Domain Theory in semantics and in the theory of C*-algebras

• Applicant: Professor Dr. Klaus Keimel (†)
• Subject Area: Theoretical Computer Science ; Mathematics
• Term: from 2011 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 210423156

The working programme is subdivided into two parts. The first part is a direct continuation of the previous project while the second part builds on recent developments.A. Models for nondeterministic phenomena in programming.In the previous project we have developed models for mixed probabilistic and purely nondeterministic features in collaboration with G. D. Plotkin (Edinburgh). These models were built in the framework of Domain Theory initiated by D. S. Scott for denotational semantics. Our models only allow to give a qualitative account of the behaviour of programs. We intend to add now quantitative aspects to these models. A motivating question is to mesure the performance of an approximate simulation of a process .The progress in Domain Theory during the past twenty years, in particular in modeling probabilistic and nondeterministic features, merits to be presented in a form that is accessible and comprehensive for non-specialists. Together with J. Goubault-Larrecq (ENS Paris-Cachan) we have started to work on a book project. Within the new project, we intend to continue this monograph and to bring it to an end.There are two minor open ends from the previous project that we intend to finalize together with A. Simpson (Ljubljana) and M. deBrecht (Kyoto), respectively. Both cases concern models for indeterminism of a topological nature beyond domain theory.B. Domain theoretic methods for the theory of C*-algebras.In 2008, Coward, Elliott und Ivanescu introduced a new invariant for C*-algebras by adding a topological feature to the Cuntz-semigroup, a semigroup associated with every C*-algebra by J. Cuntz in 1978. This additional structure is of domain theoretical nature. In the sequel, methods of domain theory where reinvented or developed in parallel to domain theory.It is a challenge for both sides, when in an area of mathematics -- here operator algebras -- methods can be applied that have in been introduced in a completely disjoint area -- there domain theory -- and for completely different purposes -- denotational semantics. Here it is our aim to bridge the gap with the help of experts in the theory of C*-algebras, in particular with H. Thiel (Universität Münster), F. Perera und R. Antoine (beide Universitat Autonoma de Barcelona). While in the theory of C*-algebras mainly the algebraic and order theoretical properties of Cuntz semigroups have been investigated, we intend to put our hands on the advantages provided by the domain theoretical features.

• DFG Programme: Research Grants