- Matthew Lorentz, Michigan State University
- Derivations and the Hochschild Cohomology of Uniform Roe Algebras, Part 3
- 10/12/2021
- 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.
Hochschild cohomology was introduced by Gerhard Hochschild in his 1945 paper “On the Cohomology Groups of an Associative Algebra”. The Hochschild cohomology of associative algebras has become a useful object of study in many fields of mathematics such as representation theory, mathematical physics, and noncommutative geometry, to name a few.
Last time we showed that all bounded derivations on uniform Roe algebras are inner. The first Hochschild cohomology measures how close derivations are to being inner. Hence, our result from last time can be restated as the first Hochschild cohomology of the uniform Roe algebra vanishing. It is then natural to ask if the higher dimensional cohomologies also vanish.
We will begin with the definition and several properties of multilinear maps which are essential to building the Hochschild complex. We then define the Hochschild complex and Hochschild cohomology as they apply to multilinear maps from a C*-algebra A to a Banach A-bimodule V. We then review many properties of these cohomologies. Lastly, we will show that if all n-linear maps have a weakly continuous representation in the Hochschild cohomology then the n’th dimensional Hochschild cohomology vanishes.