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Amenability, Approximation and Reconstruction

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.

Field of science

  • /natural sciences/physical sciences/quantum physics
  • /natural sciences/mathematics/pure mathematics/algebra/linear algebra
  • /natural sciences/mathematics/applied mathematics/dynamical systems

Call for proposal

ERC-2018-ADG
See other projects for this call

Funding Scheme

ERC-ADG - Advanced Grant

Host institution

WESTFAELISCHE WILHELMS-UNIVERSITAET MUENSTER
Address
Schlossplatz 2
48149 Muenster
Germany
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 596 017

Beneficiaries (1)

WESTFAELISCHE WILHELMS-UNIVERSITAET MUENSTER
Germany
EU contribution
€ 1 596 017
Address
Schlossplatz 2
48149 Muenster
Activity type
Higher or Secondary Education Establishments