Product of ideals in a $C^*$-algebra coincides with the intersection

Question

Let $A$ be a $C^*$-algebra and let $I_1,I_2$ be two-sided closed ideals of $A$. I want to show that $I_1I_2=I_1\cap I_2$. I can do this with approximate units when assuming $I_1I_2$ is closed. But I am not sure why it is closed...

Edit: Maybe I should take the closure in that case. However, I have seen this is true when $I_1=I_2=I$, and maybe when they're equal it suffices to show that each positive element in $I$ belongs to $I^2$ using continuous functional calculus and the fact that $I$ is a $C^*$ algebra. Right?

Thanks

Answer 1

As far as I can tell, $I_1I_2$ is defined as the closed linear span of $I_1$ and $I_2$. That's certainly how it's done in Murphy's book.