Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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2-LOCAL DERIVATIONS ON C*-ALGEBRAS

  • Wenbo Huang (School of Mathematics and Physics Jiangsu University of Technology and School of Mathematics East China University of Science and Technology) ;
  • Jiankui Li (School of Mathematics East China University of Science and Technology)
  • Received : 2023.07.26
  • Accepted : 2023.10.05
  • Published : 2024.05.31
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Abstract

In this paper, we prove that every 2-local derivation on several classes of C*-algebras, such as unital properly infinite, type I or residually finite-dimensional C*-algebras, is a derivation. We show that the following statements are equivalent: (1) every 2-local derivation on a C*-algebra is a derivation, (2) every 2-local derivation on a unital primitive antiliminal and no properly infinite C*-algebra is a derivation. We also show that every 2-local derivation on a group C*-algebra C*(𝔽) or a unital simple infinite-dimensional quasidiagonal C*-algebra, which is stable finite antiliminal C*-algebra, is a derivation.

Keywords

Acknowledgement

The first author was partially supported by National Natural Science Foundation of China (Grant No. 12026252, 12026250). The second author was partially supported by National Natural Science Foundation of China (Grant No. 11871021).

References

  1. B. H. Aupetit and H. T. Mouton, Trace and determinant in Banach algebras, Studia Math. 121 (1996), no. 2, 115-136.  https://doi.org/10.4064/sm-121-2-115-136
  2. S. Ayupov and K. Kudaybergenov, 2-local derivations on von Neumann algebras, Positivity 19 (2015), no. 3, 445-455. https://doi.org/10.1007/s11117-014-0307-3 
  3. S. Ayupov, K. Kudaybergenov, and A. Alauadinov, 2-local derivations on matrix algebras over commutative regular algebras, Linear Algebra Appl. 439 (2013), no. 5, 1294- 1311. https://doi.org/10.1016/j.laa.2013.04.013 
  4. S. Ayupov, K. Kudaybergenov, and A. M. Pereira, A survey on local and 2-local derivations on C*- and von Neumann algebras, in Topics in functional analysis and algebra, 73-126, Contemp. Math., 672, Amer. Math. Soc., Providence, RI, 2016. https://doi.org/10.1090/conm/672/13462 
  5. B. Blackadar, Operator algebras, Encyclopaedia of Mathematical Sciences, 122, Springer, Berlin, 2006. https://doi.org/10.1007/3-540-28517-2 
  6. N. P. Brown, On quasidiagonal C*-algebras, in Operator algebras and applications, 19- 64, Adv. Stud. Pure Math., 38, Math. Soc. Japan, Tokyo, 2004. https://doi.org/10.2969/aspm/03810019 
  7. M. D. Choi, The full C*-algebra of the free group on two generators, Pacific J. Math. 87 (1980), no. 1, 41-48. http://projecteuclid.org/euclid.pjm/1102780313  102780313
  8. J. M. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), no. 2, 321-324. https://doi.org/10.2307/2040004 
  9. L. Dalla, S. Giotopoulos, and N. Katseli, The socle and finite-dimensionality of a semiprime Banach algebra, Studia Math. 92 (1989), no. 2, 201-204. https://doi.org/10.4064/sm-92-2-201-204 
  10. K. R. Davidson, Nest algebras, Pitman Research Notes in Mathematics Series, 191, Longman Sci. Tech., Harlow, 1988. 
  11. J. Dixmier, C*-algebras, translated from the French by Francis Jellett, North-Holland Mathematical Library, Vol. 15, North-Holland, Amsterdam, 1977. 
  12. J. He, J. Li, G. An, and W. Huang, Characterizations of 2-local derivations and local Lie derivations on some algebras, Sib. Math. J. 59 (2018), no. 4, 721-730; translated from Sibirsk. Mat. Zh. 59 (2018), no. 4, 912-926. https://doi.org/10.1134/s0037446618040146 
  13. I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104-1110. https://doi.org/10.2307/2032688 
  14. W. Huang and J. K. Li, 2-local derivations on semisimple Banach algebras with minimal left ideals, J. Aust. Math. Soc. 110 (2021), no. 3, 321-332. https://doi.org/10.1017/S144678872000021X 
  15. W. Huang, J. K. Li, and W. H. Qian, Derivations and 2-local derivations on matrix algebras and algebras of locally measurable operators, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 1, 227-240. https://doi.org/10.1007/s40840-018-0675-0 
  16. B. E. Johnson, Local derivations on C*-algebras are derivations, Trans. Amer. Math. Soc. 353 (2001), no. 1, 313-325. https://doi.org/10.1090/S0002-9947-00-02688-X 
  17. R. V. Kadison, Local derivations, J. Algebra 130 (1990), no. 2, 494-509. https://doi.org/10.1016/0021-8693(90)90095-6 
  18. S. O. Kim and J. S. Kim, Local automorphisms and derivations on 𝕄n, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1389-1392. https://doi.org/10.1090/S0002-9939-03-07171-5 
  19. S. O. Kim and J. S. Kim, Local automorphisms and derivations on certain C*-algebras, Proc. Amer. Math. Soc. 133 (2005), no. 11, 3303-3307. https://doi.org/10.1090/S0002-9939-05-08059-7 
  20. S. Kowalski and Z. Slodkowski, A characterization of multiplicative linear functionals in Banach algebras, Studia Math. 67 (1980), no. 3, 215-223. https://doi.org/10.4064/sm-67-3-215-223 
  21. D. R. Larson and A. R. Sourour, Local derivations and local automorphisms of 𝓑(X), in Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988), 187-194, Proc. Sympos. Pure Math., 51, Part 2, Amer. Math. Soc., Providence, RI, 1990. https://doi.org/10.1090/pspum/051.2/1077437 
  22. B. Magajna, The Haagerup norm on the tensor product of operator modules, J. Funct. Anal. 129 (1995), no. 2, 325-348. https://doi.org/10.1006/jfan.1995.1053 
  23. M. Mathieu, Elementary operators on prime C*-algebras. II, Glasgow Math. J. 30 (1988), no. 3, 275-284. https://doi.org/10.1017/S0017089500007369 
  24. G. K. Pedersen, C*-algebras and their automorphism groups, second edition, Academic Press London, 2018. 
  25. P. Semrl, Local automorphisms and derivations on 𝓑(H), Proc. Amer. Math. Soc. 125 (1997), no. 9, 2677-2680. https://doi.org/10.1090/S0002-9939-97-04073-2 
  26. J. H. Zhang and H. X. Li, 2-local derivations on digraph algebras, Acta Math. Sinica (Chinese Ser.) 49 (2006), no. 6, 1411-1416.