In relation to this question of mine: C* algebra inequalities
I am wondering if it is true that if $0\leq a \leq b$ in a C* algebra, does one have $||a||\leq||b||$? If you need the C* algebra to be unital you can assume that since I could use the C* algebra unitization. If it is not true, please state any reasonable conditions under which it is true because I see things like this used all the time, and it is possible that it merely applies in those special cases.