every Markov-regular quantum Levy process on a multiplier C*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C*-bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Levy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C*-bialgebra, to locally compact quantum groups and multiplier C*-bialgebras. Strict extension results obtained by Kustermans, together with automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic convolution cocycles., Comment: 20 pages; v2 corrects some typos and no longer contains a section on quantum random walk approximations, which will now appear as a separate submission. The article will appear in the Mathematische Annalen

Idempotent states on a compact quantum group are shown to yield group-like projections in the multiplier algebra of the dual discrete quantum group. This allows to deduce that every idempotent state on a finite quantum group arises in a canonical way as the Haar state on a finite quantum hypergroup. A natural order structure on the set of idempotent states is also studied and some examples discussed., Comment: 28 pages; v3 omits the former lemma 2.1 due to a gap in the proof. This does not affect any other results. The paper will appear in the Journal of Algebra

Every quantum Levy process with a bounded stochastic generator is shown to arise as a strong limit of a family of suitably scaled quantum random walks., Comment: 7 pages

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 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 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., Comment: 32 pages, expanded introduction and updated references. The revised version will appear in Communications in Mathematical Physics

Stochastic generators of completely positive and contractive quantum stochastic convolution cocycles on a C*-hyperbialgebra are characterised. The characterisation is used to obtain dilations and stochastic forms of Stinespring decomposition for completely positive convolution cocycles on a C*-bialgebra., Comment: 20 pages; to appear in Mathematical Proceedings of the Cambridge Philosophical Society (2007)

A concept of quantum stochastic convolution cocycle is introduced and studied in two different contexts -- purely algebraic and operator space theoretic. A quantum stochastic convolution cocycle is a quantum stochastic process on a coalgebra satisfying the convolution cocycle relation and the initial condition given by the counit. The notion generalises that of quantum Levy process, which in turn is a noncommutative probability counterpart of classical Levy process on a group. Convolution cocycles arise as solutions of quantum stochastic differential equations. In turn every sufficiently regular cocycle satisfies an equation of that type. This is proved along with the corresponding existence and uniqueness of solutions for coalgebraic quantum stochastic differential equations. The stochastic generators of unital *-homomorphic cocycles are characterised in terms of structure maps on a *-bialgebra. This yields a simple proof of the Schurmann Reconstruction Theorem for a quantum Levy process; it also yields a topological version for a quantum Levy process on a C*-bialgebra. Precise characterisation of the stochastic generators of completely positive and contractive quantum stochastic convolution cocycles in the C*-algebraic context is given, leading to some dilation results. A few examples are presented and some interpretations offered for quantum stochastic convolution cocycles and their stochastic generators on different types of *-bialgebra., Comment: The article is a revised version of the PhD thesis of the author, submitted to the University of Nottingham in December 2005 and defended on the 28 July 2006