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Invertibility modulo the intersection of ideals in $C^*$-algebras

I am interested in the invertibility of $A$ modulo certain (two-sided, closed) ideals $\mathcal{J} \subseteq \mathcal{A}$, i.e. the existence of a generalized inverse $B \in \mathcal{A}$ such that $$AB … My question now concerns intersections of ideals. …
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