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Groepen van uitwendige automorfismen en bimodule categorieën van type II_1 factoren.
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Fundamentaalgroepen van II_1 factoren en equivalentie relaties
No full textOne invariant of II_1 factors and II_1 equivalence relations that has been studied extensively is the fundamental group. This notion is not related to the fundamental group of a topological space, but instead can be shown to be a subgroup of the positive real numbers. It was introduced by Murray and von Neumann for II1 factors. Murray and von Neumann were only able to compute it for the hyperfinite II_1 factor: F(R) = R*_+. Since then, much has been achieved, but it also became clear that calculating this fundamental group was very difficult. Whenever you have a countable group acting freely, ergodically and measure preservingly on a standard probability space, you can construct both a II_1 equivalence relation (the orbit equivalence relation), and a II1 factor (the group measure space construction). In this case, the fundamental group of the equivalence relation is a subgroup of the fundamental group of the II_1 factor. Many results involving these group actions show that equality holds in specific cases. However, this is not true in general. In 2006, Popa gave examples of group actions where the difference is as big as possible. In this thesis, I wil elaborate on this, giving examples where the equivalence relation can have arbitrary fundamental group, whereas the associated II1 factor has R*_+ as a fundamental group.Furthermore I give examples of another interesting phenomenon, where a II_1 factor contains a Cartan subalgebra such that the fundamental group of the equivalence relation associated to this Cartan subalgebra is non-trivial, whereas the fundamental group of the original II_1 factor is {1}. Indeed, whenever a II_1 factor has a Cartan subalgebra, Feldman and Moore showed that this gives riseto a II_1 equivalence relation and a scalar 2-cocycle. In our case, this 2-cocycle is non-trivial, and hence the equivalence relation associated to the Cartan subalgebra is twisted by a 2-cocycle inside the II_1 factor.status: Publishe
Berekenen van invarianten van II_1 factoren.
No full textThere are several invariants for type II factors, but usually they are very hard to compute. In this thesis, I provide several new computations of known invariants.One of these invariants is the fundamental group of a II factor. There is no link with the fundamental group of a topological space, and the fundamental group of a II factor is always a subgroup of \IRpos. In the paper where they introduced the fundamental group, Murray and von Neumann could compute it only for the hyperfinite type II factor: \Fundg(R)=\IRpos. They did ask if there is an intrinsic characterisation for the class of all subgroups of \IRpos that appear as the fundamental group of a II factor. In 1980, Connes showed that it is not only \IRpos itself: he could show that the group von Neumann algebras \Lg(\Gamma) of ICC property (T) groups have countable fundamental group. A breakthrough was realized by Sorin Popa. The development of his deformation/rigidity theory provided a revolution in our understanding of II factors. As a first application, he showed that there are type II factors who fundamental group is just . Later he could show that all countable subgroups of \IRpos appear as fundamental groups of type II factors. Recently, Popa and Vaes showed that many uncountable subgroups of \IRpos appear as the fundamental group of a type II factor. Their result was a pure existence result. In this thesis, I provide an explicit construction, together with a potentially larger class of subgroups of \IRpos.I also studied a pair of invariants that comes from the theory of subfactors. For an inclusion of type II factors , Jones defined a notion of index. The index of a subfactor does not have to be an integer, but Jones showed the landmark result that it is always in the set \Indi=\{4\cos2(\pi/n)\mid n=3,4,\ldots\}\cup[4,\infty]. Moreover, he showed that all these possible values of the index appear for subfactors of the hyperfinite type II factor. With the index, Jones defined two invariants for a type II factor : \AllI(M) and \AllII(M). The first, \AllI(M) is the set of all indices of finite index subfactors , and \AllII(M) is the same, but we only look at the indices of irreducible subfactors of . Very little was known about these invariants: it was known that \AllI(\Lg(\FG_\infty))=\AllII(\Lg(\FG_\infty))=\Indi and that \AllI(R)=\Indi, but we do not even know \AllII(R). Vaes showed that there are type II factors where \AllII(M)=\{1\}. In this thesis, I show that, for any countable group , there is a type II factor such that \AllII(M_\Lambda) is the set of all \nelt{G} where is a finite subgroup of .status: Publishe
Baumslag-Solitar groepen en hun von Neumann algebra's
