Additivity of maps preserving products AP ± PA* on C*-algebras
-
Ali Taghavi
Abstract
Let 𝒜 and ℬ be two prime C*-algebras. In this paper, we investigate the additivity of map Φ from 𝒜 onto ℬ that are bijective unital and satisfies
for all A ∊ 𝒜 and P ∊ {P1, I𝒜 − P1} where P1 is a nontrivial projection in 𝒜 and λ∊ {−1, +1}. Then, Φ is *-additive.
Acknowledgement
The authors are grateful to the reviewer for careful reading and valuable suggestions which improved the quality of the paper.
References
[1] Bai, Z. F.—Du, S. P.: Multiplicative Lie isomorphism between prime rings, Comm. Algebra 36 (2008), 1626–1633.10.1080/00927870701870475Search in Google Scholar
[2] Bai, Z. F.—Du, S. P.: Multiplicative *-Lie isomorphism between factors, J. Math. Anal. Appl. 346 (2008), 327–335.10.1016/j.jmaa.2008.05.077Search in Google Scholar
[3] Beidar, K. I.—Brešar, M.—Chebotar, M. A.—Martindale, W. S.: On Hersteins Lie map conjecture (I), Trans. Amer. Math. Soc. 353 (2001), 4235–4260.10.1090/S0002-9947-01-02731-3Search in Google Scholar
[4] Beidar, K. I.—Brešar, M.—Chebotar, M. A.—Martindale, W. S.: On Hersteins Lie map conjecture (II), J. Algebra 238 (2001), 239–264.10.1006/jabr.2000.8628Search in Google Scholar
[5] Beidar, K. I.—Brešar, M.—Chebotar, M. A.—Martindale, W. S.: On Hersteins Lie map conjecture (III), J. Algebra 238 (2002), 59–94.10.1006/jabr.2001.9076Search in Google Scholar
[6] Brešar, M.: Jordan mappings of semiprime rings, J. Algebra 127 (1989), 218–228.10.1016/0021-8693(89)90285-8Search in Google Scholar
[7] Brešar, M.—Foner, A.: On ring with involution equipped with some new product, Publ. Math. Debrecen 57 (2000), 121–134.10.5486/PMD.2000.2247Search in Google Scholar
[8] Cui, J.—Li, C. K.: Maps preserving product XY – Y X* on factor von Neumann algebras, Linear Algebra Appl. 431 (2009), 833–842.10.1016/j.laa.2009.03.036Search in Google Scholar
[9] Hakeda, J.: Additivity of Jordan *-maps on AW*-algebras, Proc. Amer. Math. Soc. 96 (1986), 413–420.10.2307/2046586Search in Google Scholar
[10] Herstein, I. N.: Jordan homomorphisms, Trans. Amer. Math. Soc. 81 (1956), 331–341.10.1090/S0002-9947-1956-0076751-6Search in Google Scholar
[11] Jacobson, N.—Rickart, C. E.: Jordan homomorphism of rings, Trans. Amer. Math. Soc. 69 (1950), 479–502.10.1007/978-1-4612-3694-8_7Search in Google Scholar
[12] Ji, P.—Liu, Z.: Additivity of Jordan maps on standard Jordan operator algebras, Linear Algebra Appl. 430 (2009), 335–343.10.1016/j.laa.2008.07.023Search in Google Scholar
[13] Kadison, R. V.: Isometries of operator algebras, Ann. of Math. 54 (1951), 325–338.10.2307/1969534Search in Google Scholar
[14] Li, C.—Lu, F.—Fang, X.: Nonlinear mappings preserving product XY + YX* on factor von Neumann algebras, Linear Algebra Appl. 438 (2013), 2339–2345.10.1016/j.laa.2012.10.015Search in Google Scholar
[15] Lu, F.: Additivity of Jordan maps on standard operator algebras, Linear Algebra Appl. 357 (2002), 123–131.10.1016/S0024-3795(02)00367-1Search in Google Scholar
[16] Lu, F.: Jordan maps on associative algebras, Comm. Algebra 31 (2003), 2273–2286.10.1081/AGB-120018997Search in Google Scholar
[17] Lu, F.: Jordan triple maps, Linear Algebra Appl. 375 (2003), 311–317.10.1016/j.laa.2003.06.004Search in Google Scholar
[18] Marcoux, L. W.: Lie isomorphism of nest algebras, J. Funct. Anal. 164 (1999), 163–180.10.1006/jfan.1999.3388Search in Google Scholar
[19] Molnár, L.: A condition for a subspace of B(H) to be an ideal, Linear Algebra Appl. 235 (1996), 229–234.10.1016/0024-3795(94)00143-XSearch in Google Scholar
[20] Molnár, L.: On isomorphisms of standard operator algebras, Studia Math. 142 (2000), 295–302.10.4064/sm-142-3-295-302Search in Google Scholar
[21] Martindale, W. S.: When are multiplicative mappings additive? Proc. Amer. Math. Soc. 21 (1969), 695–698.10.1090/S0002-9939-1969-0240129-7Search in Google Scholar
[22] Mires, C. R.: Lie isomorphisms of operator algebras, Pacific J. Math. 38 (1971), 717–735.10.2140/pjm.1971.38.717Search in Google Scholar
