Sheaf cohomology for C*-algebras

Lead Research Organisation: Queen's University of Belfast
Department Name: Sch of Mathematics and Physics


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).

Planned Impact

The study of any field of modern mathematics adds to the knowledge base of our society. Mathematics is fundamental to all sciences as a language in which concepts and interconnections among these can be expressed, in many situations the only adequate language. The proposed research will provide further insight in mathematical structures that interlink diverse areas such as Algebra, Analysis and Topology. These are used in quite concrete applications to networks, for example, (hence to information and communication technologies) which are modeled by graphs whose properties can be described via associated C*-algebras which in turn need to be understood in depth, for example divided into isomorphism classes for which the project offers new invariants, so-called cohomological dimension. Improved information and communication channels will contribute to economic growth in the UK.


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Description We developed a new cohomology theory for C*-algebras, based on sheaves of operator modules. This required in particular the indrouction of new techniques of a categorical nature, namely the study of exact categories. This will lead to new methods to classify non-simple C*-algebras.
Exploitation Route Our contribution leads to a better understanding of operator algebras on a fundamental level. Operator algebras are used in many diverse areas, ranging from Quantum Information Theory to Manufacturing.
Sectors Digital/Communication/Information Technologies (including Software),Education,Manufacturing, including Industrial Biotechology

Description Collaboration with Pere Ara (UAB) 
Organisation Autonomous University of Barcelona (UAB)
Country Spain 
Sector Academic/University 
PI Contribution collaborative research on EPSRC project
Collaborator Contribution collaborative research on EPSRC project
Impact work in progress is to be written up as a memoir (too large for a single research paper, not suitable to be split up into several smaller ones).
Start Year 2015
Description Invited Lecture 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact Invited talk on progress on reserach project.
Year(s) Of Engagement Activity 2018
Description Lecture Course 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact Plenary speaker at Workshop in Shiraz University, Shiraz, Iran.
Year(s) Of Engagement Activity 2017
Description invited lecture 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other audiences
Results and Impact Invited lecture on "Towards a sheaf cohomology theory for C*-algebras", a progress report on the various stages of the project.
Delievered at/in: University of Aberdeen, March 2016; Technical University Berlin, July 2016; University of New Brunswick, Fredericton, Canada, December 2016; Universidad de Oviedo, Spain, January 2017; Universitat Autonoma de Barcelona, January 2017.
Year(s) Of Engagement Activity 2016,2017