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GBMS Theme 8: Quantum Probabilistic Symmetries & Quantized Boolean Algebras

Date: 20 - 25 Aug 2015

Venue: Western Gateway Building, UCC

Scientific Advisory Committee

  • David E. Evans (Cardiff)
  • Vaughan Jones (Vanderbilt)
  • Gilles Pisier (Texas A&M)
  • Anatoly M. Vershik (St. Petersburg)
  • Dan-Virgil Voiculescu (Berkeley)

Organizers

  • Claus Köstler (UCC)
  • Stephen Wills (UCC)

Speakers Include (*=TBC)

  • Alexander Belton (Lancaster)
  • B. V. Rajarama Bhat (ISI Bangalore)
  • Collin Bleak (St. Andrew's)
  • Matthias Christandl (Copenhagen)
  • David E Evans (Cardiff)
  • Uwe Franz (Besancon)
  • Roland Friedrich (HU Berlin)
  • Malte Gerold (Greifswald)
  • Rolf Gohm (Aberystwyth)
  • Debashish Goswami (ISI Kolkata)
  • Marius Junge (UIUC)
  • Vijay Kodiyalam (Chennai IMSc)
  • Franz Lehner (Graz)
  • J. Martin Lindsay (Lancaster)
  • Weihua Liu (Berkeley)
  • Tobias Mai (Saarbrücken)
  • James Mingo (Queen's, Kingston)
  • Naofumi Muraki (Iwate)
  • Magdalena Musat (Copenhagen)
  • Mikael Rørdam (Copenhagen)
  • Vlad Sergiescu (Grenoble)
  • R. Srinivasan (Chennai IMSc)
  • Pierre Tarrago (Saarbrücken)
  • Anatoly M. Vershik (Steklov St. Petersburg)
  • Dan-Virgil Voiculescu (Berkeley)
  • Moritz Weber (Saarbrücken)
  • Ping Zhong (Lancaster)

The programme is here.

Sponsors

This event is co-sponsored by the ERC Advanced Grant with the title Non-Commutative Distributions in Free Probability.

Title & Abstracts of Talks

    • Alexander Belton, Quasi-free stochastic processes from quantum random walks

      Attal and Joye studied an operator-valued quantum random walk driven by particles in a faithful normal state. They found the quantum stochastic differential equation obeyed by its limit process, and showed that the quantum noises appearing in this Langevin equation satisfy the commutation relations for a certain quasifree state. Inspired by this example, we report on the development of a general framework to handle such quasifree random walks. The necessary theory of quantum stochastic integration builds on early work of Hudson and Lindsay, together with more recent work of Lindsay, Margetts and Weatherall. Joint work with Ping Zhong, Lancaster University.

    • B. V. Rajarama Bhat, Symmetric representations of C*-algebras and structure theorems of completely bounded maps

      A homomorphism τ of a C*-algebra is said to be symmetric if τ(a*)= J τ(a)*J, for a symmetry J (J=J*, J²=I). We study various families of symmetric homomorphisms and make use of them to get some structure theorems for completely bounded maps. This is a joint work with Nirupama Mallick and K. Sumesh.

    • Collin Bleak, On automorphisms of the Higman groups G_{n,r} and of the full one sided shift on n-letters.

      We prove that the group of automorphisms of a Higman group G_{n,r} is given as a group of homeomorphisms realisable via finite bijective transducers a-la Grigorchuk, Nekrashevych, and Suschanskii. We embed this group as a subgroup of the automorphisms of the full shift on n-letters, and show (as suggested by Hubbard), that the resulting group is isomorphic, ``on the nose,'' to the group of automorphisms of the one-sided full shift on n letters. Along the way, we will pass through the beautiful land of De Bruijn automorphisms. Summary of two projects; joint with Yonah Maissel and Andres Navas, and separately, joint with Peter Cameron.

