There are a few cases that $L^2$-homology (cohomology) that can be introduced. For example, for a manifold, it can be defined in the same way as de Rham cohomology using square integrable differential forms. However, is it possible to define it in a way analogous to homology with (abelian) coefficients, for example, we take the ordinary chain complex (e.g., simplicial or cellular) and tensor with a Hilbert space with $L^2$ functions with homology defined by kernel/closure of image? It is not easy to find such a construction in literature, is it because for a finite simplical complex, such a construction does not give anything interesting or are there any other intrinsic problems?
By the way, I noticed the construction of $L^2$-invariants (e.g., Lück, Wolfgang (2002). L2-invariants: theory and applications to geometry and K-theory.), but it is still much more complicated than the naive approach I described above where group von Neumann algebra plays an essential role.
