Derivations and homomorphisms in commutator-simple algebras
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- by J. Alaminos, M. Brešar, J. Extremera, M. L. C. Godoy and A. R. Villena
- Proc. Amer. Math. Soc. 151 (2023), 4721-4733
- DOI: https://doi.org/10.1090/proc/16483
- Published electronically: August 4, 2023
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Abstract:
We call an algebra $A$ commutator-simple if $[A,A]$ does not contain nonzero ideals of $A$. After providing several examples, we show that in these algebras derivations are determined by a condition that is applicable to the study of local derivations. This enables us to prove that every continuous local derivation $D\colon L^1(G)\to L^1(G)$, where $G$ is a unimodular locally compact group, is a derivation. We also give some remarks on homomorphism-like maps in commutator-simple algebras.References
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Bibliographic Information
- J. Alaminos
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Granada, Spain
- MR Author ID: 641559
- ORCID: 0000-0002-7857-7833
- Email: alaminos@ugr.es
- M. Brešar
- Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, and Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia
- ORCID: 0000-0001-7574-212X
- Email: matej.bresar@fmf.uni-lj.si
- J. Extremera
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Granada, Spain
- MR Author ID: 692437
- Email: jlizana@ugr.es
- M. L. C. Godoy
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Granada, Spain
- MR Author ID: 1306099
- ORCID: 0000-0001-5772-5385
- Email: mgodoy@ugr.es
- A. R. Villena
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Granada, Spain
- MR Author ID: 309482
- Email: avillena@ugr.es
- Received by editor(s): October 25, 2022
- Received by editor(s) in revised form: March 10, 2023
- Published electronically: August 4, 2023
- Additional Notes: The first, third, fourth, and fifth authors were supported by the Grant PID2021-122126NB-C31 funded by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A way of making Europe” and grant FQM-185 funded by Junta de Andalucía. The second author was supported by the Slovenian Research Agency (ARRS) Grant P1-0288. The fourth author was also supported by grant FPU18/00419 funded by MIU
- Communicated by: Javad Mashreghi
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 4721-4733
- MSC (2020): Primary 43A20, 47L10, 16W20, 16W25
- DOI: https://doi.org/10.1090/proc/16483
- MathSciNet review: 4634876