    • Matthias Christandl, On quantum channels appearing in quantum cryptography

      I will discuss some quantum channels (CP maps from M_n to M_n) motivated by quantum cryptography and discuss their curious properties.

    • David E Evans, The search for the exotic (Plenary Lecture)

      Subfactor theory provides a framework for studying modular invariant partition functions in conformal field theory, and candidates for exotic modular tensor categories. I will describe work with Terry Gannon on the search for exotic theories beyond those from symmetries based on loop groups and finite groups.

    • Uwe Franz, On conditionally positive functions and functionals

      Let A be a unital *-algebra and ε: A → C a unital *-homorphism. A functional ψ: A → C is called conditionally positive (or a generating functional) on (A,ε) if (a) ψi(1)=0, (b) ψ is hermitian, (c) ψ is positive on the kernel of ε. Examples are the linear extensions to the group algebra of negative type functions on groups. Conditionally positive functions classify Lévy processes on groups or involutive bialgebras. I will present new results about the relation between conditionally positive functions and 1-cocycles, and about the existence of a decomposition of such functions into a maximal Gaussian part and a Gauss-free remainder. Based on joint work with Biswarup Das, Malte Gerhold, Anna Kula, Adam Skalski, Andreas Thom.

    • Roland Friedrich, Formal Groups and Probability

      In this talk we present our recent results on homogeneous Lie groups and quantum probability. Further we discuss a more general approach to the subject and highlight the fundamental role of generalised pro-unipotent Lie groups which intrinsically arise in connection with any notion of independence.

    • Malte Gerold, Positive Hochschild Cocycles

      As will be discussed in the talk of Uwe Franz, the question whether all positive 2-cocycles are trivial plays a role in the study of quantum Lévy processes. More precisely, it is a sufficient condition for the possibility to decompose every generator of a quantum Lévy process into a maximal Gaussian part and a purely non-Gaussian remainder (Lévy-Khintchin decomposition). A stronger condition is the vanishing of the second cohomology group. I will present an example of a *-algebra with a character which has the following properties: 1. All positive 2-cocycles are coboundaries. 2. There is a 2-cocycle which is not a coboundary. So the two above-mentioned sufficient conditions are not equivalent. Furthermore, I will explain a certain exact sequence which helps to calculate second cohomology groups and was proved in the group algebra case by Tim Netzer and Andreas Thom. Based on joint work with Uwe Franz and Andreas Thom.

    • Rolf Gohm, Semi-cosimplicial objects and spreadability

      To a semi-cosimplicial object (SCO) in a category we associate a system of partial shifts on the inductive limit. We show how to produce an SCO from an action of the infinite braid monoid. In categories of (noncommutative) probability spaces SCOs correspond to spreadable sequences of random variables, hence SCOs can be considered as the algebraic structure underlying spreadability. This is joint work with Gwion Evans and Claus Köstler.

    • Debashish Goswami, Quantum isometry group of compact metric spaces

      After a quick introduction to the formulation of quantum isometric groups in the geometric context, we concentrate on a notion of quantum isometry in the purely metric space set-up. We define isometric actions of compact quantum groups on compact metric spaces. For a given compact metric space belonging to a large class of metric spaces including compact subsets of Euclidean spaces as well as all finite metric spaces, we prove the existence of a universal object (i.e. the quantum isometry group) in the category of compact quantum groups acting isometrically on it. Some concrete examples are given and open questions are discussed.

    • Marius Junge, Actions of q-gaussian algebras

      In joint work with Bodgan Udrea we use quantum probabilistic methods to define and investigate analogues of Shlyahktenko's A-valued semicircular algebras. Quantum symmetries and their ergodic properties are used a substitute for actions and co-actions of groups in Popa and Vaes's work on relative amenability.

Marius Junge, Analysis on noncommutative spaces (Plenary Lecture)

Noncommutative Tori are simplest examples for noncommutative mainfolds in noncommutative geometry. For these concrete spaces and their noncompact analogues, we will discuss basic concepts from noncommutative geometry and finite dimensional approximation in the Gromov-Haussdorff sense. Using a semigroup approach one can show that also many tools in classical harmonic analysis concerning convergence of Fourier series and singular integral operators remain valid in the context of noncommutative deformations of classical spaces.

