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

TL;DR

This paper characterizes when a $C^*$-algebra contains a dense nuclear Fréchet ideal, linking it to the discreteness and countability of its primitive ideal space and the finite dimensionality of its quotients.

Contribution

It provides a complete characterization of dense nuclear Fréchet ideals in $C^*$-algebras based on the structure of their primitive ideal space and quotient properties.

Findings

Dense nuclear ideals exist iff Prim$(B)$ is discrete and countable with finite dimensional quotients.

Constructs all such ideals using matrix-valued Schwartz functions on Prim$(B)$.

Extends results to general Banach algebras with two-sided ideals.

Abstract

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$(B)$.