Roland Vergnioux

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Roland Vergnioux
roland.vergnioux@unicaen.fr

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Address
« Office » Office
Freie Universität Berlin
Institut für Mathematik
Arnimallee 7, Raum 211

pub   2048D/C2B1CF1B 2011-08-25 Roland Vergnioux <roland.vergnioux@math.unicaen.fr>
Key fingerprint = B3FF 84F3 8A1E 2CC6 2DAC  034C 0646 3E52 C2B1 CF1B
http://roland.vergnioux.fr/

Activities
« Chronology» Chronology
- 2019-2020 : on leave, invited researcher at the Freie Universität, Berlin
- 2013 : Habilitation defense
- 2005- : Maître de Conférences at the University of Caen, LMNO
- 2010, 2015 : sabbatical semester at the CNRS
- 2003-2005 : Post-doc at the University of Münster, research team on Operator Algebras
- 2000-2003 : Graduate assistant at the University of Paris VII
- 1999-2002 : PhD Thesis under the supervision of G. Skandalis at the University of Paris VII
« Administrative work» Administrative work
- French head of the german-french project Quantum Groups and Quantum Probability (2019-2021)
- Member of the Laboratory Council of the LMNO (2017-2019)
- Coordinator of the ACS (HR advisory assembly) in Mathematics at the University of Caen (2017-2019)
- Member of the Academic Council of the University Community of Normandie (2015-2019)
- « More » More
- Member of the IT Committee of the LMNO (2009-)
- Organizer of the Seminar Analyse harmonique non commutative at the LMNO (2007-2019)
- Organizer of the meeting Tensor Categories and Analysis (Caen, 2018)
- Organizer of the meeting Quantum groups from combinatorics to analysis (Caen, 2016)
- Co-organizer of a Conference on Noncommutative Geometry (Marseille, 2015)
- Elected member of the National Council of Universities for Pure Mathematics (2011-2015)
- In charge of the local "L1 MASS" studies (2009-2014)
- Organizer of the conference Operator algebras, Quantum groups and Tensor categories (Caen, 2012)
- Co-organizer of the semester Quantum Groups (Caen, 2010)
- Co-organizer of the conference Quantum groups and Physics (Caen, 2010)
- Member of hiring commissions : Caen (2007, 2010, 2011), Paris 7 (2009), Clermont (2010)
- Labintel correspondent at the LMNO (2006-2015)

