# If $I$ is a prime ideal in a $C^*$-algebra $A$ and $S_1AS_2 \subseteq I$, then either $S_1 \subseteq I$ or $S_2 \subseteq I$

Consider the following fragment from Murphy's "$$C^*$$-algebras and operator theory":

Can someone explain why we have $$S_j \subseteq A S_j A$$?

I can prove this if $$S_j$$ is a sub $$C^*$$-algebra of $$A$$ or if $$A$$ is unital.

Attempt:

Let $$(u_\lambda)$$ be an approximate unit for $$A$$. If $$x\in S_j$$, then somehow we should be able to write $$x$$ as a norm-limit of some net in $$AS_jA$$. Maybe we can prove something like $$x=\lim_\lambda u_\lambda^{1/2} x u_\lambda^{1/2}$$

But I don't see why that should hold.

Since $$\|u_\lambda\|\leq1$$, as long as we approximate on the left and write simultaneously, we obtain $$x=\lim_\lambda u_\lambda xu_\lambda$$ for all $$x$$. Indeed, fix $$x\in A$$ and $$\varepsilon>0$$. There exists $$\lambda_0$$ such that $$\|xu_\lambda-x\|<\varepsilon$$ and $$\|u_\lambda x-x\|<\varepsilon$$ for $$\lambda\geq \lambda_0$$. For such $$\lambda$$, we have $$\|u_\lambda xu_\lambda-x\|\leq\|u_\lambda\|\|xu_\lambda-x\|+\|u_\lambda x-x\|<2\varepsilon.$$