the norm of positive elements in $C^*$-algebra

Question

Suppose $x,y$ are two positive elements in a unital $C^*$-algebra,if $x\leq y$,then $\|x\| \leq \|y\|$.

If $x\geq 0,y\geq 0$,$\|x\| \leq \|y\|$,can we conclude that $x\leq y$?

Answer 1

No. This is not true even if the algebra is commutative. Think of two continuous positive functions $f$ and $g$ on $[0,1]$. For example $f(x)=1$ (const) and $g(x) = 4x$ for $x\leq 0.5$ and $g(x)=-4(x-1)$ for $x\geq 0.5$. Of course $g(0.5) = 2$, so for the supremums norms $\left\lVert g\right\rVert>\left\lVert f\right\rVert$, but $g\geq f$ is NOT true.