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The goal of this short note is to prove that when $A$ is a closed*-subalgebra of a C*-algebra $B$ satisfying the ideal intersection propertyplus a mild axiom (INV), then the map $J\mapsto J\cap A$ establishes anisomorphism from the boolean algebra of all regular ideals of $B$ to theboolean algebra of all regular, invariant ideals of $A$.

Regular ideals under the ideal intersection property