NCG&T Prague

usually takes place each Tuesday at 16:00 Institute of Mathematics CAS, Blue lecture room, Zitna 25, Praha 1
Chair: Tristan Bice, Karen Stung

What is noncommutative geometry and topology? The idea stems from the Gelfand theorem which states that the category of compact Hausdorff spaces and commutative C*-algebras are dual. If we drop the condition of commutativity from our C*-algebras, we arrive at the notion of a noncommutative topological space. This can be carried further into the realm of noncommutative of geometry by equipping *-algebras with geometric structures.
Our research focusses on both quantum algebraic and operator algebraic aspects of noncommutative geometry and topology. This includes research in Hopf algebras, quantum groups, and noncommutative complex geometry, while on the operator algebra side, we study C*-algebras, with particular focus on C*-algebras arising from dynamical constructions such as minimal actions, groupoids, and semigroups.
Partially supported by GAČR project 20-17488Y Applications of C*-algebra classification: dynamics, geometry, and their quantum analogues and PRIMUS grant Spectral Noncommutative Geometry of Quantum Flag Manifolds.

On the structure of unitary groups of C*-algebras

Michal Doucha

Institute of Mathematics of the Czech Academy of Sciences

Tuesday, 11. October 2022 - 16:00 to 17:00

This talk will take place in the blue seminar room, back building, Žitná 25.

It will also be broadcast on Zoom:
https://cesnet.zoom.us/j/91975183920?pwd=SGdlRUhLS21HUnVFTXBKcE1vYXlrQT09
Meeting ID: 919 7518 3920
Passcode: 102707

I will report on a joint work in progress with Hiroshi Ando where we follow and extend the recent results of Leonel Robert on the Lie group structure of the unitary group of a unital C*-algebra. I will give some introduction to infinite-dimensional linear Lie groups and describe e.g. how the ideal structure of the algebra corresponds to the normal subgroup structure of the unitary group, how the properties such as simplicity or unique trace property are visible from the structure of the corresponding unitary group, how to characterize simplicity of reduced group C*-algebras Lie-theoretically, etc.

Quantization of linear real semisimple Lie groups

Kenny De Commer

Vrije Universiteit Brussel

Tuesday, 18. October 2022 - 16:00 to 17:00

Let L be a connected linear real semisimple Lie group, with associated real semisimple Lie algebra l. We present a novel construction of a *-algebra quantizing the universal enveloping *-algebra of l, and then show how this *-algebra can be `integrated’ into a C*-algebra which quantizes the universal group C*-algebra of L. Our construction builds on Letzter’s theory of coideal subalgebras of quantized enveloping algebras, coming from the classical theory of symmetric spaces (of compact type).

Cartan subalgebras in C*-algebras and von Neumann algebras

Jean Renault

University of Orléans

Tuesday, 18. October 2022 - 16:00 to 17:00

A generalization of the classical group measure space construction associates a von Neumann algebra to a measured groupoid. In 1977, J. Feldman and C. Moore characterized the von Neumann algebra constructed from a countable measured equivalence relation by the existence of a Cartan subalgebra. Analogous results exist in the framework of topological groupoids and C*-algebras. I will review these results and later developments, putting the emphasis on the notion of twisted inverse semigroup.

K-theoretic classification of inductive limit actions of fusion categories on AF C*-algebras

Roberto Hernandez Palomares

Texas A&M University

Tuesday, 1. November 2022 - 16:00 to 17:00

PLEASE NOTE UNUSUAL LOCATION: This talk will take place in the front building, Žitná 25, 3rd floor seminar room.

It will also be broadcast on Zoom:
https://cesnet.zoom.us/j/91975183920?pwd=SGdlRUhLS21HUnVFTXBKcE1vYXlrQT09
Meeting ID: 919 7518 3920
Passcode: 102707

I will introduce a K-theoretic complete invariant of inductive limits of finite dimensional actions of fusion categories on unital AF-algebras. This framework encompasses all such actions by finite groups on AF-algebras. Our classification result essentially follows from applying Elliott’s Intertwining Argument adapted to this equivariant context, combined with tensor categorical techniques. Time allowing, we will discuss some applications. This is joint work with Q. Chen and C. Jones.

Institute of Mathematics

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