# Calculating Norms for the Transpose

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.

• It would be nice to avoid external references and instead include the proof for which you want to have an alternative. Also, since there are many norms, it might be helpful to define which norms do you consider. Jun 11 '15 at 2:14