Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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.

References
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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