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.

simpleexample? E.g. would you be interested in: Let M be a vn alg on H, with an action of $\mathbb R$. Then we can form the cross-product, which lives on the Hilbert space $L^2(\mathbb R,H)$. Do you want to understand what an element in the resulting algebra looks like? If M is the complex numbers, then you just get the group von Neumann algebra VN(G). The predual of this is the Fourier algebra A(G): you can indeed then think of VN(G) as being "distributions" on G, but I'd have to think to find the details... $\endgroup$