# The 15th HU-SNU Joint Symposium on Mathematics: Operator Algebras

### Outline:

Dates: November 4, 2021

Venue: Online (Zoom links will be sent to participants)

Organizers: Yuhei Suzuki（HU）,  Hun Hee Lee（SNU）

### Program:

9:40-10:10

Katsunori Fujie (HU, Ph.D student)

On random interlacing sequences constructed from the eigenvalues of a random matrix and of its minor.

Abstract:In this talk, we observe a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy; for example, this class includes GUE (Gaussian Unitary Ensemble) and Wishart matrices. More precisely, if the EED of the whole matrix converges to some deterministic probability measure m, then the difference of rescaled EEDs of the whole matrix and of its principal submatrix concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with m by the Markov–Krein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. This talk is based on joint works with Takahiro Hasebe in Hokkaido university.
10:20-10:50

Sang-gyun Youn (SNU professor)

A strong Haagerup inequality for non-Kac free orthogonal quantum groups

Abstract:The Haagerup inequality on the reduced free group C*-algebras has been studied in various contexts, and one surprising fact is that a strengthened form, namely strong Haagerup inequality, can be established for `analytic’ polynomials generated only by the generators g_1,g_2,…,g_n without their inverses. Some natural analogues have been studied in the class of Kac type discrete quantum groups, but not in the non-Kac type situation. The main aim of this talk is to exhibit such a phenomenon for non-Kac free orthogonal quantum group, which is one of the most important quantum group examples.

11:00-11:30

Keisuke Yoshida (HU posdoc)

Simplicity of the universal C*-algebras associated to multispinal groups

Abstract:In 1984, Grigorchuk found the first example of a group of intermediate growth. His work also provides us an idea on how to construct interesting groups associated to symbolic dynamics. Multispinal groups are defined to give a generalization of the Grigorchuk group. By definition, we have some relations between shift maps and elements in multispinal groups. In my talk, we consider the universal C*-algebras generated by mulitspinal groups and shift maps satisfying the relations. On the hypothesis that multispinal groups are amenable, I give a necessary and sufficient condition for the universal C*-algebras to be simple.

Afternoon Session (Chairman:Keisuke Yoshida)
13:00-13:30

Hypergroups and random walks on graphs

Abstract:In this talk, I will introduce our recent works related to connections between random walks and hypergroups. These are based on joint works with Kenta Endo, Tomohiro Ikkai, Ippei Mimura and Hiromichi Ohno. Wildberger has introduced a method of constructing a hypergroup from a random walk on a graph. Any graph with a good symmetry condition called the distance regularity produces a hypergroup. His method enable us to compute a distance distribution of the random walk by the algebraic structure of the corresponding hypergroup.
13:40-14:10

Fatemeh Khosravi (SNU postdoc)

C^*-simplicity and unique trace property for unimodular discrete quantum groups.

Abstract: The action of a discrete group G on its Furstenberg boundary plays an important role in the characterization of simplicity and unique trace property of its reduced C^*-algebra, $C^*_r(G)$. This characterization shows that C*-simplicity forces $C^*_r(G)$ to admit only one tracial state. The generalized notion of the Furstenberg boundary of a discrete group to discrete quantum group is introduced and studied by Kalantar-Kasprzak-Skalski-Vergnioux. In this talk, we discuss the relation of simplicity and uniqueness of the tracial state of the C*-algebra of dual unimodular discrete quantum groups. This is based on an ongoing joint work with Mehrdad Kalantar.

14:20-14:50

Yuhei Suzuki (HU)

Simplicity and tracial weights on non-unital reduced crossed products

Abstract:We extend theorems of Breuillard-Kalantar-Kennedy-Ozawa on unital reduced crossed products to the non-unital case under mild assumptions. As a result simplicity of C*-algebras is stable under taking reduced crossed product over discrete C*-simple groups, and a similar result for uniqueness of tracial weight. Interestingly, our analysis on tracial weights involves von Neumann algebra theory.
Our generalizations have two applications. The first is to locally compact groups. We establish stability results of (non-discrete) C*-simplicity and the unique trace property under discrete group extensions. The second is to the twisted crossed product. Thanks to the Packer-Raeburn theorem, our results lead to (generalizations of) the results of Bryder-Kennedy by a different method. Based on my preprint arXiv:2109.08606
15:00-15:30

Seung-Hyeok Kye (Seoul National University)

Convex cones in mapping spaces between matrix algebras

Abstract: We introduce the notion of one-sided mapping cones of positive linear maps between matrix algebras. These are convex cones of maps that are invariant under compositions by completely positive maps from either the left or right side. The duals of such convex cones can be characterized in terms of ampliation maps, which can also be used to characterize many notions from quantum information theory—such as separability, entanglement-breaking maps, Schmidt numbers, as well as decomposable maps and k-positive maps in functional analysis. In fact, such characterizations hold if and only if the involved cone is a one-sided mapping cone. Through this analysis, we obtain mapping properties for compositions of cones from which we also obtain several equivalent statements of the PPT (positive partial transpose) square conjecture. This talk is based on the paper [Linear Alg. Appl. 608 (2021), 248–269] with Mark Girard and Erling Stormer under the same title.