Choi--Jamio{\l}kowski-type isomorphisms for von Neumann algebras
The Choi--Jamio{\l}kowski isomorphism identifies completely positive maps with bipartite states and underlies much of finite-dimensional quantum information theory. For systems with infinitely many degrees of freedom, modelled by von Neumann algebras of type III, neither traces nor density matrices are available, and the isomorphism has to be reformulated. We show that for arbitrary von Neumann algebras $\mathcal{M}$ and $\mathcal{N}$ there is a canonical order isomorphism between the space of normal completely bounded maps from $\mathcal{M}$ into the predual of $\mathcal{N}$ and the predual of the spatial tensor product of $\mathcal{M}$ with the \emph{opposite} algebra of $\mathcal{N}$; under this identification complete positivity corresponds to positivity. Replacing the opposite algebra by $\mathcal{N}$ itself requires an anti-isomorphism of $\mathcal{N}$ with itself, and we prove that this condition is not only sufficient but also necessary, provided the identification is required to be natural in $\mathcal{M}$. Some such requirement is unavoidable, since for every $\mathcal{M}$ anti-isomorphic to itself an isomorphism exists for trivial reasons. Consequently no Choi--Jamio{\l}kowski correspondence exists for the type III factors constructed by Connes. Along the way we show that the space of all normal maps, taken with the operator norm, is strictly too large for this purpose, and that complete positivity does not force complete boundedness in this setting.