Preduals of von Neumann algebras consisting of vector functionals. It is known that for a locally compact group $G$ the predual $M_{*}$ of the von Neumann algebra $M=vN(G)$ consists of vector functionals, i.e., if $\omega\in M_{*}$ then there exist vectors $\xi,\,\eta\in L^2(G)$ such that $\omega(x)=\langle{x\,\xi,\eta\rangle},\,\forall\,x\in M.$
So, a natural question is "which von Neumann algebras have this predual property?"
I found that hereditarily reflexive von Neumann algebras $M$ (i.e., every w*-closed subspace of $M$ is reflexive) and separably acting von Neumann algebras with a seperating vector (equivalently, $M^\prime$ has cyclic vector) have this property.
Does anyone else know something more? Thank you.
 A: First note that this property depends not only on the abstract von Neumann algebra (or $W^\ast$-algebra if you like), but the chosen representation.
I don't know where you found that notion of hereditarily reflexive von Neumann algebra, but it is not a useful one. Every reflexive von Neumann algebra is finite-dimensional, in which case of course every subspace is also reflexive. And even finite-dimensional von Neumann algebras don't necessarily have this property: If you look at $M_n(\mathbb C)$ in its defining representation on $\mathbb C^n$, then the trace is not a vector functional.
The property that every element of the predual is a vector functional holds if the von Neumann algebra is in standard form (see Haagerup. The standard form of von Neumann algebras). The standard representation can be obtained as GNS representation for any faithful normal state (or weight). For example, the group von Neumann algebra $\mathrm{vN}(G)$ is acting standardly on $L^2(G)$ because this action is (unitarily equivalent to) the GNS representation with respect to the Haar weight.
