Derivations and homomorphisms in commutator-simple algebras

Alaminos, J.; Brešar, M.; Extremera, J.; Godoy, M.; Villena, A.

doi:10.1090/proc/16483

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

U. Bader, T. Gelander, and N. Monod, A fixed point theorem for $L^1$ spaces, Invent. Math. 189 (2012), no. 1, 143–148. MR 2929085, DOI 10.1007/s00222-011-0363-2
V. V. Bavula, Generalized Weyl algebras and their representations, Algebra i Analiz 4 (1992), no. 1, 75–97 (Russian); English transl., St. Petersburg Math. J. 4 (1993), no. 1, 71–92. MR 1171955
Matej Brešar, Introduction to noncommutative algebra, Universitext, Springer, Cham, 2014. MR 3308118, DOI 10.1007/978-3-319-08693-4
Matej Brešar, Automorphisms and derivations of finite-dimensional algebras, J. Algebra 599 (2022), 104–121. MR 4390457, DOI 10.1016/j.jalgebra.2022.02.010
Paul R. Chernoff, Representations, automorphisms, and derivations of some operator algebras, J. Functional Analysis 12 (1973), 275–289. MR 350442, DOI 10.1016/0022-1236(73)90080-3
J. M. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), no. 2, 321–324. MR 399182, DOI 10.1090/S0002-9939-1975-0399182-5
M. A. Farinati, A. Solotar, and M. Suárez-Álvarez, Hochschild homology and cohomology of generalized Weyl algebras, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 2, 465–488 (English, with English and French summaries). MR 1990004, DOI 10.5802/aif.1950
Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
Antonio Giambruno and Mikhail Zaicev, Polynomial identities and asymptotic methods, Mathematical Surveys and Monographs, vol. 122, American Mathematical Society, Providence, RI, 2005. MR 2176105, DOI 10.1090/surv/122
Bruno Harris, Commutators in division rings, Proc. Amer. Math. Soc. 9 (1958), 628–630. MR 96697, DOI 10.1090/S0002-9939-1958-0096697-4
I. N. Herstein, Jordan homomorphisms, Trans. Amer. Math. Soc. 81 (1956), 331–341. MR 76751, DOI 10.1090/S0002-9947-1956-0076751-6
Nathan Jacobson, Structure of rings, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, 190 Hope Street, Providence, R.I., 1956. MR 81264, DOI 10.1090/coll/037
Richard V. Kadison, Local derivations, J. Algebra 130 (1990), no. 2, 494–509. MR 1051316, DOI 10.1016/0021-8693(90)90095-6
David R. Larson and Ahmed R. Sourour, Local derivations and local automorphisms of ${\scr B}(X)$, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 187–194. MR 1077437, DOI 10.1090/pspum/051.2/1077437
Viktor Losert, The derivation problem for group algebras, Ann. of Math. (2) 168 (2008), no. 1, 221–246. MR 2415402, DOI 10.4007/annals.2008.168.221
Theodore W. Palmer, Banach algebras and the general theory of $^*$-algebras. Vol. I, Encyclopedia of Mathematics and its Applications, vol. 49, Cambridge University Press, Cambridge, 1994. Algebras and Banach algebras. MR 1270014, DOI 10.1017/CBO9781107325777
Theodore W. Palmer, Banach algebras and the general theory of $*$-algebras. Vol. 2, Encyclopedia of Mathematics and its Applications, vol. 79, Cambridge University Press, Cambridge, 2001. $*$-algebras. MR 1819503, DOI 10.1017/CBO9780511574757.003
Ebrahim Samei, Approximately local derivations, J. London Math. Soc. (2) 71 (2005), no. 3, 759–778. MR 2132382, DOI 10.1112/S0024610705006496
Ebrahim Samei, Reflexivity and hyperreflexivity of bounded $n$-cocycles from group algebras, Proc. Amer. Math. Soc. 139 (2011), no. 1, 163–176. MR 2729080, DOI 10.1090/S0002-9939-2010-10454-9