# Completely bounded maps on Mn

The aim of this question is to collect nice maps on $M_n(\mathbb{C})$ with the following property:

$\phi_n:M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})$ with $||\phi||=1$ and $||\phi_n||_{cb}\rightarrow \infty$ when $n\rightarrow \infty$

Here $||\phi_n||_{cb}$ denotes completely bounded norm of $\phi_n$, i.e. in this special case $||\phi_n||_{cb}=||\phi_{n}^{(n)}||$, where $\phi_{n}^{(n)}:M_n(\mathbb{C})\otimes M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})\otimes M_n(\mathbb{C})$ such that $\phi_{n}^{(n)}([a_{ij}])=[\phi_n(a_{ij})]$ for every $[a_{ij}]\in M_n(M_n(\mathbb{C}))$.

There are many examples of such maps, the simplest one is the transposition of matrix.

• Bill: I think one can manage without assuming that spaces are exact. The proper analog for the question above is to consider spaces which are N-K-exact. Here N-K-exact space means $\phi:E\rightarrow F_n\subseteq M_n$, $\psi:F_n\rightarrow E$ has the N-th bound on the norm: $||\phi^{(N)}||\cdot ||\psi^{(N)}||\leq K$. – Kate Juschenko Apr 29 '11 at 7:46
It is well known that each Schur multiplier satisfies $||.||_{cb}=||.||$. These maps are the $D_n$-bimodule maps on $M_n$. For the right module maps, the situation is more complicated. This gives concretes exemples of maps with your property. See the following paper: