Universal characterization and explicit description of elements of the group von Neumann algebra and the crossed product 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.
 A: Okay, this is very far from being a complete answer.  But...
Firstly, if you are interested in $\mathbb R$ acting on $M$ by the modular automorphism group (which is an incredibly interesting object!) then you're getting dangerously close to trying to understand all the Type III factors.  Compare with Takesaki's "solution" to this problem (see Takesaki volume 2, or his paper in Acta, 1973.
Another place to look is two (IMHO much overlooked) papers by Haagerup, "On the dual weights for crossed products of von Neumann algebras. I." and "II." in Math. Scand. 43 (1978/79).  In particular, Section 2 of the 2nd paper is an excellent resource if you are interested in the weight on the group von Neumann algebra.
But, I doubt this will answer your original question.  Maybe a real expert will wonder by...
A: I am not sure of what you want, but let me offer a few ideas.
First, let me give you a silly answer: by the bicommutant theorem, if $M\subset B(H)$ is a von Neumann algebra, then $M = (M') '$.  Now pick some subset $F$ of $M'$ that generates it as a $W^*$-algebra.  Then a description of $M$ is of course $\{T\in B(H): Tx = xT \forall x\in F\}$.
There are, however, some instances where a more "explicit" description is available.  This is the case, for example, for von Neumann algebras related to lattices in $PSL_2(\mathbb{R})$ via Berezin quantization (the symbol is required to be equivariant with respect to the group action).  You can find more details in some papers of Radulescu about this.
Another situation where a description is available is in the case of a II$_\infty$ (or type III) factor associated to a foliation; in this case an element of the algebra ends up being a measurable field of bounded operators on Hilbert spaces associated to the leaves of the foliation.  This is described in Connes' non-commutative geometry book.
