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.