Alladi Ramakrishnan Hall

Coarse geometry for noncommutative spaces

Tathagata Banerjee

Goettingen

In the talk we shall mainly be interested in looking at quantization using Rieffel
deformation by actions of $\mathbb{R}^d$ through coarse geometry. Coarse geometry is
concerned with the large scale aspect of the topology of a space and we know that
quantum physics is equivalent to classical physics when considered at large
distances. We define noncommutative coarse structures on a non-unital
$C^\ast$-algebra in terms of its unitizations and develop a particular notion of a
noncommutative coarse map. Coarse structures given by proper metrics are the most
interesting examples of classical coarse geometry, and can be equivalently defined
in terms of its Higson compactification. We use this and our theory of
noncommutative coarse geometry to show equivalence between the metric coarse
structure of the classical plane $\mathbb{R}^{2n}$ and its Rieffel deformation, the
Moyal plane. Finally if time permits, we shall look into further examples of
noncommutative coarse maps and equivalences. This has been my Ph.D. project under
Prof. Ralf Meyer.