Topological equivariant coarse K-homology

For a C*-category with a strict G-action we construct examples of equivariant coarse homology theories. To this end we first introduce versions of Roe cate-gories of objects in C*-categories which are controlled over bornological coarse spaces, and then apply a homological functor. These equivariant coarse homology theories are then employed to verify that certain functors on the orbit category are CP-functors. This fact has consequences for the injectivity of assembly maps.