C*-algebras: structure, classification and applications

这个项目的中心概念是Haagerup性质，近年来，它的应用范围从C*-代数的k理论到刚性理论，再到几何群论。如果一个群在希尔伯特空间上有适当的仿射作用，则称其具有Haagerup性质，Higson和kasparov的一个深层结果断言Haagerup群满足Baum Connes猜想。在这个项目中，这个概念的Lp推广将被定量地研究，连同理论的上同调方面。例如，计划攻击关于约化lp上同调可调群消失的Gromov猜想。更一般地说，建议确定哪些可调群在混合中首先与值减少上同调。弱混合)lp表示消失。特别地，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，就像在许多科学领域一样，例如物理、信息论或金融。几何群理论背后的原理是，空间的许多性质，如其拓扑或其几何，可以通过研究其对称性集来揭示。这个集合具有称为群结构的数学结构。这种方法的丰富性来自于这种结构的代数性质和几何性质之间的相互作用。本课题主要研究群的新的几何不变量，并将其与代数结构联系起来。作为物理学基本对象的矩阵群将在本研究中占据中心位置。

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.