Projektbeschreibung
Die Suche nach mittelbaren Operatoren für Algebren
Algebren stetiger linearer Operatoren über Hilberträumen wurden ursprünglich als geeigneter mathematischer Rahmen konzipiert, um die Quantenmechanik zu beschreiben. In der modernen Mathematik hat sich ihr Anwendungsbereich ausgeweitet, da Operatoralgebren äußerst flexibler Natur sind. Von besonderem Interesse sind sie in der Analyse von Gruppen und ihren Wirkungen. Bei dem Begriff „mittelbar“ handelt es sich um eine Endlichkeitseigenschaft mit zahlreichen äquivalenten Formulierungen. Das EU-finanzierte Projekt AMAREC wird eine Analyse mittelbarer Gruppen durchführen, um die Approximationseigenschaften abstrakter C*-Algebren, topologischer dynamischer Systeme und diskreter Gruppen auszuloten. Approximationseigenschaften werden als Brücke zwischen diesen Gebilden dienen und verwendet werden, um systematisch Informationen über die zugrunde liegenden Strukturen zu gewinnen.
Ziel
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.
Wissenschaftliches Gebiet
- natural sciencesmathematicspure mathematicsalgebralinear algebra
- natural sciencesphysical sciencesquantum physics
- natural sciencesmathematicspure mathematicstopology
- natural sciencesmathematicsapplied mathematicsdynamical systems
- natural sciencesmathematicspure mathematicsmathematical analysisfunctional analysisoperator algebra
Schlüsselbegriffe
Programm/Programme
Thema/Themen
Finanzierungsplan
ERC-ADG - Advanced GrantGastgebende Einrichtung
48149 MUENSTER
Deutschland