[23] Mires, C. R.: Lie isomorphisms of factors, Trans. Amer. Math. Soc. 147 (1970), 5–63.10.1090/S0002-9947-1970-0273423-7Search in Google Scholar
[24] Qi, X.—Hou, J.: Additivity of Lie multiplicative maps on triangular algebras, Linear and Multilinear Algebra 59 (2011), 391–397.10.1080/03081080903582094Search in Google Scholar
[25] Šemrl, P.: Quadratic functionals and Jordan *-derivations, Studia Math. 97 (1991), 157–165.10.4064/sm-97-3-157-165Search in Google Scholar
© 2017 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- State hoops
- On derivations of partially ordered sets
- Interior and closure operators on commutative basic algebras
- When is the cayley graph of a semigroup isomorphic to the cayley graph of a group
- Sequences of cantor type and their expressibility
- δ-Fibonacci and δ-lucas numbers, δ-fibonacci and δ-lucas polynomials
- Law of inertia for the factorization of cubic polynomials – the case of primes 2 and 3
- Closed hereditary coreflective subcategories in epireflective subcategories of Top
- Method of upper and lower solutions for coupled system of nonlinear fractional integro-differential equations with advanced arguments
- Difference of two strong Światkowski lower semicontinuous functions
- Fejér-type inequalities (II)
- Representation of maxitive measures: An overview
- On a conjecture of Y. H. Cao and X. B. Zhang
- On the generalized orthogonal stability of the pexiderized quadratic functional equations in modular spaces
- Almost everywhere convergence of some subsequences of Fejér means for integrable functions on some unbounded Vilenkin groups
- Homological properties of banach modules over abstract segal algebras
- Variable Hajłasz-Sobolev spaces on compact metric spaces
- Commuting pairs of self-adjoint elements in C*-algebras
- Additivity of maps preserving products AP ± PA* on C*-algebras
- A note on derived connections from semi-symmetric metric connections
- Lindelöf P-spaces need not be Sokolov
- Strong convergence properties for arrays of rowwise negatively orthant dependent random variables
- Least absolute deviations problem for the Michaelis-Menten function
- Congruence pairs of principal p-algebras
Articles in the same Issue
- State hoops
- On derivations of partially ordered sets
- Interior and closure operators on commutative basic algebras
- When is the cayley graph of a semigroup isomorphic to the cayley graph of a group
- Sequences of cantor type and their expressibility
- δ-Fibonacci and δ-lucas numbers, δ-fibonacci and δ-lucas polynomials
- Law of inertia for the factorization of cubic polynomials – the case of primes 2 and 3
- Closed hereditary coreflective subcategories in epireflective subcategories of Top
- Method of upper and lower solutions for coupled system of nonlinear fractional integro-differential equations with advanced arguments
- Difference of two strong Światkowski lower semicontinuous functions
- Fejér-type inequalities (II)
- Representation of maxitive measures: An overview
- On a conjecture of Y. H. Cao and X. B. Zhang
- On the generalized orthogonal stability of the pexiderized quadratic functional equations in modular spaces
- Almost everywhere convergence of some subsequences of Fejér means for integrable functions on some unbounded Vilenkin groups
- Homological properties of banach modules over abstract segal algebras
- Variable Hajłasz-Sobolev spaces on compact metric spaces
- Commuting pairs of self-adjoint elements in C*-algebras
- Additivity of maps preserving products AP ± PA* on C*-algebras
- A note on derived connections from semi-symmetric metric connections
- Lindelöf P-spaces need not be Sokolov
- Strong convergence properties for arrays of rowwise negatively orthant dependent random variables
- Least absolute deviations problem for the Michaelis-Menten function
- Congruence pairs of principal p-algebras
