In a $C^*$-algebra, does $b^*a^*ab = 0$ for all $b$ imply $a = 0$?

Question

If an element $a$ of a $C^*$-algebra $A$ satisfies $$(\forall b ∈ A)(b^*a^*ab = 0), $$ does it follow that $a = 0$?

Answer 1

Put $b=a^\ast$, we get $0=\|aa^\ast aa^\ast\| =\|(aa^\ast)^\ast(aa^\ast)\| =\|aa^\ast\|^2 =\|a\|^4$. Therefore $\|a\|=0$ and $a=0$.

Answer 2

The assumption implies that $ab=0$ for every $b.$ In particular we can take $b=e$ if the algebra contains a unit, or $b=a^*.$ In both cases we obtain $a=0.$