Taiwanese Journal of Mathematics


Xiuping Xiuping and Fangyan Lu

Full-text: Open access


Banach spaces. Let $\phi$ be a bijection from $B(X)$ onto $B(Y)$ satisfying ${\phi}([A,B])=[\phi(A),\phi(B)]$ for all $A, B\in B(X)$. Then $\phi=\psi+\tau$, where $\psi$ is a ring isomorphism or a negative of a ring anti-isomorphism from $B(X)$ onto $B(Y)$, and $\tau$ is a map from $B(X)$ into $\Bbb CI$ satisfying $\tau([A,B])=0$ for all $A, B\in B(X)$.

Article information

Taiwanese J. Math., Volume 12, Number 3 (2008), 793-806.

First available in Project Euclid: 21 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B49: Transformers, preservers (operators on spaces of operators) 47L10: Algebras of operators on Banach spaces and other topological linear spaces

Lie product Banach spaces ring isomorphism


Xiuping, Xiuping; Lu, Fangyan. MAPS PRESERVING LIE PRODUCT ON B(X). Taiwanese J. Math. 12 (2008), no. 3, 793--806. doi:10.11650/twjm/1500602436. https://projecteuclid.org/euclid.twjm/1500602436

Export citation


  • J. Alaminos, M. Mathieu and A. R. Villena, Symmetric amenability and Lie derivations, Math. Proc. Cambridge Philos. Soc., 137 (2004), 433-439.
  • K. I. Beidar and M. A. Chebotar, On lie derivations of Lie ideals of prime rings, Israel J. Math., 123 (2001), 131-148.
  • K. I. Beidar, M. Bre\u sar, M. A. Chebotar and W. S. Mardindale III, On Herstein's Lie Map conjectures I, Trans. Amer. Math. Soc., 353 (2001), 4235-4260.
  • M. Bresar and P. Semrl, Commuting traces of biadditive maps revisited, Commun. Algebra, 31 (2003), 381-388.
  • D. \u Z. Dokovi\' c, Automorphisms of the Lie algebra of upper triangular matrices over a connected commutative rings, J. Algebra, 170 (1994), 101-110.
  • J. Hakeda and K. Sait\^ o, Additivity of Jordan *-maps on operators, J. Math. Soc. Japan, 38 (1986),403-408.
  • L. Hua, A theorem on matrices over an s-field and its applications, J. Chinese Math. Soc. (N.S.), 1 (1951), 110-163.
  • R. E. Johnson, Rings with unique addition, Proc. Amer. Math. Soc., 9 (1958), 57-61.
  • B. E. Johnson, Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math Proc. Cambridge Philos. Soc., 120 (1996), 455-473.
  • F. Lu, Additivity of Jordan maps on standard operator algebras, Linear Algebra Appl., 357 (2002), 123-131.
  • F. Lu, Jordan triple maps, Linear Algebra Appl., 375 (2003), 311-317.
  • L. W. Marcoux and A. R. Sourour, Lie isomorphisms of nest algebras, J. Funct. Anal., 164 (1999), 163-180.
  • W. S. Martindale III, Lie isomorphisms of primitive rings, Michigan Math. J., 11 (1964), 183-187.
  • W. S. Martindale III, When are multiplicative mappings additive, Proc. Amer. Math. Soc., 21 (1969), 695-698.
  • M. Mathieu and A. R. Villena, The structure of Lie derivations on C*-algebras, J. Funct. Anal., 202 (2003), 504-525.
  • C. R. Miers, Lie isomorphisms of factors, Trans. Amer. Math. Soc., 147 (1970), 55-63.
  • C. R. Miers, Lie derivations of von Neumann algebras, \em Duke Math. J., 40 (1973), 403-409.
  • L. Molnár, Jordan maps on standard operator algebras, in: Functional Equations-Results and Advances, Kluwer Academic Publishers, Dordrecht, 2002.
  • L. Molnár, On isomorphisms of standard operator algebras, Stud. Math., 142 (2000), 295-302.
  • C. E. Rickart, One-to-one mapping of rings and lattices, Bull. Amer. Math. Soc., 54 (1948), 759-764.
  • C. E. Rickart, Representations of certain Banach algebras on Hilbert space, Duke Math. J., 18 (1951), 27-39.
  • P. \u Semrl, Isomorphisms of standard operator algebras, Proc. Amer. Math. Soc., 123 (1995), 1851-1855.
  • G. A. Swain, Lie derivations of the skew elements of prime rings with involution, J. Algebra, 184 (1996), 679-704.
  • G. A. Swain and P. S. Blau, Lie derivations in prime rings with involution, Canad. Math. Bull., 42 (1999), 401-411.
  • R. A. Villena, Lie derivations on Banach algebras, J. Algebra, 226 (2000), 390-409.