"Identity tensor transpose" as a map $M_n \hat{\otimes} M_n \to M_n \overline{\otimes} M_n$

Well I guess I should write something quickly, even if it doesn't have all the details, otherwise I'll keep putting it off. And maybe someone will spot a mistake...

Fix Hilbert spaces $V$ and $W$, which we think of as having column OSS. Equip $B(V)$ and $B(W)$ with their usual OS structures. Then the linear map $$ \iota\otimes \top : B(V) \otimes B(W) \to B(V \otimes_2 W)$$ extends to a contractive linear map $B(V) \ptp B(W) \to B(V\otimes_2 W)$.

The proof goes in stages.

Step 1. If $x = \sum_i a_i \otimes b_i \in B(V)\otimes B(W)$, then $$ \Vert \sum\nolimits_i a_i \otimes b_i^\top \Vert_{B(V\otimes_2 W)} = \sup \left\{ \Vert \sum\nolimits_i a_i c b_i \Vert_{HS(W,V)} \;\colon\; c \in HS(W,V), \Vert c\Vert_{HS(W,V)} \leq 1 \right\} $$

(This is not hard to hack out by hand, but with hindsight can also be found in various places, for instance I think it is in Pisier's book somewhere early on.)

Step 2. Note that there are have natural completely contractive maps$\newcommand{\ptp}{\widehat{\otimes}}$ $$ B(V) \ptp V \to V\quad,\quad W^* \ptp B(W) \to W^* $$ where the first is the usual action $a\otimes v \mapsto av$ and the second is the transposed action $\psi\otimes b \mapsto \psi\circ L_b$, $L_b$ being the action $w\mapsto bw$. Therefore by general stuff on operator space tensor products, we have complete contractions$\newcommand{\itp}{\otimes_{\rm min}}$ $$ B(V) \ptp( V\ptp W^*) \ptp B(W) \to V\ptp W^* $$ $$ B(V) \ptp( V\itp W^*) \ptp B(W) \to V\itp W^* $$

where $\itp$ denotes injective tensor product of operator spaces. Note that if we identify $V\otimes W^*$ with the space of finite rank operators $W\to V$, then the two maps above just correspond to $a\otimes c \otimes b \mapsto acb$.

Step 3. Under the natural identification of $V\otimes W^*$ with the finite-rank operators $W\to V$, we have isometric isomorphisms $V\ptp W^*\cong S_1(W,V)$ and $V\itp W^* \cong S_\infty(W,V)$, where $S_1$ denotes trace-class operators and $S_\infty$ the compact operators.

Step 4. Let $x$ be as in Step 1 and WLOG assume its norm in $B(V)\ptp B(W)$ is $1$. Let $E_x$ denote the elementary operator on $B(W,V)$ defined by $c\mapsto \sum_i a_icb_i$. We wish to show that the norm of $E_x$ as a map $HS(W,V)\to HS(W,V)$ is $\leq 1$. But now by Steps 2 and 3, we know that $E_x$ is (completely) contractive from $S_1(W,V)$ to $S_1(W,V)$, and (completely) contractive from $S_\infty(W,V)$ to $S_\infty(W,V)$. By classical complex interpolation results, $E_x$ is therefore contractive on all the intermediate Schatten classes, in particular on the Hilbert-Schmidt operators, and we are done.

Remark. Of course we can interpolate in the category of operator spaces. The argument above, if correct, seems to actually show that $E_x$ is completely contractive on $HS(W,V)$ when we equip this space with the "self-dual" OSS. If I denote this operator space by OH temporarily, then we can rephrase this as: $\iota\otimes \top$ is contractive from $B(V)\ptp B(W)$ to $CB(OH)$. It seems plausible that we actually get a complete contraction, but I haven't yet done the book-keeping required to check this.