We consider a unitary cocycle or Schürmann triple on the non-commutative unitary group fixed by a complex matrix which induces an additive free white noise or an additive free Lévy process on the tensor algebra over the full Fock space. A Lévy process on a Voiculescu dual semi-group is given by a generator or Schürmann triple. We will show how a free Lévy process on the non-commutative unitary group fixed by a complex matrix can be obtained by infinitesimally convolving the additive free white noise.
It is shown how to use non-commutative stopping times in order to stop the CCR flow of arbitrary index and also its isometric cocycles, i.e., left operator Markovian cocycles on Boson Fock space. Stopping the CCR flow yields a homomorphism from the semigroup of stopping times, equipped with the convolution product, into the semigroup of unital endomorphisms of the von Neumann algebra of bounded operators on the ambient Fock space. The operators produced by stopping cocycles themselves satisfy a cocycle relation.
It is proved that the (volume and orientation-preserving) quantum isometry group of a spectral triple obtained by deformation by some dual unitary 2-cocycle is isomorphic with a similar twist-deformation of the quantum isometry group of the original (undeformed) spectral triple. This result generalizes similar work by Bhowmick and Goswami for Rieffel-deformed spectral triples in [Comm. Math. Phys. 285 (2009), 421-444, arXiv:0707.2648].
This paper concerns the structure of the group of local unitary cocycles, also called the gauge group, of an E_0-semigroup. The gauge group of a spatial E_0-semigroup has a natural action on the set of units by operator multiplication. Arveson has characterized completely the gauge group of E_0-semigroups of type I, and as a consequence it is known that in this case the gauge group action is transitive. In fact, if the semigroup has index k, then the gauge group action is transitive on the set of k+1-tuples of appropriately normalized independent units. An action of the gauge group having this…
From Kümmerer's investigations on stationary Markov processes has emerged an operator algebraic definition of white noises which captures many examples from classical as well as from non-commutative probability. Within non-commutative $L^p$-spaces associated to a white noise, the role of (non-)commutative Lévy processes is played by additive cocycles for the white noise shift, and moreover, the notion for exponentials of classical Lévy processes is generalized by unitary cocycles. As a main result we report a bijective correspondence between additive and unitary cocycles for white noise shifts…
In this article, we give a class of examples of compact quantum groups and unitary 2-cocycles on them, such that the twisted quantum groups are non-compact, but still locally compact quantum groups (in the sense of Kustermans and Vaes). This also provides examples of cocycle twists where the underlying C*-algebra of the quantum group changes.
Bosons and fermions are described by using canonical generators of Cuntz algebras on any permutative representation. According to branching laws associated with these descriptions, a certain representation of the Cuntz algebra $\co{2}$ induces Fock representations ${\cal H}_{B}$ and ${\cal H}_{F}$ of bosons and fermions simultaneously. From this, a unitary operator $U$ from ${\cal H}_{B}$ to ${\cal H}_{F}$ is obtained. We show the explicit formula of the action of $U$ on the standard basis of ${\cal H}_{B}$. It is shown that $U$ preserves the particle number of ${\cal H}_{B}$ and ${\cal H}_{F}…
For the family Delta_q, q in [0,1), of the SU_q(2)-comultiplications on the C*-algebra A\cong C(SU_q(2)), we show that there exist unitary operators Omega_q such that Delta_q(x)=Omega_q Delta_0(x) (Omega_q)* for x in A.
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 …
We initiate a study of infinite tensor products of projective unitary representations of a discrete group G. Special attention is given to regular representations twisted by 2-cocycles and to projective representations associated with CCR-representations of bilinear maps. Detailed computations are presented in the case where G is a finitely generated free abelian group. We also discuss an extension problem about product type actions of G, where the projective representation theory of G plays a central role.
For $φ$ a normalized positive definite function on a locally compact abelian group $G$, we consider on the one hand the unitary representation $π_φ$ associated to $φ$ by the GNS construction, on the other hand the probability measure $μ_φ$ on the Pontryagin dual $\hat{G}$ provided by Bochner's theorem. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,π_φ)$ and reduced 1-cohomology $\bar{H}^1(G,π_φ)$. For example, $\bar{H}^1(G,π_φ)=0$ if and only if either $Hom(G,\mathbb{C})=0$ or $μ_φ(1_G)=0$, where $1_G$ is the trivial character of $G$.
Quantization of locally compact groups associated with essentially bijective $1$-cocycles
Université Catholique de Louvain（天主教鲁汶大学）；Université de Reims Champagne-Ardenne（兰斯香槟-阿登大学）；University of Oslo（奥斯陆大学）；OsloMet - storbyuniversitetet（奥斯陆城市大学）
In this article, we investigate the notion of a Galois object for a locally compact quantum group M. Such an object consists of a von Neumann algebra N equipped with an ergodic integrable coaction of M on N, such that the crossed product is a type I factor. We show how to construct from such a coaction a new locally compact quantum group P, which we call the reflection of M along N. By way of application, we prove the following statement: any twisting of a locally compact quantum group by a unitary 2-cocycle is again a locally compact quantum group.
We study existence, uniqueness and triviality of path cocycles in the quantum Cayley graph of universal discrete quantum groups. In the orthogonal case we find that the unique path cocycle is trivial, in contrast with the case of free groups where it is proper. In the unitary case it is neither bounded nor proper. From this geometrical result we deduce the vanishing of the first L^2-Betti number of A_o(I_n).
The q-commutation relations in the title are those that have recently received much attention, and that for -1<q<1 provide an interpolation between Bosonic and Fermionic statistics, passing through free statistics at q=0. We look at the C*-algebra R^q generated by the representation of these relations on the twisted Fock space of Bozejko-Speicher, and we find a canonical unitary U_q from the twisted Fock space to usual full Fock space, such that U_q R^q (U_q)^* contains the Cuntz algebra R^0, and such that we have equality for |q|<0.44.
