Descripción del proyecto
Estudio de operadores promediables de álgebras
Las álgebras de operadores lineales y continuos en espacios de Hilbert se idearon originalmente como un marco matemático adecuado para describir la mecánica cuántica. En la matemática moderna, se ha ampliado su alcance debido a la gran versatilidad de las álgebras de operadores. Los temas de interés especial son el análisis de grupos y sus acciones. La promediabilidad («amenability» en inglés) es una propiedad de finitud que posee un gran número de formulaciones equivalentes. El proyecto financiado con fondos europeos AMAREC llevará a cabo un análisis de promediabilidad en términos de propiedades de aproximación en el contexto de las C*-álgebras abstractas, los sistemas dinámicos topológicos y los grupos discretos. Las propiedades de aproximación servirán como puente entre estas configuraciones y se utilizarán para recuperar sistemáticamente información geométrica sobre las estructuras subyacentes.
Objetivo
Algebras of operators on Hilbert spaces were originally introduced as the right framework for the mathematical description of quantum mechanics. In modern mathematics the scope has much broadened due to the highly versatile nature of operator algebras. They are particularly useful in the analysis of groups and their actions. Amenability is a finiteness property which occurs in many different contexts and which can be characterised in many different ways. We will analyse amenability in terms of approximation properties, in the frameworks of abstract C*-algebras, of topological dynamical systems, and of discrete groups. Such approximation properties will serve as bridging devices between these setups, and they will be used to systematically recover geometric information about the underlying structures. When passing from groups, and more generally from dynamical systems, to operator algebras, one loses information, but one gains new tools to isolate and analyse pertinent properties of the underlying structure. We will mostly be interested in the topological setting, and in the associated C*-algebras. Amenability of groups or of dynamical systems then translates into the completely positive approximation property. Systems of completely positive approximations store all the essential data about a C*-algebra, and sometimes one can arrange the systems so that one can directly read of such information. For transformation group C*-algebras, one can achieve this by using approximation properties of the underlying dynamics. To some extent one can even go back, and extract dynamical approximation properties from completely positive approximations of the C*-algebra. This interplay between approximation properties in topological dynamics and in noncommutative topology carries a surprisingly rich structure. It connects directly to the heart of the classification problem for nuclear C*-algebras on the one hand, and to central open questions on amenable dynamics on the other.
Ámbito científico
- natural sciencesmathematicspure mathematicsalgebralinear algebra
- natural sciencesphysical sciencesquantum physics
- natural sciencesmathematicspure mathematicstopology
- natural sciencesmathematicsapplied mathematicsdynamical systems
- natural sciencesmathematicspure mathematicsmathematical analysisfunctional analysisoperator algebra
Palabras clave
Programa(s)
Régimen de financiación
ERC-ADG - Advanced GrantInstitución de acogida
48149 MUENSTER
Alemania