The usual definition of complete positivity (e.g. Stinespring (1955), or Holevo's Statistical Structure of Quantum Theory) is that a linear map between (sub $C^*$ algebras of) the bounded operators on some Hilbert spaces $\phi:\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathcal{K})$ is $k$ positive if the map
\begin{align}
\left(\mathrm{id}_k\otimes\phi\right):\mathcal{L}(\mathbb{C}^{k})\otimes\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathbb{C}^{k})\otimes\mathcal{L}(\mathcal{K}),
\end{align}
you get by tensoring with the identity on $\mathcal{L}(\mathbb{C}^{k})$ is positive, and $\phi$ is *completely positive* if it is $k$ positive for all $k\in\mathbb{N}$.

My general question is why do we consider all *finite* dimensional auxillary spaces $\mathbb{C}^k$, rather than infinite dimensional dimensional spaces? In particular from the point of view of quantum information theory we use complete positivity to ensure we can "act locally" on our system of interest whilst the global state remains positive. It seems natural to want this to happen even if the global state is infinite dimensional. Note that this is only an interesting question if $\mathcal{H}$ and $\mathcal{K}$ are infinite dimensional.

For convenience I will call maps $\phi$ such that
\begin{align}
\left(\mathrm{id}_{\mathcal{S}}\otimes\phi\right):\mathcal{L}(\mathcal{S})\otimes\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathcal{S})\otimes\mathcal{L}(\mathcal{K}),
\end{align}
is positive for every Hilbert space $\mathcal{S}$, where $\mathrm{id}_{\mathcal{S}}$ is the identity on $\mathcal{L}(\mathcal{S})$, *extra completely positive*. The question may make more sense if we restrict this to separable $\mathcal{S}$.

It is relevant that the Stinespring factorisation is possible if, and only if the map is completely positive in the usual sense.

My specific questions are

- Is there already a name for, and work on, the extra completely positive maps in the literature? After some searching, and asking colleagues I have not found anything about them.
- Is there a $\phi$ which is completely positive but not extra completely positive? Conversely I would be very interested in a proof that all completely positive maps are extra completely positive. I have tried to come up with an example of the former, and a proof of the latter but with no success.
- If the extra completely positive maps are a proper subset of the completely positive maps is there a nice characterisation of them (e.g. a "Stinespring-esque" factorisation)?