C*-algebras: structure, classification and applications

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The central notion of this project is the Haagerup property which, in recent years,has found applications ranging from K-theory of C*-algebras, to rigidity theory,to geometric group theory. A group is said to have the Haagerup property if itadmits a proper affine action on a Hilbert space, and a deep result of Higson andKasparov asserts that a Haagerup group satisfies the Baum Connes conjecture.In this project, Lp generalizations of this concept will be studied quantitatively,together with cohomological aspects of the theory. It is for instance planned toattack a Gromov conjecture about the vanishing of reduced Lp-cohomology onamenable groups. More generally, it is proposed to determine for which amenable groups the first reduced cohomology with values in mixing (resp. weakly mixing) Lp-representation vanishes. In particular, this will produce new geometric invariants for a large class of groups.Looking for quantities that remain invariant under certain transformationsis one the main objectives in mathematics as in many scientific fields like forexample, physics, information theory or finance. The principle behind geometricgroup theory is that many properties of a space, such as its topology or itsgeometry, can be revealed by studying its set of symmetries. This set has amathematical structure called a group structure. The wealth of this approachresults from the interactions between the algebraic properties of this structureand its geometric properties. In this project, it is proposed to study new geometricinvariants of groups, and to relate them with their algebraic structure. Groupsof matrices, that appear as fundamental objects in physics, will have a centralposition in this study.