Isoradial embeddings and non-commutative geometry

Abstract: In this talk, we describe a framework to study non-commutative geometry as a relative version of differential geometry. More precisely, given a C*-algebra A, we would like to make sense of a "smooth" subalgebra $A^\infty \subseteq A$, and deduce properties about A using such a subalgebra. Such a smooth subalgebra should be analogous to the Frechet algebra $C^\infty(M) \subseteq C(M)$ for a smooth manifold M, in the world of commutative C*-algebras. We shall review the fundamental properties and applications of such embeddings, called $\textit{isoradial embeddings}$, due to Ralf Meyer. If time permits, I will mention an ongoing research program with Meyer, Corti\~nas and Cuntz, that uses such embeddings to develop noncommutative geometry over finite fields.

I will not assume that the audience has any background beyond familiar examples of C*-algebras. A lot of the motivation would however be clearer to those familiar with cyclic homology or operator algebraic K-theory.

functional analysisK-theory and homologyoperator algebras

Audience: researchers in the topic

Webinars on Operator Theory and Operator Algebras