Project description
Exploring amenable operators of algebras
Algebras of continuous linear operators on Hilbert spaces were originally devised as a suitable mathematical framework for describing quantum mechanics. In modern mathematics, the scope has broadened due to the highly versatile nature of operator algebras. Topics of particular interest include the analysis of groups and their actions. Amenability is a finiteness property that has a large number of equivalent formulations. The EU-funded AMAREC project will conduct an analysis of amenability in terms of approximation properties in the context of abstract C*-algebras, topological dynamical systems and discrete groups. Approximation properties will serve as a bridge between these setups and will be used to systematically recover geometric information about the underlying structures.
Objective
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.
Fields of science
- natural sciencesmathematicspure mathematicsalgebralinear algebra
- natural sciencesphysical sciencesquantum physics
- natural sciencesmathematicspure mathematicstopology
- natural sciencesmathematicsapplied mathematicsdynamical systems
- natural sciencesmathematicspure mathematicsmathematical analysisfunctional analysisoperator algebra
Keywords
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Funding Scheme
ERC-ADG - Advanced GrantHost institution
48149 MUENSTER
Germany