June 8, Tuesday | June 9, Wednesday | June 10, Thursday | |
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16-16:45 | Ivan Bardet | Anna Jencova | |

17-17:45 | Ion Nechita | Fabio Cipriani | |

OPENING | BREAK | BREAK | |

18-18:50 | Arthur Jaffe | Alice Guionnet | Joachim Cuntz |

19-19:50 | Sorin Popa | Gilles Pisier | George Elliott |

BREAK | BREAK | BREAK | |

20:30-21:15 | Marcelo Laca | Kristin Courtney | |

21:30-22:15 | Hans Wenzl | Ryszard Nest | |

22:30-23:15 | Florin Radulescu |

Meeting ID: 989 8140 2795, Passcode: 887019

Meeting ID: 973 6283 0735, Passcode: 833208

The Coffee breaks will be between 19:50-20:30 (Tuesday, June 8, Wednesday, June 9 and Thursday, June 10) and between 17:45-18:00 ( Wednesday, June 9 and Thursday, June 10).

I will present a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. The latter inequality, which we call approximate tensorization of the relative entropy, can be expressed as a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems.

We provide a new construction of Dirichlet forms on von Neumann algebras associated to eigenvalues of the modular operator of f.n. non tracial states. We describe their structure in terms of derivations and prove coercivity bounds, from which the spectral growth rate are derived. We also introduce a regularizing property of Markovian semigroups (superboundedness) stronger than hypercontractivity, in terms of noncommutative Lp(M)spaces. We also prove superboundedness for the Markovian semigroups associated to the class of Dirichlet forms introduced above, for type I factors M. We then apply this tools to provide a general construction of the quantum Ornstein-Uhlembeck semigroups of the CCR and some of their non-perturbative deformations.

One of Alain Connes' seminal results establishes that any semi-discrete (or injective or amenable) von Neumann algebra can be written as a direct limit of dimensional von Neumann algebras. In the C*-setting however, such a concise characterization is not possible: the direct C*-analogue of semi-discreteness is nuclearity, and most nuclear C*-algebras do not arise as the direct limits of finite dimensional C*-algebras. Nonetheless, by generalizing the notion of inductive limits of C*-algebras, Blackadar and Kirchberg were able to characterize quasidiagonal nuclear C*-algebras as those arising as (generalized) inductive limits of finite dimensional C*-algebras. In joint work with Wilhelm Winter, we give a further generalization of this construction, which gives us a complete characterization of separable nuclear C*-algebras as those arising from a (generalized) inductive limit of finite dimensional C*-algebras.

The property of an operator s on a Hilbert space to be isometric can be characterized by the algebraic condition s*s = 1. Many interesting and important C*-algebras can be generated by isometries or obtained by constructions involving isometries. We give a (partly historical) survey of various results in which the author has been involved and which are based on such constructions.

A brief survey will be given of the classification of simple separable amenable C* algebras which are Jiang-Su stable and (possibly redundant) satisfy the Universal Coefficient Theorem (UCT). There are many examples of such algebras, but note that, if a given simple UCT separable amenable C* algebra is not known to be stable under tensoring with the Jiang-Su algebra, this is assured just by tensoring it anyway with this algebra. Furthermore, the invariant can be formulates in a way that is insensitive to this operation. (Of course, it is only complete after tenderization).

In this lecture, I will discuss the remarkable connection between random matrices, the enumeration of maps and some applications to operator algebras and physics. This talk will be based on joint works with Vaughan Jones and Dima Shlyakhtenko as well as work in progress with Edouard Maurel Segala.

There are several quantum versions of the Renyi relative entropies, which are fundamental in quantum information theory. Some of these quantities were extended to the general context of normal states of a von Neumann algebra. We concentrate on the class of sandwiched quantum Renyi relative entropies. We show that this class can be defined in terms of the interpolation Lp spaces due to Kosaki. We discuss some properties of these quantities, especially the connection to the Araki relative entropy and the data processing inequality (monotonicity) with respect to positive unital normal maps. In the second part of the talk, it is shown that reversibility of a 2-positive unital normal map with respect to a set of normal states is characterized by equality in the data processing inequality.The talk is based on the papers A. Jencova: Renyi relative entropies and noncommutative Lp spaces I and II, Annales H. Poincare, 2018 and 2021 (to appear).

I will start by reviewing classical work of Coburn, Douglas, and Cuntz about C*-algebras generated by isometries, and then present universal models for the Toeplitz algebras of submonoids of groups and for their boundary quotients, discussing their uniqueness and simplicity properties. This is recent joint work with Camila F. Sehnem that generalizes previous results of Nica, Li, and Raeburn and myself.