  • Vijay Kodiyalam, Planar algebras, cabling and the Drinfeld double

    We produce explicit embeddings of the planar algebra of a finite-dimensional semisimple and cosemisimple Hopf algebra into the two-cablings of the planar algebras of the dual and opposite Hopf algebras and characterise the images. This work is joint with Sandipan De.

  • Franz Lehner, Cumulants, spreadability and Hausdorff series

    We extend the notion of cumulants to spreadability systems, including e.g. monotone cumulants. The combinatorics are based on ordered set partitions and it turns out that instead of being additive, cumulants of independent sums involve the Campbell-Baker-Hausdorff series. This is joint work with T. Hasebe.

  • J. Martin Lindsay, KMS-Symmetry and quantum Markov semigroups

    Two related problems will be discussed: the symmetry associated with time-reversal for quantum sub-Markov semigroups, and the generation of such semigroups. The former calls for an appropriate notion of `adjoint' for maps between von Neumann algebras with faithful normal semifinite weights; the latter for a `quantum' notion of Dirichlet form. They both involve inducing maps on associated noncommutative Lp-spaces; specifically, interpolating between the algebra, and its predual, and conversely extrapolating from its standard Hilbert space to the von Neumann algebra. Our profound debt to Uffe Haagerup through his pioneering works will be manifest in this talk. Joint work with Stanislaw Goldstein and Adam Skalski.

  • Weihua Liu, Extended de Finetti theorems for Boolean independence and monotone independence.

    In this talk, we will define noncommutative spreadability for Boolean and monotone independence. We will show Ryll-Nardzewski type theorems for monotone and Boolean independence: Roughly speaking, an infinite bilateral sequence of random variables is monotonically (Boolean) spreadable if and only if the variables are identically distributed and monotone (Boolean) with respect to the conditional expectation onto its tail algebra. For an infinite sequence of noncommutative random variables, Boolean spreadability is equivalent to Boolean exchangeability.

  • Tobias Mai, Regularity of distributions of Wigner integrals

    With their seminal work in 1998, P. Biane and R. Speicher founded a non-commutative counterpart of classical stochastic calculus and Malliavin calculus in the realm of free probability. In particular, they introduced the so-called Wigner integrals as the free analogue of the classical Wiener integrals. In my talk, I will discuss how recent results that were obtained in joint work with R. Speicher and M. Weber, which under certain conditions allowed to exclude atoms in the distributions of non-constant polynomials in finitely many non-commutative random variables, can be extended and applied to Wigner integrals.

  • James Mingo, Freeness and the Partial Transpose

    Wishart matrices can be used to describe a random state. In 2012 Aubrun showed that the partial transpose of a Wishart matrix converges to a shifted semi-circular operator which may or may not be positive. In recent work with M. Popa, we showed that a Wishart matrix, its left and right partial transpose, and its full transpose form an asymptotically free family. I will show that the same applies to a Haar distributed random unitary operator, with the partial transpose converging to a circular operator.

  • Naofumi Muraki, q-Deformation of free independence

    Although there is the no-go theorem of Leeuwen-Maassen on the existence of q-convolution, we can construct, in a sense, a notion of q-deformed free independence and the associated notions of q-convolution and q-cumulatnts. The construction is based on the q-product operation for non-commutative probability spaces. In a sense, our construction of q-convolution and q-cumulants is consistent with Anshelevich' s construction of q-Levy processes.

