Sheaf cohomology for C*-algebras

拓扑学是对拓扑空间和连续变形的研究，也就是说，它是一种抽象的形状概念，以及如何在不破坏它的情况下变形。拓扑空间的一个例子是欧氏空间的任意子集。在拓扑空间中，可以存在全局现象和非常不同的局部现象，这些现象只在空间中某一点的附近有效。束理论为我们提供了控制从局部属性到全局属性的过渡的工具。层上同调增加了代数（计算）性质的额外技术，使我们能够处理区分拓扑空间的不变量（即变形下的不变量），否则这些拓扑空间可能难以彼此区分。它还将其他上同论相互联系起来，是一种高度复杂的方法，它从范畴论中汲取了很多力量，范畴论是纯数学的一个非常抽象的领域。非交换拓扑作为量子物理的数学语言已经使用了一段时间，并且最近在许多其他数学领域（如数论）中发现了多种，有时是意想不到的应用。拓扑空间的概念被C*-代数（Hilbert空间上有界线性算子的自伴随闭子代数）所取代，C*-代数之间的连接（“变形”）是*-同态的，有时是保持相关结构的映射。开放子集被理想取代；因此，在这种更普遍的情况下，一组C*代数非常适合处理局部和全局现象之间的差异。根据我们与Pere Ara （Barcelona）合作开发的局部乘数理论，这些束的一些基本例子的梗现在已经很好地理解了，截面函子是可用的，并且各种重要的结果已经发表。下一步，自然的一步将是发展C*-代数的层上同调理论，这将使我们能够利用同调理论中强大的代数工具。在克服了解析对象范畴的行为（有些典型的令人不快的）所产生的基本困难之后，我们将获得C*-代数的新不变量，这些不变量再次可以区分那些以前无法处理的不变量（艾略特的非简单C*-代数程序）。

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).