Final Report Summary - VNALG (Von Neumann algebras, group actions and discrete quantum groups)
The research of the ERC project VNALG is situated at the crossroads of three areas of mathematics: group theory, functional analysis and ergodic theory. The mathematical concept of a group encodes symmetry. A well known example is the group of all possible transformations of a Rubik's cube. Another one is the group of all distance preserving transformations of the three-dimensional space. This is already a quite complicated group, containing all rotations, translations and mirror symmetries.
Secondly, functional analysis has been developed in the early 20th century in order to provide a solid mathematical framework for particle physics. It turned out that quantum mechanical observables do not behave like functions, but rather like matrices. Furthermore these matrices were infinite in size. Addition of two infinite matrices is easy, but in order to multiply two infinite matrices, the theory of Hilbert spaces is needed. A Hilbert space can be viewed as an infinite-dimensional version of our ambient three-dimensional space. Exactly as with matrices, operators can be added and multiplied so that they can form algebras of operators. A particular class of operator algebras are von Neumann algebras introduced by Murray and von Neumann in the 1930's.
Finally, ergodic theory is the study of area (or length or ‘measure’) preserving dynamical systems. In ergodic theory, one studies the long time behavior when such a length preserving transformation is iterated. When in average, over many iterations, the dynamical system visits all states an approximately equal number of times, the system is called ergodic.
The above three areas of mathematics meet each other in the theory of von Neumann algebras through the so-called group measure space construction of Murray and von Neumann. This construction associates a von Neumann algebra M to a group G of measure preserving symmetries of a space X. The isomorphism class of the von Neumann algebra M depends in subtle ways on the group G and the nature of its action on X. And vice-versa, some of the most interesting aspects of the dynamics of measure preserving symmetry groups are revealed by the study of the associated group measure space von Neumann algebras.
The main achievements of the ERC project VNALG were the following. In an article in Inventiones Mathematicae, Sorin Popa (UCLA) and Stefaan Vaes (KU Leuven) proved for the first time that in specific circumstances, one can entirely recover a group G and its action on a space X from the ambient group measure space von Neumann algebra M. This so-called ‘superrigidity’ theorem is a highly surprising result since very different groups and actions have the tendency to give rise to the same (isomorphic) von Neumann algebras.
Another breakthrough established in the ERC project has been the construction of the first von Neumann algebras M with fractal symmetry groups. More precisely, in an article in the Journal of the American Mathematical Society, Popa and Vaes proved that the fundamental group of a von Neumann algebra M can be fractal and can have any Hausdorff dimension between 0 and 1.
In an article in the Annals of Mathematics, Adrian Ioana (UCSD), Popa and Vaes proved the first superrigidity theorem for group von Neumann algebras, by constructing countable groups G such that the group von Neumann algebra L(G) entirely remembers G.
The most recent collaboration of Popa and Vaes lead to a range of unique Cartan decomposition theorems for II_1 factors and will be published in Acta Mathematica. In these results, one can entirely recover the orbits of an action of a group G on a space X from the ambient group measure space von Neumann algebra.
The research results established in the VNALG project got a broad recognition and PI Stefaan Vaes was an invited speaker at the 2010 International Congress of Mathematicians and became a Fellow of the American Mathematical Society in 2012.
Secondly, functional analysis has been developed in the early 20th century in order to provide a solid mathematical framework for particle physics. It turned out that quantum mechanical observables do not behave like functions, but rather like matrices. Furthermore these matrices were infinite in size. Addition of two infinite matrices is easy, but in order to multiply two infinite matrices, the theory of Hilbert spaces is needed. A Hilbert space can be viewed as an infinite-dimensional version of our ambient three-dimensional space. Exactly as with matrices, operators can be added and multiplied so that they can form algebras of operators. A particular class of operator algebras are von Neumann algebras introduced by Murray and von Neumann in the 1930's.
Finally, ergodic theory is the study of area (or length or ‘measure’) preserving dynamical systems. In ergodic theory, one studies the long time behavior when such a length preserving transformation is iterated. When in average, over many iterations, the dynamical system visits all states an approximately equal number of times, the system is called ergodic.
The above three areas of mathematics meet each other in the theory of von Neumann algebras through the so-called group measure space construction of Murray and von Neumann. This construction associates a von Neumann algebra M to a group G of measure preserving symmetries of a space X. The isomorphism class of the von Neumann algebra M depends in subtle ways on the group G and the nature of its action on X. And vice-versa, some of the most interesting aspects of the dynamics of measure preserving symmetry groups are revealed by the study of the associated group measure space von Neumann algebras.
The main achievements of the ERC project VNALG were the following. In an article in Inventiones Mathematicae, Sorin Popa (UCLA) and Stefaan Vaes (KU Leuven) proved for the first time that in specific circumstances, one can entirely recover a group G and its action on a space X from the ambient group measure space von Neumann algebra M. This so-called ‘superrigidity’ theorem is a highly surprising result since very different groups and actions have the tendency to give rise to the same (isomorphic) von Neumann algebras.
Another breakthrough established in the ERC project has been the construction of the first von Neumann algebras M with fractal symmetry groups. More precisely, in an article in the Journal of the American Mathematical Society, Popa and Vaes proved that the fundamental group of a von Neumann algebra M can be fractal and can have any Hausdorff dimension between 0 and 1.
In an article in the Annals of Mathematics, Adrian Ioana (UCSD), Popa and Vaes proved the first superrigidity theorem for group von Neumann algebras, by constructing countable groups G such that the group von Neumann algebra L(G) entirely remembers G.
The most recent collaboration of Popa and Vaes lead to a range of unique Cartan decomposition theorems for II_1 factors and will be published in Acta Mathematica. In these results, one can entirely recover the orbits of an action of a group G on a space X from the ambient group measure space von Neumann algebra.
The research results established in the VNALG project got a broad recognition and PI Stefaan Vaes was an invited speaker at the 2010 International Congress of Mathematicians and became a Fellow of the American Mathematical Society in 2012.