Group von Neumann algebras and crossed products for a locally compact group G can be constructed in many different ways. For example, one can take the von Neumann algebra generated by certain operators on a certain Hilbert space.
However, none of these constructions give an explicit description of elements of the group algebra or the crossed product.
I am looking for such an explicit description. I suspect that distributions with bounded Fourier transform might be involved, but I am not entirely sure about this.
I am also looking for a more abstract description of these constructions. Can we characterize the group von Neumann algebra and the crossed product by some universal property?
Any references on this matter will be appreciated.