  • Magdalena Musat, Quantum error correction and the Connes embedding problem

    Work on quantum error correction led J. Smolin, F. Verstraete and A. Winter to formulate in 2005 a restoration in the asymptotic limit of Birkhoff’s classical theorem. More precisely, they conjectured that every unital quantum channel might always be well approximated by a convex combination of unitarily implemented ones. In earlier joint work with U. Haagerup we disproved this conjecture by showing that so-called non-factorizable quantum channels, which we construct in all dimensions greater than or equal to 3, are counterexamples. In recent work, we exhibit an asymptotic property of factorizable quantum channels which leads to a reformulation of the Connes embedding problem. I will further discuss recent work with U. Haagerup and M.-B. Ruskai, where we study the convex structure of factorizable quantum channels.

  • Mikael Rørdam, Just infinite groups and C*-algebras

    A (discrete) group is called just infinite if it is infinite and all its non-trivial normal subgroups have finite index. There is a well-established theory for just infinite groups, and there are interesting examples of just infinite groups (including, for example, the Grigorchuk groups). In a similar way one can define a (unital) C*-algebra to be just infinite if it is infinite dimensional and all its proper quotients are finite dimensional. Infinite dimensional simple C*-algebras and essential extensions of simple C*-algebras by finite dimensional C*-algebras are just infinite (for trivial reasons). We show that there exist residually finite dimensional just infinite C*-algebras (that can be chosen to be AF-algebras), and we explain some structure results for just infinite C*-algebras. The construction of a just infinite residually finite dimensional AF-algebra can be done using an old result by Bratteli and Elliott which says that each totally disconnected spectral space arises as the primitive ideal space of an AF-algebra. We discuss possible connections to just infinite groups. This is work in progress joint with R. Grigorchuk and M. Musat.

  • Vlad Sergiescu, Braided Thompson groups

    The purpose of this talk is to present recent progress on the interaction between Artin braid groups and Thompson's groups of pieces linear homeomorphisms of respectively the interval, the circle and the Cantor set.

  • R. Srinivasan, Cohomology for spatial super-product systems

    Super-product system is a generalization of product system of Hilbert spaces introduced by Bill Arveson. They arise naturally in the theory of E₀-semigroups on factors. We propose a cohomology theory for spatial super-product systems, and describe the 2-cocycles for basic examples. This consequently classifies, up to cocycle conjugacy, a family of E₀-semigroups on type III factors associated with canonical anti-commutation relations, and its restrictions to some well-known subalgebras. This is a joint work with Oliver T. Margetts.

  • Pierre Tarrago, Free wreath product and spin planar algebras

    I will present some recent results on the representations of some particular free wreath products. This description involves the free product of spin planar algebras. In a second part I will give some probabilistic applications of this construction.

  • Anatoly M. Vershik, The filtrations of Boolean σ-algebras, standardness or hierarchical independence, and virtual metric spaces with measure

    (Plenary Lecture)

    We consider a new theory of filtrations: classification/notion of standard filtration and generalization of the notion of independence. The applications of the theory include stochastic processes, Bratteli-Vershik diagrams, and C*-algebras.

  • Dan-Virgil Voiculescu, Free Probability for Pairs of Faces (Plenary Lecture)

    We will discuss the recent extension of free probability to systems with left and right variables. This includes the analogues, in the simplest cases, of operations on independent variables and extreme values.

  • Moritz Weber, Unitary easy quantum groups

    The class of easy quantum groups (introduced by Banica and Speicher in 2009) consists of compact matrix quantum groups of combinatorial nature. Many of their quantum algebraic properties are visible in their combinatorial data, in the categories of (set theoretical) partitions. We will briefly introduce easy quantum groups and then focus on their unitary extensions.

  • Ping Zhong, Remarks on the Cauchy-Stieltjes transform of freely infinitely divisible distributions

    Biane found some useful properties of the Cauchy-Stieltjes transform of freely infinitely divisible distributions. This provides a nice tool to study the marginal distributions of free additive Brownian motion and free additive convolution semigroups. Belinschi extended Biane's result further to operator-valued free probability. We report some analogue results for free multiplicative convolutions and their applications.

    Learn more about the George Boole 200 programme at: www.georgeboole.com