An inequality concerning positive elements in a C* algebra

Question

Let $A$ be a complex unital C*- algebra. Let $a\in A$ be a self adjoint element. Let $|a|$ denote $\sqrt{a^*a}$. Can any one tell how the below inequality is true? $$0\leq (|a|-a)^2\leq (2|a|)^2$$

Answer 1

This fails even in the case $A=\mathbb C$ (and thus in every unital C$^*$-algebra), by taking $a=-1$.

Answer 2

Hint: If $A = C(X)$ for some compact Hausdorff space, then the inequality is obvious.

If $A$ is a general unital $C^*$-algebra, note that the $C^*$-algebra generated by $a$ is commutative, i.e. $C^*(1,a)\cong C(X)$ for some compact Hausforff space which reduces the question to the commutative case.