Dense nuclear Fr\'echet ideals in $C^\star$-algebras

We show that a $C^\star$-algebra $B$ contains a dense left or right Fr\'echet ideal $A$, with $A$ a nuclear locally convex space, if and only if the primitive ideal space Prim$(B)$ of $B$ is discrete and countable, and $B/I$ is finite dimensional for each $I \in$ Prim$(B)$. We show the forward implication holds for a general Banach algebra $B$, if the ideal is assumed two-sided. For $C^\star$-algebras, we construct all two-sided dense nuclear ideals by defining a set of matrix-valued Schwartz functions on the countable discrete space Prim$(B)$.