Can a $C^*$-algebra $A$ be a prime algebra?

Question

I'm reading Jordan derivation, I'd like to know

Can a $C^*$-algebra $A$ be a prime algebra?

my attempt: Let $A$ be $C^*$-algebra so that I have to prove, for $a, b\in A$ if $aAb=0$ then either $a=0$ or $b=0$, since $A$ has an approximate identity $(e_\alpha)$ , we get $a e_\alpha b=0$

I'm stuck here give me a hint to show either $a=0$ or $b=0$. Thanks in advance

Answer 1

This is not true in general. Consider $A=C([0,1])$, and define $f,g\in C([0,1])$ by $$f(t)=\max\left\{\frac12-t,0\right\},\qquad g(t)=\max\left\{t-\frac12,0\right\}.$$ Then $f\cdot C([0,1])\cdot g=0$, but $f\neq0$ and $g\neq 0$.