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

Gardella, Eusebio; Thiel, Hannes

doi:10.1090/proc/16808

Prime ideals in C*-algebras and applications to Lie theory
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by Eusebio Gardella and Hannes Thiel;

Proc. Amer. Math. Soc. 152 (2024), 3647-3656

DOI: https://doi.org/10.1090/proc/16808

Published electronically: July 1, 2024

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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.

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Proceedings of the American Mathematical Society

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