Sheaf cohomology for C*-algebras

C* 代数的层上同调

• 批准号: EP/M02461X/1
• 负责人: Martin Mathieu
• 金额: $ 5.93万
• 依托单位: Queen's University Belfast
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2015
• 资助国家: 英国
• 起止时间: 2015 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FM02461X%2F1
• 关键词: Sheaf; cohomology; algebras

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).

拓扑学是拓扑空间和连续变形的研究，也就是说，形状的抽象概念以及它如何变形而不分裂。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，可以有全局现象和非常不同的局部现象，这些局部现象仅在空间中的一点附近有效。层理论为我们提供了控制从局部性质到全局性质的过渡的工具。Sheaf上同调增加了代数（计算）性质的额外技术，并使我们能够处理不变量（即，变形下不变的性质），区分拓扑空间，否则它们可能难以彼此区分。它也将其他的上同调理论相互联系起来，是一种高度复杂的方法论，从范畴论（一个非常抽象的纯数学领域）中汲取了大量的力量。非对易拓扑已经被用作量子物理的适当数学语言有一段时间了，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闭子代数）所取代，C*-代数之间的连接（“变形”）是 *-同态或有时保持相关结构的映射。开子集被理想所取代;因此C*-代数的层非常适合于在这种更一般的设置中处理局部和全局现象之间的差异。基于我们与Pere Ara合作开发的局部乘数理论（巴塞罗那），秸秆的一些基本的例子，这些层现在很好地理解，部分函子是可用的，和各种重要的结果已经出版。下一个，自然的一步将是发展一个层上同调理论的C*-代数，这将使我们能够使用来自同调理论的强大代数工具。在基本的困难所产生的（有点典型的不愉快）行为的范畴的分析对象已被克服，我们将获得新的不变量的C*-代数，再次，可以区分那些除了以前不能处理（埃利奥特的计划，非简单的C*-代数）。

项目成果

Towards a sheaf cohomology theory for C*-algebras

走向 C* 代数的层上同调理论

• 作者: Mathieu, M