Prime ideals in C*-algebras and applications to Lie theory

TL;DR

This paper explores the structure of ideals in C*-algebras, showing how prime ideals relate to dense ideals, and applies these findings to transfer Lie algebra results to the setting of C*-algebras, revealing new algebraic properties.

Contribution

It establishes a connection between prime ideals and dense ideals in C*-algebras and applies this to extend Lie theory results from prime rings to C*-algebras.

Findings

Every proper, dense ideal in a C*-algebra is contained in a prime ideal.

A subset generates a C*-algebra as an ideal iff it is not contained in any prime ideal.

If a C*-algebra is generated by its commutator subspace, then the double commutator equals the commutator subspace.

Abstract

We show that every proper, dense ideal in a C*-algebra is contained in a prime ideal. It follows that a subset generates a C*-algebra as a not necessarily closed ideal if and only if it is not contained in any prime ideal. This allows us to transfer Lie theory results from prime rings to C*-algebras. For example, if a C*-algebra $A$ is generated by its commutator subspace $[A,A]$ as a ring, then $[[A,A],[A,A]] = [A,A]$. Further, given Lie ideals $K$ and $L$ in $A$, then $[K,L]$ generates $A$ as a not necessarily closed ideal if and only if $[K,K]$ and $[L,L]$ do, and moreover this implies that $[K,L]=[A,A]$. We also discover new properties of the subspace generated by square-zero elements and relate it to the commutator subspace of a C*-algebra.