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Derivations and Related Maps

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Zero Product Determined Algebras

Part of the book series: Frontiers in Mathematics ((FM))

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Abstract

There are many parallels between homomorphisms and derivations. Having just about any result on homomorphisms, one may ask whether a similar result, possibly under milder assumptions, is true for derivations. It is therefore not surprising that some of the results on homomorphisms from the preceding chapter have their counterparts for derivations. This will be shown in Sect. 8.1.

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References

  1. B.A.E. Ahlem, A.M. Peralta, Linear maps on C -algebras which are derivations or triple derivations at a point. Linear Algebra Appl. 538, 1–21 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Alaminos, M. Brešar, J. Extremera, A.R. Villena, Characterizing homomorphisms and derivations on C -algebras. Proc. Roy. Soc. Edinburgh Sect. A 137, 1–7 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Alaminos, M. Brešar, J. Extremera, A.R. Villena, Maps preserving zero products. Studia Math. 193, 131–159 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Alaminos, M. Brešar, J. Extremera, A.R. Villena, Characterizing Jordan maps on C -algebras through zero products. Proc. Edinb. Math. Soc. 53, 543–555 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Alaminos, J. Extremera, A.R. Villena, Hyperreflexivity of the derivation space of some group algebras. Math. Z. 266, 571–582 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Alaminos, J. Extremera, A.R. Villena, Hyperreflexivity of the derivation space of some group algebras, II. Bull. London Math. Soc. 44, 323–335 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Alaminos, M. Brešar, J. Extremera, Š. Špenko, A.R. Villena, Determining elements in C -algebras through spectral properties. J. Math. Anal. Appl. 405, 214–219 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Alaminos, M. Brešar, J. Extremera, Š. Špenko, A.R. Villena, Derivations preserving quasinilpotent elements. Bull. Lond. Math. Soc. 46, 379–384 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. B. Aupetit, A Primer on Spectral Theory. Universitext (Springer, New York, 1991)

    Book  MATH  Google Scholar 

  10. K.I. Beidar, M. Brešar, Extended Jacobson density theorem for rings with derivations and automorphisms. Israel J. Math. 122, 317–346 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Benkovič, M. Grašič, Generalized derivations on unital algebras determined by action on zero products. Linear Algebra Appl. 445, 347–368 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Bourhim, J. Mashreghi, A. Stepanyan, Maps between Banach algebras preserving the spectrum. Arch. Math. 107, 609–621 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Braatvedt, R. Brits, Uniqueness and spectral variation in Banach algebras. Quaest. Math. 36, 155–165 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Braatvedt, R. Brits, H. Raubenheimer, Spectral characterizations of scalars in a Banach algebra. Bull. Lond. Math. Soc. 41, 1095–1104 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Brešar, Characterizations of derivations on some normed algebras with involution. J. Algebra 152, 454–462 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Brešar, Jordan derivations revisited. Math. Proc. Cambridge Philos. Soc. 139, 411–425 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Brešar, Characterizing homomorphisms, derivations and multipliers in rings with idempotents. Proc. Royal Soc. Edinb. Sect. A 137, 9–21 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Brešar, Introduction to Noncommutative Algebra. Universitext (Springer, Berlin, 2014)

    Book  MATH  Google Scholar 

  19. M. Brešar, M. Mathieu, Derivations mapping into the radical, III. J. Funct. Anal. 133, 21–29 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Brešar, P. Šemrl, On locally linearly dependent operators and derivations. Trans. Amer. Math. Soc. 351, 1257–1275 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Brešar, Š. Špenko, Determining elements in Banach algebras through spectral properties. J. Math. Anal. Appl. 393, 144–150 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Brešar, B. Magajna, Š. Špenko, Identifying derivations through the spectra of their values. Integral Equ. Oper. Theory 73, 395–411 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Brits, F. Schulz, C. Touré, A spectral characterization of isomorphisms on C -algebras. Arch. Math. 113, 391–398 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Burgos, F. J. Fernández-Polo, A. M. Peralta, Local triple derivations on C -algebras and JB -triples. Bull. Lond. Math. Soc. 46, 709–724 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. L. Catalano, On maps characterized by action on equal products. J. Algebra 511, 148–154 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. M.A. Chebotar, W.-F. Ke, P.-H. Lee, Maps characterized by action on zero products. Pacific J. Math. 216, 217–228 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Y. Chen, J. Li, Mappings on some reflexive algebras characterized by action on zero products or Jordan zero products. Studia Math. 206, 121–134 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. L. Chen, F. Lu, T. Wang, Local and 2-local Lie derivations of operator algebras on Banach spaces. Integral Equ. Oper. Theory 77, 109–121 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. P.M. Cohn, The range of derivations on a skew field and the equation ax − xb = c. J. Indian Math. Soc. 37, 61–69 (1973)

    Google Scholar 

  30. R.L. Crist, Local derivations on operator algebras. J. Funct. Anal. 135, 76–92 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  31. J. Duncan, A.W. Tullo, Finite dimensionality, nilpotents and quasinilpotents in Banach algebras. Proc. Edinburgh Math. Soc. 19, 45–49 (1974/1975)

