# Hereditary C*-subalgebras of the unitization of a hereditary C*-subalgebra

Let $$A$$ be a unital C*-algebra with unit $$1$$, and $$B$$ a non-unital hereditary C*-subalgebra of $$A$$. Suppose $$J$$ is a proper hereditary C*-subalgebra of $$B\oplus C1$$ (the unitization of B). Is $$J$$ also a hereditary C*-subalgebra of $$A$$?