Noncommutative covering dimension for $C^*$-algebras and dynamical systems

Joachim Zacharias  (University of Glasgow)

14:00 pm to 15:00 pm, May 11th, 2015    Science Building A1510

Abstract:

Various noncommutative generalisations of dimension have been considered and studied in the past decades. Recently nuclear dimension, a new dimension concepts for noncommutative nuclear $C^*$-algebras, has been introduced based on previous work in this direction. The basic idea is that open covers may be regarded as approximations of spaces but also encode the covering dimension of a space. Thus nuclear dimension relates covering dimension and approximations. It has turned out to be important in the classification programme of simple nuclear C*-algebras. More recently, Rokhlin dimension, a closely related dimension concept for dynamical systems, has been introduced. In case of actions on spaces it corresponds to a kind of equivariant covering dimension. Actions with finite Rokhlin dimension preserve finiteness of nuclear dimension. There are interesting connections between coarse geometry and Rokhlin dimension. We will give an introduction to these concepts and survey some applications and connections between them.

This survey is based on work in collaboration with Hirshberg, Szabo, Winter, Wu and further recent developments.

Joachim Zacharias is a reader at University of Glasgow.

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