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517.986.6 --- #WWIS:STAT --- 517.986.6 Harmonic analysis of functions of groups and homogeneous spaces --- Harmonic analysis of functions of groups and homogeneous spaces

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To 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.

517.986 <043> --- Topological algebras. Theory of infinite-dimensional representations--Dissertaties --- Theses --- Academic collection

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Ordered algebraic structures --- 517.986 --- Topological algebras. Theory of infinite-dimensional representations --- 517.986 Topological algebras. Theory of infinite-dimensional representations --- Hopf algebras --- Hopf, Algèbres de --- Hopf, Algèbres de. --- Hopf, Algèbres de.

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Operator theory --- Analytical spaces --- Algebraic topology --- Operator algebras --- K-theory --- 517.986 --- Algebras, Operator --- Topological algebras --- Homology theory --- Topological algebras. Theory of infinite-dimensional representations --- 517.986 Topological algebras. Theory of infinite-dimensional representations

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Topological groups. Lie groups --- Harmonic analysis. Fourier analysis --- 517.986.6 --- Harmonic analysis of functions of groups and homogeneous spaces --- 517.986.6 Harmonic analysis of functions of groups and homogeneous spaces