No full textWe examine both the group von Neumann algebras of the Baumslag-Solitar groups and the crossed product von Neumann algebras of some of their actions. In the case of the group von Neumann algebras, we show that the rational number |n/m| is an invariant of L(BS(n,m)). Concretely, if L(BS(n,m)) and L(BS(p,q)) are isomorphic and nonamenable, then |n/m|=|p/q|^±1. In the case of the crossed products, we show that BS(n,m) is an invariant of L^\infty(X) \rtimes BS(n,m) whenever the canonical almost normal, abelian subgroup acts aperiodically. More precisely, let BS(n,m) \curvearrowright X and BS(p,q) \curvearrowright Y be two ergodic essentially free probability measure preserving actions of nonamenable Baumslag-Solitar groups whose canonical almost normal abelian subgroups act aperiodically, then an isomorphism between the crossed products of the actions forces BS(n,m) to be isomorphic with BS(p,q) when |n| \neq |m| and BS(n,m) to be isomorphic with BS(p,±q) when |n|=|m|.status: Publishe
Type II_1 factoren met overaftelbaar veel niet-geconjugeerde Cartan deelalgebra's
No full textWe construct a class of type II_1 factors M that admit uncountably many Cartan subalgebras that are not conjugate by a stable automorphism of M. Our results lead to a class of II_1 factors with unclassifiably many Cartan subalgebras in the sense that the equivalence relation of "being conjugate by an automorphism of M" is complete analytic, in particular non-Borel. Finally we present the first example of a II_1 factor that admits uncountably many non-isomorphic group measure space decompositions, all involving the same group G. So G is a group that admits uncountably many non-stably orbit equivalent actions whose crossed product II_1 factors are all isomorphic.status: Publishe
Superrigiditeit van groeps-von-Neumannalgebra's
No full textTo any countable discrete group one can associate the group von Neumann algebra, which is generated by the image of the left regular representation of the group. More generally, to any action of a countable group on a probability measure space by probability measure preserving transformations one can associate the group measure space von Neumann algebra, which is an object that encodes information about the group, the space and the action. Over the last years, Popa's deformation/rigidity theory led to a lot of progress in the classification of group measure space von Neumann algebras associated with free, ergodic, probability measure preserving actions of countable groups. In comparison, our understanding of group von Neumann algebras is much more limited.One of the fundamental problems in the theory of von Neumann algebras is to classify group von Neumann algebras in terms of the group. More precisely, we want to know how much the group von Neumann algebra remembers about the group. A celebrated theorem of Alain Connes from 1976 says that whenever G is an amenable group with infinite conjugacy classes (i.c.c.), its group von Neumann algebra does not remember anything about the group, except its amenability. The opposite phenomenon, when the group von Neumann algebra remembers everything about the group, is called W*-superrigidity. Connes' rigidity conjecture from 1980 says that i.c.c. groups with Kazhdan's property (T) are W*-superrigid, but this remains wide open even for classical groups like SL(n, Z), with n≥3. The fundamental idea of Popas deformation/rigidity theory is to speculate the tension between these two extreme phenomena. More precisely, we study von Neumann algebras that have, at the same time, rigid parts and strong deformation properties. A countable group G is W*-superrigid if whenever there exists another countable group that yields the same group von Neumann algebra as G, then the two groups must be isomorphic. The first example of such W*-superrigid groups was given only in 2010 by Adrian Ioana, Sorin Popa and Stefaan Vaes. They proved that for a large class of generalized wreath products G, the group von Neumann algebra associated to G completely remembers the group. Motivated by the work of Ioana, Popa and Vaes, we find in this thesis more examples of W*-superrigid groups. Given a countable group G, we consider the action of the direct product G x G on G by left-right multiplication and we define a generalized wreath product group associated to this action. We prove that the resulting generalized wreath product is W*-superrigid whenever the starting group G belongs to a large class of non-amenable groups, containing free groups, hyperbolic groups, non-trivial free products, certain groups with positive first l2-Betti number, etc. We follow the same strategy as Ioana, Popa and Vaes, but our methods of proof are different. As a consequence, we can prove W*-superrigidity also for a number of subgroups of generalized wreath product groups.status: Publishe
Classificatie van type III factoren gegeven door Bernoulli-acties
