Banach Journal of Mathematical Analysis

Characterizations of Jordan left derivations on some algebras

Guangyu An, Yana Ding, and Jiankui Li

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


A linear mapping δ from an algebra A into a left A-module M is called a Jordan left derivation if δ(A2)=2Aδ(A) for every AA. We prove that if an algebra A and a left A-module M satisfy one of the following conditions—(1) A is a C-algebra and M is a Banach left A-module; (2) A=AlgL with {L:LJL}=(0) and M=B(X); and (3) A is a commutative subspace lattice algebra of a von Neumann algebra B and M=B(H)—then every Jordan left derivation from A into M is zero. δ is called left derivable at GA if δ(AB)=Aδ(B)+Bδ(A) for each A,BA with AB=G. We show that if A is a factor von Neumann algebra, G is a left separating point of A or a nonzero self-adjoint element in A, and δ is left derivable at G, then δ0.

Article information

Banach J. Math. Anal., Volume 10, Number 3 (2016), 466-481.

Received: 1 April 2015
Accepted: 18 August 2015
First available in Project Euclid: 13 May 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B47: Commutators, derivations, elementary operators, etc.
Secondary: 47L35: Nest algebras, CSL algebras 47C15: Operators in $C^*$- or von Neumann algebras

$C^{*}$-algebra Jordan left derivation left derivable point left separating point


An, Guangyu; Ding, Yana; Li, Jiankui. Characterizations of Jordan left derivations on some algebras. Banach J. Math. Anal. 10 (2016), no. 3, 466--481. doi:10.1215/17358787-3599675.

Export citation


  • [1] J. Alaminos, M. Brešar, J. Extermera, and R. Villena, Characterizing Jordan maps on $C^{*}$-algebras through zero products, Proc. Edinb. Math. Soc. (2) 53 (2010), no. 3, 543–555.
  • [2] A. Bikchentaev, On representation of elements of a von Neumann algebra in the form of finite sums of products of projections, Sibirsk. Mat. Zh. 46 (2005), no. 1, 24–34.
  • [3] M. Brešar, Jordan derivations on semiprime rings, Bull. Aust. Math. Soc. 104 (1988), no. 4, 1003–1006.
  • [4] M. Brešar and J. Vukman, Jordan derivations on prime rings, Bull. Aust. Math. Soc. 37 (1988), no. 3, 321–322.
  • [5] M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc. 110 (1990), no. 1, 7–16.
  • [6] J. Cuntz, On the continuity of Semi-Norms on operator algebras, Math. Ann. 220 (1976), no. 2, 171–183.
  • [7] J. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), no. 2, 321–324.
  • [8] H. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monog. Ser. 24, Oxford Univ. Press, New York, 2000.
  • [9] H. Dales, F. Ghahramani, and N. Grønbæk, Derivations into iterated duals of Banach algebras, Studia Math. 128 (1998), no. 1, 19–54.
  • [10] Q. Deng, On Jordan left derivations, Math. J. Okayama Univ. 34 (1992), 145–147.
  • [11] S. Goldstein and A. Paszkiewicz, Linear combinations of projections in von Neumann algebras, Proc. Amer. Math. Soc. 116 (1992), no. 1, 175–183.
  • [12] J. Guo and J. Li, On centralizers of reflexive algebras, Aequationes Math. 84 (2012), no. 1, 1–12.
  • [13] I. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104–1110.
  • [14] R. Kadison and J. Ringrose, Fundamentals of the Theory of Operator Algebras, I, Pure Appl. Math. 100, Academic Press, New York, 1983.
  • [15] I. Kosi-Ulbl and J. Vukman, A note on $(m,n)$-Jordan derivations on semiprime rings and Banach algebras, Bull. Aust. Math. Soc. 93 (2016), no. 2, 231–237.
  • [16] J. Li and J. Zhou, Jordan left derivations and some left derivable maps, Oper. Matrices 4 (2010), 127–138.
  • [17] W. Longstaff, Strongly reflexive lattices, J. Lond. Math. Soc. (2) 11 (1975), no. 4, 491–498.
  • [18] W. Longstaff and O. Panaia, $\mathcal{J}$-subspace lattices and subspace $\mathcal{M}$-bases, Studia Math. 139 (2000), no. 3, 197–212.
  • [19] T. Ogasawara, Finite dimensionality of certain Banach algebras, J. Sci. Hiroshima Univ. Ser. A 17 (1954), 359–364.
  • [20] Z. Pan, Derivable maps and derivational points, Linear Algebra Appl. 436 (2012), no. 11, 4251–4260.
  • [21] K. Park, Jordan higher left derivaitons and commutativity in prime rings, J. Chungcheong Math. Soc. 23 (2010), 741–748.
  • [22] J. Ringrose, Automatically continuous of derivations of operator algebras, J. Lond. Math. Soc. (2) 5 (1972), no. 3, 432–438.
  • [23] J. Vukman, On left Jordan derivations of rings and Banach algebras, Aequations Math. 75 (2008), no. 3, 260–266.
  • [24] J. Vukman, On $(m,n)$-Jordan derivations and commutativity of prime rings, Demonstr. Math. 41 (2008), no. 4, 773–778.