If $A$ is a C\*-algebra, we say that a subset $I\subseteq A$ is hereditary if
$$
 0\leq x \leq y \in I \Rightarrow x\in I.
 $$
It is is well known that closed 2-sided ideals are hereditary.


Would it also be true for arbitrary 2-sided ideals? What about self-adjoint 2-sided ideals?



Is every 2-sided ideal in a C*-algebra hereditary?
No. Take $A = C[0,1]$ and let $I$ be the (unclosed) ideal generated by the function $f(t) = t$. This ideal is self-adjoint, but it does not contain the function $g(t) = t\sin^2(\frac{1}{t})$, so it is not hereditary. (Example II.5.2.1 (iii) in Bruce Blackadar's fantastic book [*Operator Algebras: Theory of C${}^\*$-Algebras and von Neumann Algebras*](https://packpages.unr.edu/media/1224/cycr.pdf).)