    Article  MathSciNet  MATH  Google Scholar 

  32. D. Hadwin, J. Li, Local derivations and local automorphisms. J. Math. Anal. Appl. 290, 702–714 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. N. Jacobson, C. Rickart, Jordan homomorphisms of rings. Trans. Amer. Math. Soc. 69, 479–502 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  34. P. Ji, W. Qi, Characterizations of Lie derivations of triangular algebras. Linear Algebra Appl. 435, 1137–1146 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. W. Jing, S. Lu, P. Li, Characterization of derivations on some operator algebras. Bull. Austral. Math. Soc. 66, 227–232 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. B.E. Johnson, Symmetric amenability and the nonexistence of Lie and Jordan derivations. Math. Proc. Cambridge Philos. Soc. 120, 455–473 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. B.E. Johnson, Local derivations on C -algebras are derivations. Trans. Amer. Math. Soc. 353, 313–325 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  38. B.E. Johnson, A.M. Sinclair, Continuity of derivations and a problem of Kaplansky. Amer. J. Math. 90, 1067–1073 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  39. R.V. Kadison, Local derivations. J. Algebra 130, 494–509 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  40. A. Katavolos, C. Stamatopoulos, Commutators of quasinilpotents and invariant subspaces. Studia Math. 128, 159–169 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  41. D.R. Larson, A.R. Sourour, Local derivations and local automorphisms of B(X), in Proceedings of Symposia in Pure Mathematics, vol. 51, Part 2 (American Mathematical Society, Providence, 1990), pp. 187–194

    Google Scholar 

  42. T.-K. Lee, Generalized skew derivations characterized by acting on zero products. Pacific J. Math. 216, 293–301 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  43. C. Le Page, Sur quelques conditions entraînant la commutativité dans les algèbres de Banach. C. R. Acad. Sc. Paris Sér. A-B 265, A235–A237 (1967)

    MATH  Google Scholar 

  44. C.-K. Liu, P.-K. Liau, Generalized derivations preserving quasinilpotent elements in Banach algebras. Linear Multilinear Algebra 66, 1888–1908 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  45. F. Lu, W. Jing, Characterizations of Lie derivations of B(X). Linear Algebra Appl. 432, 89–99 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Mackey, Local derivations on Jordan triples. Bull. Lond. Math. Soc. 45, 811–824 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  47. M. Mathieu, Where to find the image of a derivation. Functional analysis and operator theory, polish academy of sciences. Banach Cent. Publ. 30, 237–249 (1994)

    Google Scholar 

  48. M. Mathieu, G. Murphy, Derivations mapping into the radical. Arch. Math. 57, 469–474 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  49. G. Murphy, Aspects of the theory of derivations. functional analysis and operator theory, polish academy of sciences. Banach Cent. Publ. 30, 267–275 (1994)

    Google Scholar 

  50. A. Nowicki, On local derivations in the Kadison sense. Colloq. Math. 89, 193–198 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  51. V. Pták, Derivations, commutators and the radical. Manuscripta Math. 23, 355–362 (1977/1978)

    Article  MathSciNet  MATH  Google Scholar 

  52. E. Samei, Approximately local derivations. J. London Math. Soc. 71, 759–778 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  53. E. Samei, Reflexivity and hyperreflexivity of bounded n-cocycles from group algebras. Proc. Amer. Math. Soc. 139, 163–176 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  54. E. Samei, J. Soltan Farsani, Hyperreflexivity of bounded N-cocycle spaces of Banach algebras. Monatsh. Math. 175, 429–455 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  55. E. Samei, J. Soltan Farsani, Hyperreflexivity constants of the bounded n-cocycle spaces of group algebras and C -algebras. J. Aust. Math. Soc. 109, 112–130 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  56. F. Schulz, R. Brits, Uniqueness under spectral variation in the socle of a Banach algebra. J. Math. Anal. Appl. 444, 1626–1639 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  57. P. Šemrl, Local automorphisms and derivations on B(H). Proc. Amer. Math. Soc. 125, 2677–2680 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  58. V.S. Shulman, Operators preserving ideals in C -algebras. Studia Math. 109, 67–72 (1994)

    Article  MathSciNet  Google Scholar 

  59. A.M. Sinclair, Continuous derivations on Banach algebras. Proc. Amer. Math. Soc. 20, 166–170 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  60. I.M. Singer, J. Wermer, Derivations on commutative normed algebras. Math. Ann. 129, 260–264 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  61. J. Soltan Farsani, Hyperreflexivity of the bounded n-cocycle spaces of Banach algebras with matrix representations. Studia Math. 250, 35–55 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  62. M.P. Thomas, The image of a derivation is contained in the radical. Ann. Math. 128, 435–460 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  63. C. Touré, F. Schulz, R. Brits, Truncation and spectral variation in Banach algebras. J. Math. Anal. Appl. 445, 23–31 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  64. Yu.V. Turovskii, V.S. Shulman, Conditions for the massiveness of the range of a derivation of a Banach algebra and of associated differential operators. Mat. Zametki 42, 305–314 (1987)

    MathSciNet  Google Scholar 

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Brešar, M. (2021). Derivations and Related Maps. In: Zero Product Determined Algebras. Frontiers in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-80242-4_8

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