Abstract
We study the relation of mutual strong Birkhoff–James orthogonality in two classical $$C^*$$ -algebras: the $$C^*$$ -algebra $${\mathbb {B}}(H)$$ of all bounded linear operators on a complex Hilbert space H and the commutative, possibly nonunital, $$C^*$$ -algebra. With the help of the induced graph it is shown that this relation alone can characterize right invertible elements. Moreover, in the case of commutative unital $$C^*$$ -algebras, it can detect the existence of a point with a countable local basis in the corresponding compact Hausdorff space.
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