The problem of enumerating meanders is a long-standing open problem in combinatorics. Many different techniques have been used to provide bounds on the asymptotic growth rate of the number of meanders. Here, we present some of the old methods and some new ones, coming from three (related) points of view. First, as noted by Fukuda and Sniady, meanders appear in relation to the partial transposition operation in quantum information theory. A second model for meandric numbers comes from random matrix theory: we shall review some old models due to di Francesco and present some new ones. Finally, I shall present a joint work with Motohisa Fukuda (arXiv:1609.02756 and arXiv:2103.03615) on a third point of view, that of non-commutative probability. Using the operations of free and boolean moment-cumulant transforms, we enumerate large sub-classes of meanders, generalizing previous work of Goulden, Nica, and Puder.

The torsion phenomena play important role in the construction of the assembly map in the context of Baum-Connes conjecture. The corresponding case of quantum groups is more involved, since the torsion phenomena are not necessarily associated to torsion subgroups.An important role in this context is played by projective representations of quantum groups. We will describe the general structure of projective representations, associated twisted group C*-algebras and the related torsion phenomena for compact quantum groups.We will also describe the role that these results play in the context of the assembly map for compact quantum groups.This is ''work in progress'' joint with Kenny De Commer and Ruben Martos.

A von Neumann algebra M is called injective if there is a projection P:B(H) -> M with ||P||= 1. This is the analogue for von Neumann algebras of amenability for discrete groups, and it notoriously fails when M = M(F) is the von Neumann algebra of a non-commutative free group F. We will introduce the class of ''seemingly injective'' von Neumann algebras. This includes M(F). We show that M is seemingly injective iff it has the (matricial) weak* positive metric approximation property (AP in short). This is parallel to Connes's characterization of injectivity by the weak* completely positive AP. We show that M(F) is isomorphic to B(H) as Banach spaces when F is countable. Lastly we discuss several open questions that might be related to Kazhdan's property (T) for groups.

One of the most fascinating aspects in the analysis of non-commutative spaces (aka von Neumann algebras), is the way their building data, which is often geometric in nature, impacts on their generalized (or virtual) symmetry picture. This is particularly the case for II_1 factors, where virtual symmetries are encoded by subfactors of finite Jones index, a numerical invariant that can be quantized in intriguing ways. I will discuss some results and open problems that illustrate the unique interplay between analysis and algebra/combinatorics entailed by this interdependence, that's specific to subfactor theory.

Abstract: Many years ago (almost 30) Vaughan Jones initiated an approach to a new program of understanding of automorphic forms as Operator Algebra objects. There are naturally associated II-1 factors ( in the free groups factors series) His original motivation was to understand if Hecke subgroups could be possibly related to a more natural construction of non-integer index subfactors in free group factors.. There is one "mystery trace vector (s)" which one would like to understand, and this showed up again in his late work. I will discuss these topics and their relations to other problems in number theory that have a natural Operator Algebra counterpart.

A diagonalizable matrix has linearly independent eigenvectors. Since the set of non diagonalizable matrices has measure zero, every matrix is a limit of diagonalizable matrices. We prove a quantitative version of this fact: every n x n complex matrix is within distance delta in the operator norm of a matrix whose eigenvectors have condition number poly(n)/delta, confirming a conjecture of E. B. Davies. The proof is based adding a complex Gaussian perturbation to the matrix and studying its pseudospectrum. Joint work with J. Banks, A. Kulkarni, S. Mukherjee

Szegedy developed a generic method for quantizing classical algorithms based on random walks. One of the contribution of his work was the construction of walk unitary for any reversible (also called detailed balanced) random walk. Such walk unitary posses two crucial properties: its eigenvector with eigenphase 0 is a quantum sample of the limiting distribution of the random walk and its eigenphase gap is quadratically larger than the spectral gap of the random walk. In this presentation we solve the question asking whether it is possible to generalize Szegedy's quantization method from stochastic maps to quantum maps. We answer the question in the affirmative by providing a construction of a Szegedy walk unitary for detailed balanced Lindbladians -- generators of quantum Markov semigroups. In particular, we prove that our Szegedy walk unitary has a purification of the fixed point of the Lindbladian as eigenvector with eigenphase 0 and that its eigenphase gap is quadratically larger than the spectral gap of the Lindbladian. Moreover, we give an efficient quantum algorithm for implementing Davies generators that described many important open-system dynamics. Our algorithm extends known techniques for simulating quantum systems on a quantum computer.

When Vaughan Jones started to think about the notion of an index for subfactors, the only known examples came from groups and their representations and from the embedding of groups H < G. In both cases the indices were integers. Vaughan's surprising examples with non-integer index were later connected to representations of the quantum group U_q(sl2) and to representations of the loop group LSU(2). This was subsequently generalized to the construction of a sequence of subfactors for every representation of a semisimple Lie algebra.The question remains to construct subfactors corresponding to analogs of subgroups of Lie groups; in modern language this amounts to classifying module categories of certain tensor categories. Again, Vaughan made important contributions for solving the problem for the sl_2 case constructing what is generally referred to as Goodman-de la Harpe-Jones subfactors. While complete classifications are known for several Lie groupsof small rank, the general problem is still far from being solved. We give an overview of the current state of knowledge, and present some explicit examples.