# Norm inequality for stochastic maps

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.

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I'm confused about your first claim. If stochastic just means that it maps the identity to the identity then this does not imply that $\Lambda$ has norm 1. On the 2x2 matrices consider, $\Lambda$= $2(x)-Tr(x)I$. Where Tr is the normalized trace. This sends I to I but has norm larger than 2. You're statement requires that $\Lambda$ be a positive operator. Under this situation (that $\Lambda$ is positive) then this should still work for all p-norms. – Owen Sizemore Sep 13 '10 at 21:05
Yes! Sorry, I've been working only with positive maps, so that I forgot to specify. – Mateus Araújo Sep 13 '10 at 21:33
Could you also clarify what $\Lambda$ is acting on. Is it all of B(H)? Or is it some subalgebra?. The reason I ask this is to clarify what you mean by the 2-norm. Do you mean the Hilbert-Schmidt norm? If so this means that we must restrict to those B with finite Hilbert Schmidt norm. – Owen Sizemore Sep 14 '10 at 1:02
Reworded question. Thanks. – Mateus Araújo Sep 14 '10 at 1:26

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.