No full textCrossed products with noncommutative Bernoulli actions were introduced by Connes as the first examples of full factors of type III. In this thesis, we provide a complete classification of the factors (P,φ)^Fn \rtimes Fn , where Fn is the free group and P is an amenable factor with a normal faithful state φ that either is almost periodic, or has a weakly mixing modular automorphism group. We show that the family of factors (P,φ)^Fn \rtimes Fn with φ almost periodic, is completely classified by the rank n of the free group Fn and Connes’s Sd-invariant; and that the family of factors (P,φ)^Fn \rtimes Fn with φ a weakly mixing state, is classified by n and the action of Fn on (P,φ)^Fn , up to state-preserving conjugation of the action. We prove similar results for free product groups, as well as for classes of generalized Bernoulli actions.status: Publishe
Categorieën van representaties en classificatie van von Neumann algebra's en kwantum groepen
No full textWe present several classification results and calculation of categories of representations for von Neumann algebras and quantum groups. The work is structured according to its previous publication as preprints or journal articles and grouped as two blocks, the first one dealing with quantum groups, the second one with II_1 factors.The first chapter gives a brief introduction to the background of these subjects. We recommend the reader who is not familiar withthe subjects of this work, to consult directly this chapter.In the first article, the fusion rules of the category of corepresentations of several free quantum groups are calculated and we prove a theorem relating fusion rules of a free complexification of an orthogonal quantum group with the fusion rules of the original quantum group. The next article (joint work with Moritz Weber) contains classification results for easy quantum groups. We classify a large subclass of easy quantum groups in terms of reflection groups. This allows us to prove that easy quantum groups form a rich and complex object of study. The work also exhibits a fairly large class of non-classical quantum isometry groups.In the first article on von Neumann algebras (joint work with Niels Meesschaert and Stefaan Vaes), we give a new proof for stable orbit equivalence of arbitrary Bernoulli actions of finite rank free groups - a result earlier shown by Bowen. Moreover, we can extend Bowens work and prove orbit equivalence with some quotients of Bernoulli actions. This implies stable isomorphism of the associated group measure space II1 factors. The second article on von Neumann algebras (joint work with Sébastien Falguières) contains our work on bimodule categories of II_1 factors. We prove that for a tensor C*-category from a fairly large class, including finite tensor C*-categories, there is a II_1 factor whose category of finite index bimodules is equivalent to this category. We also include consequences for the calculation of other invariants of II1 factors. The last article contains partial classification results for free Bogoliubov crossed products by the integers. These include isomorphism as well as non-isomorphism results. We also conjecture a characterisation of strong solidity for free Bogoliubov crossed products and support it with our results.Our work is complemented by an introduction to the history of the subject and a list of open problems illustrating the common direction of our research.status: Publishe
L²-Bettigetallen voor deelfactoren en rigide C*-tensorcategorieën
No full textPioneered by Vaughan Jones in the 1980s, subfactors provide a powerful framework to encode quantum symmetries, with applications in various areas of mathematics, including knot theory. Any subfactor gives rise to a group-like structure, the so-called 'standard invariant'. Popa's work on the classifcation of subfactors with amenable standard invariant illustrates the need for a representation-theoretic framework to better understand the structure of this object.
This framework, expressed in the language of rigid C*-tensor categories, was recently developed by Popa and Vaes.
Using this new technology, they were able to generalise various properties of groups to the setting of general rigid C*-tensor categories, including L²-Betti numbers. These are important numerical group invariants, related to various structural properties of groups.
During my PhD, I made several contributions to the theory of L²-Betti numbers in this context. Together with Kyed, Raum and Vaes, we related the L²-Betti numbers of representation categories of compact quantum groups with those of their discrete duals. As an application, we were able to compute the L²-Betti numbers of various discrete quantum groups. I later also proved a vanishing result for L²-Betti numbers of rigid C*-tensor categories, generalising a result of Bader, Furman and Sauer from the discrete group setting.
In a joint article with Vaes, we also proved a spectral criterion for a rigid C*-tensor category to have property (T), generalising a theorem of Żuk. As an application, we gave the first 'genuinely quantum' examples of discrete quantum groups with property (T).status: Publishe
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