Paweł Józiak: Uniqueness of Cartan subalgebras (21.02.2017, 22.02.2017, 28.02.2017, 29.02.2017, 14.03.2017, 4.04.2017, and 11.04.2017)
The aim of the next meeting will be to introduce key ingredients of a proof of Ozawa and Popa that certain crossed product II_1 von Neumann algebras have unique Cartan subalgebra. We will cover relative amenability and complete metric approximation property for von Neumann algebras. If time permits, we will start discussing weakly compact actions.
Adam Skalski: Amenable equivalence relations (1.02.2017, 8.02.2017, and 15.02.2017)
The plan for the next two-three meetings is as follows: we will prove
the theorem of Dye (following Hajian, Ito and Kakutani, and using
purely ergodic theoretic methods) establishing the uniqueness of an
ergodic singly generated - equivalently hyperfinite - equivalence
relation. We will then discuss the theorem of Connes, Feldman and
Weiss on amenability=hyperfiniteness for equivalence relations and
outline the von Neumann algebraic consequences.
the theorem of Dye (following Hajian, Ito and Kakutani, and using
purely ergodic theoretic methods) establishing the uniqueness of an
ergodic singly generated - equivalently hyperfinite - equivalence
relation. We will then discuss the theorem of Connes, Feldman and
Weiss on amenability=hyperfiniteness for equivalence relations and
outline the von Neumann algebraic consequences.
Mateusz Wasilewski: Simple applications of intertwining by bimodules (25.01.2017)
We will present applications of Popa's intertwining techniques. We will show that rigid Cartan subalgebras of certain crossed products are unique.
Damian Orlef: Popa's intertwining techniques (21.12.2016, 4.01.2017, and 18.01.2017)
We will discuss Popa's method of intertwining by bimodules, which in some cases can be used to show that one subalgebra can be conjugated into another using a unitary. This will be a crucial tool for obtaining rigidity results, e.g. conjugacy of actions from isomorphism of crossed products.
Mateusz Wasilewski: Introduction to von Neumann algebras (9.11.2016, 23.11.2016, 30.11.2016, and 7.12.2016)
We will present examples of von Neumann algebras coming from spectral theory and group representation theory. Then we will mostly talk about finite von Neumann algebras and II1 factors. We plan to discuss examples most relevant to group theory: group algebras, group-measure space constructions (crossed products) and von Neumann algebras associated with countable equivalence relations. This will be a natural setting for introducing Cartan subalgebras. We will also discuss how properties of groups and group actions are reflected in the properties of the associated operator algebras.
Damian Sawicki: A primer on group actions (26.10.2016 and 2.11.2016)
We will present the ergodic decomposition theorem for group actions. We will discuss topological amenability of a continuous group action and the spectral gap property for a probability measure preserving action. These properties are implied by, respectively, amenability and property (T) of the acting group, but there are many other examples. Moreover, amenable actions can be used to characterise exactness aka property A.
Mateusz Wasilewski: An overview (19.10.2016)
This talk will be a non-technical introduction. We will explain how a group action can be studied on three levels: the group action itself, the orbit equivalence relation, and the von Neumann algebra associated with the action (the crossed product). We will see that some properties of the group action (e.g. having an invariant probability measure, ergodicity) depend only on the orbit equivalence relation or, under the presence of freeness, on the von Neumann algebra. We will stress the importance of freeness in connecting the three levels. We will also give examples of theorems that we plan to discuss later.