Let $A$ be a
$C^*$-algebra. The Stinespring construction shows that a completely positive contraction $T:A\rightarrow B(H)$ has the form
$T(x) = U^* \pi(x) U$ where $U:H\rightarrow K$ is a contraction, and $\pi:A\rightarrow B(K)$ is a
$*$-homomorphism. The Paulsen 2x2 matrix trick allows us to characterise complete contractions similarly, by replacing
$U^*$ with $V$, another contraction.
Similarly, if $F$ is a Hilbert
$C^*$-module over $B$ then a strict completely positive contraction $T:A\rightarrow L(F)$ is of the form
$T(x) = U^* \pi(x) U$ where now $\pi:A\rightarrow L(G)$ is a
$*$-homomorphism, $G$ being some Hilbert
$C^*$-module over $B$, and $U\in L(F,G)$ is a contraction. (Recall, from e.g. Lance's book on Hilbert
$C^*$-modules, that "strict" means that $(T(e_i))$ is a Cauchy net in the strict topology on $L(F)$, for some approximate unit $(e_i)$ for $A$. So, if you wish, you can just restrict to unital algebras, and then "strict" is a vacuous condition).
It seems to me that: (a) The Paulsen 2x2 trick should also work here (although I haven't thought very long about this at all, because...) (b) if so, surely someone has already written this down (although a bit of MathSciNet work didn't find anything). So my question is:
Is there a known characterisation of strict complete contractions? And what's a good reference?
Edit: Thinking more about, the Paulsen 2x2 matrix technique uses Arveson's extension theorem-- and this is likely to be somewhat more delicate in the Hilbert module / adjointable operator setting. Still, I'd be interested if anything is known...