Order-preserving operator norms Let us regard the $n\times n$ matrices as operators on the $n$-dimensional $\ell_p$ space; that is, we consider them as linear operators $\ell_p^n\to \ell_p^n$. When $p=2$, $M_n$ is a C*-algebra and we have 
$0 \leqslant A \leqslant B \implies \|A\|\leqslant \|B\|$.
Here $\|A\|$ denotes the operator norm of a map $A\colon \ell_2^n\to \ell_2^n$. What about  other $p\in [1,\infty]$? 

Fix $p\in [1,\infty]$. Is it true that there exists $K>0$ such that for every $n$ and for all $A,B\colon \ell_p^n\to \ell_p^n$ with $0\leqslant A\leqslant B$ (meaning that $A$ and $B$ are self-adjoint and non-negative semi-definite) we have $$\|A\|_{\ell_p^n\to\ell_p^n}\leqslant K\|B\|_{\ell_p^n\to\ell_p^n}?$$

My feeling is that it should be true for $p\in (1,\infty)$. For $p=1$ or $p=\infty$ there is an easy counter-example with $K=1$. Take 
$$A=\left[\begin{smallmatrix}2&1\\1&  \tfrac{1}{2}\end{smallmatrix}\right],\;\;B = \left[\begin{smallmatrix}\tfrac{5}{2}&0\\0&  \tfrac{5}{2}\end{smallmatrix}\right].
$$
Then $0\leqslant A\leqslant B$ yet for $p\in \{1,\infty\}$ we have $\|A\|_{\ell_p^2\to\ell_p^2} = 3$ whereas $\|B\|_{\ell_p^2\to\ell_p^2}=\tfrac{5}{2}$.
In the language of this thread: Monotone matrix norms, I ask whether the operator $\ell_p$-norms are monotone, that is, if we can take $K=1$. user147215 cleverly shows that this is not the case when $p\neq 2$.
Possible approach: It is not inconceivable that using some Riesz–Thorin-type argument we could show that the operator $\ell_p$-norms are indeed monotone for $p$ is some neighbourhood of 2.
 A: The answer is no, Tomek.  Use a Kashin decomposition of $L_p^n$, $1\le p < 2$, to see that there are orthogonal projections whose norms as operators on $L_p^n$ are of order $Cn^{|1/p-1/2|}$.  (Kashin proved that for $1\le p < 2$ there is an orthogonal decomposition $A+B$  of $n$-space s.t. if $x \in A \cup B$, then $\|x\|_p \le  \|x\|_2 \le C\|x\|_p$, where I am using the uniform probability measure on $\{1,\dots,n\}$ rather than counting measure to define the norms.) 
The case $p>2$ follows by duality.  
B. S. Kashin, Sections of some
  finite-dimensional bodies and classes of smooth functions. 
  Izv. Acad. Nauk SSSR41 (1997), 334--351.
A: The original version of the question, which asked for $\|A\|_{\ell_p^n\to\ell_p^n}\leqslant K\|B\|_{\ell_p^n\to\ell_p^n}$ with $K = K(p)$, was more interesting (and I don't have an answer to that one yet). 
But if you insist on $K=1$, then the inequality fails for all $p\ne 2$. Let $n=3$, $$A=\begin{pmatrix} 2/3 & -1/3 & -1/3 \\ -1/3 & 2/3 & -1/3 \\ -1/3 & -1/3 & 2/3 \end{pmatrix},\qquad B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
That is, $B=I$ and $A=I-P$ where $P$ is the orthogonal projection onto the line $x_1=x_2=x_3$. What $A$ does is subtract the mean of $x_i$ from each coordinate. 
Subtracting the mean does not increase the $\ell_2$ norm of a vector, but it may increase $\ell_p$ for any other $p$. I will  show that $\|A\|_{\ell_p^n\to\ell_p^n}>1= \|B\|_{\ell_p^n\to\ell_p^n}$.
Fix $p\in (1,\infty)\setminus \{2\}$ and let  $x = (2^{1/(p-1)},-1,-1)^T$. For $p\in (1,\infty)$ the function 
$$\phi(t) = \sum_i |x_i-t|^p$$
is strictly convex, and attains its minimum when $\phi'(t)=0$, namely at the point with 
$$ \sum_i |x_i-t|^{p-1}\operatorname{sign}(x_i-t)=0 \tag{1}$$
For $x$ as above, (1) is satisfied with $t=0$. Therefore, $\phi(t)>\|x\|_p^p$ for all $t\ne 0$. Applying $A$ amounts to replacing each $x_i$ with $x_i-t$, where $t= (2^{1/(p-1)}-2)/3$. When $p\ne 2$, we have $t \ne 0$, hence $\|Ax\|_p> \|x\|_p$.
