## Mathematical Physics and Operator Algebras

•  Matthew Lorentz, Michigan State University
•  Derivations and the Hochschild Cohomology of Uniform Roe Algebras, Part 1
•  09/14/2021
•  11:10 AM - 12:00 PM
•  C304 Wells Hall
•  Brent Nelson (banelson@msu.edu)

In this series of talks we show a necessary and sufficient condition for the vanishing of the Hochschild cohomology of a uniform Roe algebra. Specifically, the n-dimensional continuous Hochschild cohomology vanishes if and only if every norm continuous n-linear map from the uniform Roe algebra to itself is equivalent to a weakly continuous n-linear map. In our first talk we will begin defining and discussing derivations as they are an important building block of Hochschild cohomology. Motivated by the needs of mathematical physics and the study of one-parameter automorphism groups, it is interesting to study whether all derivations are inner (i.e. given by the commutator bracket) for a particular C*-algebra. In the 1970s, a complete solution to this problem was obtained in the separable case via the work of several authors. For non-separable C*-algebras the picture is murkier. Our main goal in this talk is to give a new class of examples that only have inner derivations: uniform Roe algebras, which are separable only in the trivial finite dimensional case. Uniform Roe algebras were originally introduced for index-theoretic purposes but are now studied for their own sake as a bridge between C*-algebra theory and coarse geometry, as well as having interesting applications to single operator theory. Lastly, we will briefly explain how the uniform Roe algebra only having inner derivations is equivalent to the first Hochschild cohomology vanishing.

## Contact

Department of Mathematics
Michigan State University