On the cohomology of uniform Roe algebras with coefficients in uniform Roe modules

The uniform Roe algebra of a metric space is an important object of study in noncommutative geometry, reflecting the geometric properties of the corresponding space. It was recently shown by Lorentz and Willett that all derivations of the uniform Roe algebra are inner. We have also recently singled out an important class of uniform Roe modules, which are responsible for the mutual positioning of two copies of the corresponding metric space. It was a natural task to calculate the space of outer derivations of uniform Roe algebras with coefficients in these modules. It turned out that this space is almost always non-trivial. This also allowed us to show the nontriviality of odd cohomology with coefficients in these modules.