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.
Best Answer
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.