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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 $A$?
Required, but never shown
1 month ago