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Rigidity and classification of von Neumann algebras

Final Report Summary - RIGIDITY (Rigidity and classification of von Neumann algebras)

(A version of this text appeared in the ERC brochure "Mathematics - Spotlight on ERC projects -

Group theory, functional analysis and ergodic theory are three distinct areas of mathematics that
meet within the theory of von Neumann algebras. The RIGIDITY project aims to classify families of
such von Neumann algebras.

Each of these three areas of mathematics has an origin in physics. Groups not only describe
symmetries of physical systems, but the representation theory of compact groups plays a key role
in the standard model of particle physics. Hilbert space operators - one of the main concepts in
functional analysis - play the role of quantum mechanical observables, while ergodic theory
provides the necessary tools to describe and understand the long term and global behavior of a
physical system.

Whenever a group acts on a measure space, there is an associated crossed product von Neumann
algebra. The most basic question is when two such von Neumann algebras are isomorphic? To
which extent do they remember the initial data that they were constructed from? In earlier joint
work of Popa and Vaes, we obtained the first families of group actions that can be entirely
recovered from their ambient von Neumann algebra. In other words, these von Neumann
algebras can be decomposed in exactly one way as a crossed product. The first main result
obtained in the RIGIDITY project was the construction of von Neumann algebras that have
precisely two and more generally, precisely n, crossed product decompositions.

Von Neumann algebras also arise naturally in Voiculescu's free probability theory, a
noncommutative or quantum version of classical probability theory. The classification of these
free Araki-Woods von Neumann algebras is wide open and only very partial results were known.
In the RIGIDITY project, we established definitive classification theorems for large families of free
Araki-Woods von Neumann algebras, in terms of their defining spectral measure.

A third focus of the RIGIDITY project has been on the quantum symmetries of von Neumann
algebras. These arise in the form of inclusions of von Neumann algebras, known as Jones'
subfactors. To every such subfactor is associated a group-like invariant, which is a very intricate
combinatorial structure with connections to many areas of mathematics, most notably knot
theory. In the RIGIDITY project, we developed several aspects of harmonic analysis for these
group-like invariants. Very recently, this led to the first truly quantum instances of Kazhdan's
property (T), which is a rigidity property that is for example known for its usage in the
construction of expander graphs and thus widely used in theoretical aspects of computer science.

The research achievements of the RIGIDITY project were broadly recognized and led to the award
of the Francqui Prize to PI Stefaan Vaes and to his election as a member of the Royal Academy of