How does one show that the transpose mapping $T: \mathcal{M}_{n} \to \mathcal{M}_{n}$, given by $T(a)=a^{t}$, has norm $||T||=1$ but completely bounded norm $||T||_{CB}=n$?
Notation. $\mathcal{M}_{n}$ denotes the space of $n \times n$ matrices.
Reference: Example 1.1.6 from http://arxiv.org/pdf/1410.7188v3.pdf
There's a proof included in the reference. However, I am interested in alternate ways to calculate the norms above.