Articles

« Noncommutative Furstenberg boundary » Noncommutative Furstenberg boundary with Mehrdad Kalantar, Paweł Kasprzak and Adam Skalski
Files | arxiv : ps, pdf | local : pdf
In this article we extend the theory of Furstenberg boundaries to the case of discrete quantum groups. We study the kernel of the action on the maximal boundary by means of various notions of relative amenability and when the action is faithful we shows that the reduced C*-algebra of the quantum group under consideration admits at most one KMS state with respect to the scaling group, hence a unique trace in the unimodular case. We illustrate these constructions and results by showing that the Gromov boundaries of non amenable orthogonal free quantum groups are Furstenberg boundaries and that the corresponding actions are faithful. The associated crossed products are simple and lie in between the reduced C*-algebra and its injective envelope.
« Property RD and hypercontractivity for orthogonal free quantum groups » Property RD and hypercontractivity for orthogonal free quantum groups with Michael Brannan and Sang-Gyun Youn
International Mathematics Research Notices, published online
Files | arxiv : ps, pdf | journal : html | local : pdf
In this article we pursue the study of the Property of Rapid Decay (RD) for discrete quantum goups, particularly in the non-unimodular case where we show that orthogonal free quantum groups do not satisfy the twisted Property RD of Bhowmick, Voigt and Zacharias. Then we study applications to ultra- and hypercontractivity of a quantum analogue of the heat semigroup on the duals of these groups. We obtain first results in the non-unimodular case, and we improve the known estimates on optimal time to hypercontractivity in the unimodular case.
« Orthogonal free quantum group factors are strongly 1-bounded » Orthogonal free quantum group factors are strongly 1-bounded with Michael Brannan
Advances in Mathematics, 329 (2018) 133-156
Files | arxiv : ps, pdf | journal : html | local : pdf
In this article we prove that the orthogonal free quantum group factors are stronly 1-bounded in the sense of Jung. This allows to answer in the negative the important question whether these factors are isomorphic to free group factors. The proof relies on a recent rank theorem for free entropy dimension due to Jung and Shlyakhtenko, and on a spectral regularity result for the edge reversing operator in the quantum Cayley graph.
« The Connes embedding property for quantum group von Neumann algebras » The Connes embedding property for quantum group von Neumann algebras with Michael Brannan and Benoît Collins
Transactions of the AMS, 369 (2017) 3799-3819
Files | arxiv : ps, pdf | journal : html, pdf | local : pdf
In this article we show that the von Neumann algebras of orthogonal free quantum groups FO(n), endowed with the Haar state, satisfy Connes' embedding property for n≠3. We work at the level of the dual compact quantum groups and proceed by induction over the dimension n. The main technical result consists in proving that the universal orthogonal quantum group O(n,+) is generated by two copies of O(n-1,+) seen as stabilizer subgroups.
« The radial MASA in free orthogonal quantum groups » The radial MASA in free orthogonal quantum groups with Amaury Freslon
Journal of Functional Analysis, 271 (2016) 2776-2807
Files | arxiv : ps, pdf | journal : html | local : pdf
In this article we prove that the subalgebra generated by the fundamental character in the von Neumann algebra of an orthogonal free quantum group FO(n) is maximal abelian for n≥3, mixing, and we study the corresponding bimodule. In order to do so we establish new properties of the Jones-Wenzl projections in the category of corepresentations of FO(n).
« A cocycle in the adjoint representation of the orthogonal free quantum groups » A cocycle in the adjoint representation of the orthogonal free quantum groups with Pierre Fima
International Mathematics Research Notices, 2015 (2015) 10069-10094
Files | arxiv : ps, pdf | journal : html, pdf | local : pdf
In this article we construct a deformation by automorphisms of the von Neumann algebra M associated to a unimodular orthogonal free quantum group inside M⊗M, and we use it to give a new proof of Haagerup's property for these groups. Moreover we show that the bimodule M⊗M is associated to the adjoint representation, and we prove that the non-trivial part of it is weakly contained in the regular representation. This proves non inner amenability, Ozawa-Popa's property strong (HH), and yields a new proof of strong solidity of M.
« The K-theory of free quantum groups » The K-theory of free quantum groups with Christian Voigt
Mathematische Annalen, 357 (2013) 355-400
Files | arxiv : ps, pdf | journal : html, pdf | local : pdf
In this article we study the K-theory of free products of unitary and orthogonal free quantum groups, using the formulation of the Baum-Connes conjecture in terms of triangulated categories. We show the stability of the Baum-Connes conjecture under free products for torsion-free discrete quantum groups by constructing Dirac and dual-Dirac elements from the action on the quantum Bass-Serre tree. We deduce in particular K-amenability of unitary free quantum groups and compute their K-theory.
« Paths in quantum Cayley trees and L²-cohomology » Paths in quantum Cayley trees and L²-cohomology
Advances in Mathematics, 229 (2012) 2686-2711
Files | arxiv : ps, pdf | journal : html, pdf | local : pdf
There was a first version about the orthogonal case. The second version has been submitted in april 2010.
In this article I go on with the study of quantum Cayley trees, focusing in particular on the existence and uniqueness of paths to the origin. For the free group F_n, these paths define a proper cocycle, which proves Haagerup's Property. In the case of universal orthogonal quantum groups, I prove on the contrary that the unique path cocycle is trivial. In the unitary case, it is neither bounded nor proper. Finally, I use this geometrical result to establish the vanishing of the first L^2 Betti number for A_o(I_n) — whereas the one of F_n equals n-1.
« Invariants of the half-liberated orthogonal group » Invariants of the half-liberated orthogonal group with Teodor Banica
Annales de l'Institut Fourier, 60 (2010) 2137-2164
Files | arxiv : ps, pdf | journal : html, pdf | local : pdf
The half-liberated orthogonal quantum group O_n^* is a intermediate compact quantum group between the compact group O_n and its free version O_n^+ associated with the Woronowicz algebra A_o(n). We discuss in this article the basic algebraic properties of O_n^* and we classify its irreducible representations. This classification relies upon the use of a noncommutative weight lattice and a comparison with the representation theory of U_n. We show in particular that O_n^* has noncommutative fusion rules and has the same exponent of polynomial growth as SU_n.
« Fusion rules for quantum reflection groups » Fusion rules for quantum reflection groups with Teodor Banica
Journal of Noncommutative Geometry, 3 (2009) 327-359
Files | arxiv : ps, pdf | journal : html, pdf | local : pdf
