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$\newcommand\calH{\mathcal H} \newcommand\calK{\mathcal K} \newcommand\tr{\operatorname{Tr}}$I am looking for a (citable) reference for the following fact:

  • Bounded linear maps $g:T(\calH)\to T(\calK)$ (bounded w.r.t. trace norm) stand in 1-1 correspondence with normal bounded linear maps $f:B(\calK)\to B(\calH)$ via $\tr g(\rho)a=\tr \rho f(a)$ for all $\rho,a$. (Here $T(\calH)$ are trace-class operators and $B(\calH)$ are bounded operators. And $\calH,\calK$ are Hilbert spaces.)
  • And then $f$ is completely positive iff $g$ is.

This is mentioned as a standard result e.g. in this comment. I have seen $f$ referred to as the Schrödinger-Heisenberg dual of $g$ but all references were for finite-dimensional Hilbert spaces $\calH,\calK$.

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  • $\begingroup$ I'm no expert on this, but this correspondence looks suspicious to me. Suppose $\mathcal H$ is 1-dimensional and $\mathcal K=\ell_2$. Then your $g$ will just amount to an element of $\ell_2$, while your $f$ amounts a linear functional on $B(\ell_2)$. The latter can be wild, for example sending any bounded linear operator $X$ on $\ell_2$ to the limit, along your favorite non-principal ultrafilter, of $(Xe_n,e_n)$ where the $e_n$ form an orthonormal base for $\ell_2$. $\endgroup$
    – Andreas Blass
    Commented Jun 16, 2021 at 20:13
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    $\begingroup$ @AndreasBlass the important word is "normal", which (I guess) means continuous for the weak-* topologies. $\endgroup$
    – Mikael de la Salle
    Commented Jun 16, 2021 at 23:56
  • $\begingroup$ Yes, "normal" means weak*-continuous. Equivalently, "normal" means that SOT-limits of increasing nets are preserved. (Conway, A course in operator theory, Def 46.1) In particular, normal linear functionals on von-Neumann algebras (as in @AndreasBlass example) are of the form $a\mapsto\mathrm{tr}\, ab$ (Conway, Thm 46.4) which would exclude "wild" $f$'s. $\endgroup$
    – Dominique Unruh
    Commented Jun 17, 2021 at 9:31
  • $\begingroup$ @DominiqueUnruh Unfortunately I don't have the time to write a proper answer, and probably won't for quite some time. What I had in mind is what is in section IV.2 of Schaefer's Topological Vector Spaces: If $E_1,F_1$ are in a dual pairing and so are $E_2,F_2$, then a linear map $f : E_1 \rightarrow E_2$ is weak-* continuous (i.e. continuous from $\sigma(E_1,F_1)$ to $\sigma(E_2,F_2)$) iff there is a linear map $g : F_2 \rightarrow F_1$ that is an adjoint with respect to the pairings. $\endgroup$
    – Robert Furber
    Commented Jun 18, 2021 at 20:46
  • $\begingroup$ It follows using some reasoning with the bipolar theorem that if, for each $i \in \{1,2\}$, $F_i$ is a Banach space and $E_i = F_i^*$ (the continuous dual space) then the adjoint mapping $g : F_2 \rightarrow F_1$ is bounded. If you want a reference from Schaefer, see IV.7.4, and use the fact that Banach spaces are Mackey, but it's actually easier to prove the special case of Banach spaces directly. $\endgroup$
    – Robert Furber
    Commented Jun 18, 2021 at 20:58

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