Concrete representation of the annihilating algebra Suppose $\mathfrak{M} = A^{\prime\prime}$, where $A$ is a concretely described subalgebra of $\mathcal{B}(\ell^{2}(\mathbb{N}))$. In some instances, it is possible to provide a concrete description of the Operators in $A''$. Is the same possible for $A^{\prime}$?
Consider, e. g. $G$ a fixed group, and view $L(G)$ as acting on $\mathcal{B}(\ell^{2}(G))$. Can the operators in $L(G)^{\prime}$ be more concretely described (other than simply commuting with every $L_{g}$ for $g\in G$ --- where $L_{g} : \mathbf{e}_{h} \mapsto \mathbf{e}_{gh}$ for each $g, h\in G$)?

The case you mention is one of the very few where the commutant can be expressed explicitly.
Namely,
$
L(G)'=R(G), $ where
$R(G) $ is the von Neumann algebra of the right regular representation $R_g:e_h\mapsto e_{hg} $.
