C*-algebras: structure, classification and applications

这个项目的中心概念是Haagerup性质，近年来，它得到了广泛的应用，从C*-代数的K-理论，到刚性理论，再到几何群论。一个群被称为具有Haagerup性质，如果它在Hilbert空间上有一个适当的仿射作用，并且Higson和Kasparov的一个深刻结果断言一个Haagerup群满足Baum Connes猜想.在这个项目中，我们将定量地研究这个概念的Lp推广，以及该理论的上同调.例如，它计划攻击Gromov猜想，即关于顺从群上的约化LP上同调消失的猜想。更一般地，建议确定哪些顺从基团的第一约简上同调具有混合值(分别为。弱混合)Lp-表示消失。特别是，这将为一大类群产生新的几何不变量。寻找在某些变换下保持不变的量是数学的主要目标之一，如在许多科学领域，例如，物理、信息论或金融。几何群论背后的原理是，一个空间的许多性质，如它的拓扑或几何，可以通过研究它的对称集来揭示。这个集合有一种称为群结构的数学结构。这种方法的丰富源于这种结构的代数性质和它的几何性质之间的相互作用。在这个项目中，我们建议研究群的新的几何不变量，并将它们与它们的代数结构联系起来。在物理学中作为基本物体出现的一组矩阵，将在这项研究中处于中心位置。

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.