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Irreversibility and Non-commutative Convexity in C*-algebra Theory
Provider: Faculty of Science

Activity no.: 5564-20-07-32 
Enrollment deadline: 01/09/2020
Date and time21.09.2020, at: 08:00 - 25.09.2020, at: 16:00
Regular seats30
ECTS credits2.50
Contact personNina Weisse    E-mail address: weisse@math.ku.dk
Enrolment Handling/Course OrganiserSøren Eilers    E-mail address: eilers@math.ku.dk
Written languageEnglish
Teaching languageEnglish
Course workload
Course workload categoryHours
Preparation / Self-Study40.00
Course hours25.00
Sum65.00
Aim and content
Originally a branch of functional analysis, the field of operator algebras has grown to stand on its own and influence diverse areas of mathematics and physics such as operator theory, complex analysis, dynamical systems, group theory, convex geometry and quantum information theory. The study was initiated by von-Neumann in the 30s, motivated by mathematics related to quantum mechanics. One key goal is to study algebras of bounded operators on Hilbert space that are closed with respect to an appropriate topology, and provide new useful perspectives.

General operator structures such as non-self-adjoint operator algebras and operator systems provide useful devices for studying non-commutative convexity, dynamical systems, operator representations of directed graphs and C*-algebras in their own right. A recent example where operator systems played a pivotal role is in the work of Kennedy and Kalantar on simplicity of reduced discrete group C*-algebras. This work provided insight for the solution of a multitude of open problems and conjectures in group theory. Another instance is recent work of Katsoulis and Ramsey where non-self-adjoint operator algebra techniques resolved group Hao-Ng isomorphism problems for Cuntz-Pimsner C*-algebras. This leads to automatic structure results for many natural non-commutative dynamics on such C*-algebras. Classification of C*-algebras with additional structure has seen great strides recently with new interactions with non-self-adjoint classification recently discovered. In work of Li many C*-algebras associated to semigroups were studied and classified, but the relationship to their non-self-adjoint counterparts is still quite mysterious.

This Masterclass aims to deal with interactions between operator structures and C*-algebras, with an eye towards applications in other areas of Mathematics. Katsoulis, Kennedy and Li are leading experts in their respective areas and have published in some of the highers level journals in Mathematics. They have organized, participated in and taught in many international conferences, workshops and masterclasses. Furthermore, they are known for their enthusiasm, preparedness, clarity and precision when delivering lectures. This will clearly benefit young students and researchers who are exposed to the topics of the masterclass for the first time.

E. Katsoulis (East Carolina University) has written three monographs and has more than 50 publications. He has given many plenary talks in conferences and workshops and is the author of several works relating non-self-adjoint operator algebras to C*-algebra theory. Katsoulis' work exactly on the interactions between irreversible and reversible operator algebras. M. Kennedy (University of Waterloo) has won many NSERC grants, fellowships and awards over the years, and is currently a University Research Chair at his institution. Kennedy has several works connecting non-commutative convexity to the study of C*-algebras, dynamical systems and group theory. His works feature in the highest caliber journals in Mathematics. X. Li (Queen Mary University at London) has over 30 publications and has given many colloquia, and attended many BIRS and Oberwolfach workshops. He held an ERC consolidator grant as well as an EPSRC first grant. His work uses C*-algebraic invariants to classify irreversible objects such as semigroups coming from number theory, graph theory and one-sided dynamical systems.

Course 1 (E. Katsoulis). This course will start with an introduction to dilation theory for operator algebras focusing on examples coming from multivariable operator theory. Next, operator algebras arising from C*-correspondences will be discussed, focusing on examples arising from directed graphs and dynamical systems. Results about classification, C*-envelope and the Hao-Ng isomorphism problem will be explained with current applications to C*-algebra theory.

Course 2 (M. Kennedy). This course will focus on the theory of nc sets recently introduced by Davidson and Kennedy, with a view towards applications in non-commutative dynamics. It will start with the basic theory of nc sets and their categorical duality with operator systems. This will link back to Course 1 where connections with Arveson's boundary theory and dilation will be discussed and given geometric intuition. The course will finish with applications of the theory of nc sets to the structure of C*-algebras arising from non-commutative dynamics in papers of Kennedy--Schafhauser and Kennedy-Shamovich.

Course 3 (X. Li). This course will deal with classification of C*-algebras arising from semigroups and their actions with a viewpoint towards connections with non-self-adjoint algebras. Semigroups C*-algebras and their K-theory will be explained with emphasis on examples coming from graph theory and number theory. The course will finish with recent results on rigidity of semigroup C*-algebras that have the potential to resolve new non-self-adjoint isomorphism problems.

The program of this masterclass with the three proposed courses will cover many new and useful techniques for drawing new connections between operator algebras and other areas of Mathematics. After the masterclass, students will be equipped with state-of-the art techniques for attacking several classification and structure problems in operator algebras. The event will bring together young researchers and students from different perspectives of operator algebra theory, and will promote collaboration and the exchange of ideas.

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