I know that if $\Lambda$ is a stochastic positive linear map, i.e., $\Lambda(I) = I$, it is true that


`\[ \|\Lambda(B)\| \leq \| B \| \]`


For any operator $B$, where $\|\cdot\|$ is the standard operator norm $\|B\| := \max\_{|v|= 1} |Bv|$. Is it true for any other $p$-norm? Specifically, I want to prove it for the $2$-norm


`\[ \|B\|^2_2 := \operatorname{tr} (B^*B)\]`


also known as Hilbert-Schmidt norm, and I'm only interested in self-adjoint operators.


Naturaly, this question only makes sense if these operators have well-defined norms, so $\Lambda$ can be taken to act in this subalgebra.


It would be nice if the infinite-dimensional case could be done, but the main focus is on the finite-dimensional case.



Norm inequality for stochastic maps
Positivity is not enough (complete positivity is). 


Indeed, in $M\_2(\mathbb{C})$, let $\Lambda\left(\left[\begin{array}{cc}a&b \\\\ c&d\end{array}\right]\right)=\left[\begin{array}{cc}a&0 \\\\ 0&a\end{array}\right]$. Then $\Lambda$ is positive and $\Lambda(I)=I$; but if $B=\left[\begin{array}{cc}1&0 \\\\ 0&0\end{array}\right]$ we have, for any $1\leq p<\infty$, 
$\|B\|\_p=1$, $\|\Lambda(B)\|\_p=2^{1/p}$.


For a unital completely positive map, Stinespring leads to an easy proof that the inequality holds for $p=2$. For other $p$ it looks trickier, but I think it might work too. 


