Non-closed ideals in $C^*$-algebras

Question

What is an example of an ideal in a commutative $C^*$-algebra that is not closed? If by chance every ideal in a commutative $C^*$-algebra is closed, how about in non-commutative $C^*$-algebras?

(Ideal here means two-sided ideal.)

Answer 1

Consider $C[0,1]$ and $I$ the set of function that vanish at all the points $1/2,1/3,...,0$ that fall inside some interval $[0,a]$, for some $0<a<1$.

The function $f(x)=x$ is not in $I$ but is the uniform limit of function in $I$.

Answer 2

Another simple example.

Consider $C[0,1]$ and the set of functions $I_0=\{xf(x)\mid f\in C[0,1]\}.$ The closure of $I_0$ is the set of functions vanishing in $0.$ But $\sqrt{x}\notin I_0.$

Answer 3

You know that every (non unital) commutative C*-algebra is in the form of $C_0(\Omega)$. The set of all continuous functions on $\Omega$ whose support are compact forms a non closed ideal.

In non commutative case, the trace class and Hilbert Schmidt operators are two well-known non closed ideals in $B(H)$.