Sheaf cohomology for C*-algebras

拓扑学是对拓扑空间和连续变形的研究，也就是对形状以及如何在不破坏形状的情况下进行变形的抽象概念。拓扑空间的一个例子是欧几里得空间的任何子集。在拓扑空间中，可以有全局现象，也可以有非常不同的局部现象，这些现象只在空间中的一个点附近有效。Sheaf理论为我们提供了控制从局部性质到全局性质的通道的工具。层上同调增加了代数(计算)性质的额外技术，并使我们能够处理区分拓扑空间的不变量(即，在变形下不变的性质)，否则这些拓扑空间可能很难区分开来。它还将其他上同调理论相互联系起来，是一种高度复杂的方法论，它从范畴理论中汲取了大量的力量，范畴理论是纯数学的一个非常抽象的领域。非交换拓扑学作为量子物理的适当数学语言已经使用了一段时间，最近在许多其他数学领域，如数论，发现了多种有时意想不到的应用。拓扑空间的概念被C*-代数(Hilbert空间上有界线性算子的自伴闭子代数)所取代，C*-代数之间的联系(“变形”)是*-同态，有时是保持相关结构的映射。开子集被理想所取代；因此，在这种更一般的设置下，C*-代数的束非常适合于处理局部和全局现象之间的差异。基于我们与Pere Ara(巴塞罗那)合作发展的局部乘子理论，到目前为止，我们已经很好地理解了这些层的一些基本例子的柄，截面函子是可用的，并且已经发表了各种重要的结果。下一步，自然地将发展C*-代数的层上同调理论，这将使我们能够使用同调理论中的强大的代数工具。在克服了分析对象范畴(有些典型的令人不快的)行为引起的基本困难之后，我们将获得C*-代数的新的不变量，它可能再次区分那些以前不能处理的不变量(非简单C*-代数的Elliott程序)。

Topology is the study of topological spaces and continuous deformations, that is, an abstract notion of shape and how it can be deformed without breaking it apart. An example of a topological space is any subset of Euclidean space. In a topological space, there can be global phenomena and very different local ones, those that are only valid in the vicinity of a point in the space. Sheaf theory provides us with tools to control the passage from local to global properties. Sheaf cohomology adds additional techniques of an algebraic (computational) nature and enables us to treat invariants (i.e., properties invariant under deformation) that distinguish between topological spaces which may otherwise be difficult to tell apart from each other. It also connects other cohomology theories with each other and is a highly sophisticated methodology drawing a lot of its strength from Category Theory, a very abstract field of Pure Mathematics.Non-commutative Topology has been in use as the adequate mathematical language for Quantum Physics for some time and has lately found manifold, sometimes unexpected applications in numerous other areas of mathematics, such as Number Theory. The concept of a topological space is replaced by a C*-algebra (a self-adjoint closed subalgebra of the bounded linear operators on Hilbert space), the connections between C*-algebras (the "deformations") are *-homomorphisms or sometimes mappings preserving related structure. Open subsets are replaced by ideals; therefore a sheaf of C*-algebras is well suited to handle the differences between local and global phenomena in this more general setting. Based on the theory of local multipliers, which we developed in collaboration with Pere Ara (Barcelona), stalks of some fundamental examples of these sheaves are by now well understood, the section functors are available, and various important results have been published.The next, natural step will be to develop a sheaf cohomology theory for C*-algebras which will put us in a position to employ the powerful algebraic tools from Homology Theory. After basic difficulties which arise from the (somewhat typical unpleasant) behaviour of categories of analytic objects have been overcome, we shall obtain new invariants for C*-algebras that, once again, may tell those apart that previously could not be handled (Elliott's programme for non-simple C*-algebras).