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 $A$?
2026-04-03 08:27:23.1775204843
Hereditary C*-subalgebras of the unitization of a hereditary C*-subalgebra
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No. As a simple example, take $A = C([0,1])$, $B = C_0([0,1))$ and $J = C_0((0,1])$.