Equipping $M\_n$ with its usual operator space structure,
$\newcommand{\ptp}{\widehat{\otimes}}$
 we can form the projective tensor product of operator spaces $M\_n\ptp M\_n$. In particular this puts a Banach space norm on the algebraic tensor product $M\_n\otimes M\_n$.

Now consider the transpose map $T:M\_n \to M\_n$. It is a standard calculation to show that $T$ is not completely bounded,
$\newcommand{\stp}{\overline{\otimes}}$
 and in particular one can show that the map $\iota \otimes T : M\_n \stp M\_N \to M\_n\stp M\_n$ has norm $n$, where $\stp$ denotes the spatial tensor product (in this setting the same as the injective tensor product of operator spaces).

> 
> **Question 1.** What is the asymptotic behaviour (as $n\to \infty$) of $\Vert \iota \otimes T : M\_n \ptp M\_N \to M\_n\stp M\_n\Vert$?
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> 
> 

Note that I'm asking merely about the norm as a map between two Banach spaces, not about the cb norm. If precise asymptotics are tricky, how about the following sub-question:

> 
> **Question 2.** In particular, does the norm of this map tend to infinity as $n\to\infty$?
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> 
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This feels like something that should follow by tweaking a standard example or exercise in one of the usual books on Operator Spaces, but I couldn't succeed in converting the usual examples to get something that answers the question above.

**Remark:** the usual way to get a lower bound on
$\Vert\iota \otimes T : M\_n \stp M\_N \to M\_n\stp M\_n \Vert$ is to consider what this map does to the tensor $x = \sum\_{i,j=1}^n E\_{ij} \otimes E\_{ji}$, the point being that $x$ has norm $1$ when viewed as an element of $M\_{n^2}$ while $(\iota\otimes T)(x)$ has norm $n$ as an element of $M\_{n^2}$. However, since matrix multiplication gives a complete contraction $M\_n \ptp M\_n \to M\_n$, I think it can be shown that $x$ has norm $n$ as an element of $M\_n\ptp M\_n$.

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**Update 2016-05-04:** I think I've now found a proof that this map is contractive for all $n$,
which moreover works if you replace proj tp with Haagerup tp. Previously I thought that this stronger claim (with the Haagerup tp) was false by adapting the usual argument to show the claim fails for min tp; however, this was based on a stupid miscalculation.
based on an interpolation argument. If the details work then I'll leave them as an answer.