In my research of operator algebras and their connection with machine learning I of course use the well know result:



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> For the map $ tr:M\_n \to M\_n $ denoting the transpose map of matrices (meaning that $ tr(A)=A^{tr} $ we know for the completely bounded norm (cb-norm) that $ ||tr||\_{cb}=n $ 
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I keep using this result freely but to be honest it had just occurred to me that I know not a proof of this nor do I have a reference for the proof or an idea for proving it that would actually work, could operator algebraists out there please help in providing a proof or a reference to a nice (or any...) functioning proof of this useful fact?



A nice proof that completely bounded (cb) norm of transpose map on $ M_n $ is n
I literally just googled "cb norm of transpose" and got a link to [this math.stackexchange question](https://math.stackexchange.com/questions/1320435/calculating-norms-for-the-transpose) which links to a proof.


Edit: the answer which contained the link disappeared, so I used Google magic again and found [Tomiyama's original paper](http://www.jstor.org/stable/2045453?seq=1#page_scan_tab_contents) with this calculation.