In this article we find the fusion rules of the quantum reflection groups A_h^s, which are free versions of the complex reflection groups G(s,s,n). We describe the category of corepresentation by means of certain subsets of indexed non-crossing partitions, and we show that irreducible corepresentations can be indexed by the words over Z/sZ so as to obtain explicit and "local" fusion rules. We also compute the dimensions of irreducible corepresentations.
« Growth estimates for discrete quantum groups » Growth estimates for discrete quantum groups with Teodor Banica
Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12 (2009) 321-340
Files | arxiv : ps, pdf | journal : html, pdf | local : pdf
We give in this article estimates and exact formulae for the main known classes of discrete quantum groups, including exponential and polynomial examples. In the case of duals of real compact Lie groups we establish a relation between a quantum random walk and a classical random walk on the weight lattice. This allows to check in this dual case the quantum analogue of Gromov's result relating growth to the probability of return to the origin for discrete groups.
« The boundary of universal discrete quantum groups, exactness and factoriality » The boundary of universal discrete quantum groups, exactness and factoriality with Stefaan Vaes
Duke Mathematical Journal, 140 (2007) 35-84
Files | arxiv : ps, pdf | journal : html, pdf | local : pdf
In this article we construct and study the Gromov boundary of Wang-Banica orthogonal free quantum groups. It is a unital C*-algebra, constructed as an inductive limit, which is naturally endowed with an action of the discrete quantum group. We show that this action is amenable, in an appropriate quantum sense, and we deduce the exactness and Property AO for these quantum groups. We also prove simplicity results and hence we see that the von Neumann algebras associated to the quantum groups under consideration are full and prime factors. We finally establish connections with the notions of Poisson and Martin boundaries, and give a direct proof of exactness based on the notion of monoidal equivalence.
« The Property of Rapid Decay for Discrete Quantum Groups » The Property of Rapid Decay for Discrete Quantum Groups
Journal of Operator Theory, 57 (2007) 303-324
Files | arxiv : ps, pdf | journal : html, pdf | local : pdf
In this article I introduce the Property of Rapid Decay for discrete quantum groups by equivalent characterizations that generalize the classical ones, and I show that the application to the K-theory of the reduced C*-algebras of these groups carries over to the quantum case. I also investigate examples, proving in particular the Property of Rapid Decay for unimodular free quantum groups. I turns out that only unimodular quantum groups can have Property RD.
« Orientation of quantum Cayley trees and applications » Orientation of quantum Cayley trees and applications
Journal für die reine und angewandte Mathematik 580 (2005) 101-138
Files | arxiv : ps, pdf | journal : html | local : pdf
In this article I introduce the notion of quantum Cayley graph, and I study it in the case of trees, which corresponds to the case of free quantum groups. I focus in particular on the notion of ascending orientation for these quantum trees, and I describe the associated space of edges at infinity, which is an outcome of the non-involutivity of the edge returning operator and vanishes in the classical case. I give two applications to conclude: a proof of Property AO for free quantum groups, and the construction of their KK-theoretic gamma element. The results concerning the space of edges at infinity have been generalized and ameliorated since my PhD thesis, and the application to Property AO is new.
« K-amenability for amalgamated free products of amenable discrete quantum groups » K-amenability for amalgamated free products of amenable discrete quantum groups
Journal of Functional Analysis 212 (2004) 206-221
Files | arxiv : ps, pdf | journal : html, pdf | local : pdf
In this article I present the results of my thesis about the equivariant KK-theory with respect to locally compact quantum groups, and I demonstrate the K-amenability of amalgamated free products of amenable discrete quantum groups, by constructing a quantum analogue of the Bass-Serre trees. Some proofs have been ameliorated since my PhD thesis.
Academic Works
« Groupes quantiques discrets et algèbres d'opérateurs» Groupes quantiques discrets et algèbres d'opérateurs
Habilitation Thesis
Files | local : pdf
This mémoire has been written on the occasion of my Habilitation, that I defended in June 2013. It presents and synthesizes the results I obtained since the end of my PhD Thesis.
« Haar integrals on finite and compact quantum groups» Haar integrals on finite and compact quantum groups
Notes of a graduate course given at Caen in 2010
Files | local : pdf
These notes were written for a short graduate course given at the University of Caen in September 2010 as part of a “quantum group” semester. It turned out that it was necessary to include some introductory material about tensor products and group algebras. Then we tried to present some nice features of the elementary theory of Hopf algebras, both in the algebraic and the C*-algebraic settings. Our main goal was the existence and unicity of “Haar integrals”, whose proofs are quite different in both settings, and the presentation of some “real-life” examples where computation skills can be trained. There are also some additional exercices (in french).
« KK-théorie équivariante et opérateur de Julg-Valette pour les groupes quantiques» KK-théorie équivariante et opérateur de Julg-Valette pour les groupes quantiques
PhD Thesis supervised by G. Skandalis
Files | theses directory of the IMJ : dvi, pdf (zip), ps (bz2)
After three chapters of reminders, I study the crossed product construction for locally compact quantum groups, and in particular, its fonctorial properties and its use in equivariant KK-theory (descent, Green-Julg isomorphism, K-amenability). Then I study examples of actions on quantum trees: quantum Bass-Serre and Cayley trees. In the first case, it allows to demonstrate the K-amenability of the amalgamated free products under consideration. The second case is much more complicated, and it is in particular necessary to introduce a space of edges at infinity. One can then define the gamma-elements associated to these quantum Cayley trees, in the case of the orthogonal free quantum groups.
« Magistère's Thesis » Magistère's Thesis
Files | local : pdf, dvi
This thesis contains the texts I wrote at the Ecole Normale Supérieure of Paris: Equivalence between fermions and bosons (Master's Thesis under the supervision of B. Julia), Matrix Compact Quantum Groups (Diplôme d'Etudes Approfondies under the supervision of G. Skandalis), and an introduction to the links between particle physics and non-commutative geometry.
See also...
...my publications on MathSciNet, my preprints on arXiv, my page on HAL.

Univ. Caen > Sciences > LMNO > Roland Vergnioux

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