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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Bibliographic Information
- Eusebio Gardella
- Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg SE-412 96, Sweden
- MR Author ID: 1118291
- Email: gardella@chalmers.se
- Hannes Thiel
- Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg SE-412 96, Sweden
- MR Author ID: 930802
- ORCID: 0000-0003-0388-6495
- Email: hannes.thiel@chalmers.se
- Received by editor(s): August 10, 2023
- Received by editor(s) in revised form: January 18, 2024
- Published electronically: July 1, 2024
- Additional Notes: The first author was partially supported by the Swedish Research Council Grant 2021-04561. The second author was partially supported by the Knut and Alice Wallenberg Foundation (KAW 2021.0140).
- Communicated by: Matthew Kennedy
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3647-3656
- MSC (2020): Primary 16N60, 46L05; Secondary 16W10, 47B47
- DOI: https://doi.org/10.1090/proc/16808
- MathSciNet review: 4781962