Let $B$ be a $C^{*}$-algebra and $\mathcal{B}$ a dense *-subalgebra stable under holomorphic functional calculus and $C^{1}$-functional calculus for selfadjoint elements. Also, $\mathcal{B}$ is a Banach algebra in a norm $\|\cdot\|_{1},$ satisfying
$\|\cdot\|\leq\|\cdot\|_{1}$.
Also, there is a countable bounded approximate unit $u_{n}$ for $\mathcal{B}$ such which is a contractive, increasing approximate unit for $B$. Let $\mathcal{I}$ be a closed two sided ideal in $\mathcal{B}$, and denote by $I$ its closure in $B$.
Is it true that $\mathcal{I}=I \cap \mathcal{B}$ ?
The pertinent examples are Lipschitz functions on the circle and on the real line, both with norm $\|f\|_{1}=\|f\|+\|\partial f\|$.
