The students who participate in the seminar give talks on functional analytic aspects of noncommutative geometry. Talks devoted to noncommutative algebraic geometry and to "pure" functional analysis (preferably with an algebraic or geometric flavour) are also welcome. The topics of talks are usually taken from the literature, but sometimes the participants present their own results. Occasionally, talks are given by the seminar advisor or by an invited speaker.
Starting from November 19, the seminar takes place online on Thursdays, 14:40-16:00.
If you wish to give a talk, please send you suggestions to the seminar advisor (pirkosha at gmail dot com).
Abstract. The talk summarises some results of the paper "A new knot polynomial and von Neumann algebras" by Professor Vaughan Jones (Notices of the AMS, March 1986). The goal is to explore the relation between polynomial invariants for links and von Neumann algebras in order to construct the celebrated Jones polynomial. The definition and basic properties of von Neumann algebras, subfactors and their trace will be stated. The definition of knots and their invariants, following the Artin presentation for braid groups in connection with knots will be explored. Using the relations obtained, the Jones polynomial would be defined.
The talk is dedicated to the memory of Professor Vaughan Jones who passed away on September 6, 2020.
Abstract. We discuss an algebraic approach to compact quantum groups due to Koornwinder. A distinctive feature of this approach is that C*-algebras appear only on the final stage of the construction, in contrast to the traditional approach of Woronowicz. We consider algebras of polynomial functions on classical matrix groups and, grounding on some of their properties, come to the general notion of compact Hopf *-algebras, or CQG algebras. Then we consider corepresentaitons of such algebras, construct the Haar weight, and, by taking the C*-completion, come to compact quantum groups in the sense of Woronowicz.
Abstract. Given a family (Ix) of homogeneous ideals in the noncommutative disk algebra A, we construct a continuous Banach algebra bundle with fibers isomorphic to the quotients A/Ix. Such objects naturally appear in noncommutative complex analysis, and they can also be useful in some problems of nonformal deformation quantization. Our main tool is a theorem due to Orr Shalit and Baruch Solel, which yields a construction of completely isometric representations of the above quotients on the full Fock space.
Abstract. Bost-Connes systems are special quantum statistical dynamical systems, which provide a surprising connection between C*-algebras and number theory. We will give an overview of this area and discuss generalizations of original Bost-Connes construction to arbitrary number fields as well as results on K-theory of corresponding C*-algebras.
Abstract. The James space is the first example of a nonreflexive Banach space isomorphic to its double dual. There is a generalisation of the James space due to James and Lindenstrauss. They showed that for every separable Banach space X one can construct a separable Banach space Z such that Z**/Z is isomorphic to X. Before constructing these spaces I will discuss bases in Banach spaces and give a criterion for reflexivity of a Banach space in terms of properties of its bases.
Abstract. At the previous meetings we discussed C*-envelopes of Nica tensor algebras associated to product systems. Now we apply this theorem to describing C*-envelopes of graph algebras. We will look at concrete examples, and we will show that the respective C*-envelopes are isomorphic to Cuntz-Krieger algebras.
Abstract. This time we will analyze several tensor product constructions of locally convex operator spaces, introduce the concept of a nuclear locally convex operator space, and prove several interesting facts related to it.
Abstract. This time we will deal with the classical matrix approach to defining operator spaces, analyze several tensor product constructions, introduce the concept of a nuclear locally convex operator space, and prove several interesting facts related to it.
Abstract. Our original goal is to understand whether or not the C*-envelope of the free product A1*A2*...of operator algebras is isomorphic to the free product of their C*-envelopes. It turns out that a similar result holds for Nica tensor algebras associated to semigroups. At the previous meeting, following https://arxiv.org/pdf/1804.10546.pdf and https://arxiv.org/pdf/1801.07296.pdf, I introduced the basic notions and formulated the main result. Now I will explain some corollaries of this result, using the Cuntz-Krieger algebra as the motivating example, and will then prove the main theorem.
Abstract. Our original goal is to understand whether or not the C*-envelope of the free product A1*A2*...of operator algebras is isomorphic to the free product of their C*-envelopes. It turns out that a similar result holds for Nica tensor algebras associated to semigroups. Following https://arxiv.org/pdf/1804.10546.pdf and https://arxiv.org/pdf/1801.07296.pdf, I will define Nica tensor algebras, Senhem C*-algebras, and will show that the C*-envelopes of Nica tensor algebras are isomorphic to Senhem algebras. All the necessary concepts will be introduced along the way.
Abstract. In my talk, I will speak about a coordinate-free approach to the theory of operator (or quantum) locally convex spaces and Arens-Michael operator algebras. We will analyze the basic concepts and theorems related to the theory of operator spaces. I will also try to provide the audience with a set of examples necessary to understand the usefulness of this area.
