PDF(465 KB)
PDF(465 KB)
PDF(465 KB)
素环上强保持2-Jordan乘积的映射
Strong 2-Jordan Product Preserving Maps on Prime Rings
对于给定的正整数k ≥ 1，环R上的元x，y的k-Jordan乘积定义为{x，y}k={{x，y}k-1，y}1，其中{x，y}0=x，{x，y}1=xy+yx.假设R是包含有单位元与一非平凡幂等元的素环.本文证明了R上的满射f满足{f（x），f（y）}2={x，y}2对所有x，y ∈ R成立当且仅当存在λ ∈ C（R的可扩展中心）且λ3=1，使得下列之一成立：（1）若R的特征不为2，则f（x）=λx对所有x ∈ R成立；（2）若R的特征为2，则f（x）=λx+μ（x）对所有x ∈ R成立，其中μ：R → C是一个映射.作为应用，得到了因子von Neumann代数上保持上述性质映射的结构.
For any k ≥ 1, k-Jordan product of two elements x,y in a ring R is defined by {x, y}k={{x, y}k-1, y}1, where {x, y}0=x and {x, y}1=xy + yx. Assume that R is a unital prime ring with a nontrivial idempotent. It is shown that a surjective map f:R → R satisfies {f(x), f(y)}2={x, y}2 for all x, y ∈ R if and only if there exists some λ ∈ C (the extended centroid of R) with λ3=1 such that one of the following statements holds:(1) if the characteristic of R is not 2, then f(x)=λx holds for all x ∈ R; (2) if the characteristic of R is 2, then f(x)=λx + μ(x) holds for all x ∈ R, where μ:R → C is a map. As an appication, such maps on factor von Neumann algebras are characterized.
因子von Neumann代数 / 素环 / Jordan乘积 {{custom_keyword}} /
factor von Neumann algebras / prime rings / Jordan products {{custom_keyword}} /
[1] Ara P., The extended centroid of C*-algebras, Arch. Math., 1990, 54(4):358-364.
[2] Bell H. E., Daif M. N., On commutativity and strong commutativity-preserving maps, Canad. Math. Bull., 1994, 37(4):443-447.
[3] Brešar M., Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Ameri. Math. Soc., 1993, 335(2):525-546.
[4] Brešar M., Chebotar M. A., Martindale Ⅲ W. S., Functional Identities, Birkhäuser Basel, 2007.
[5] Brešar M., Miers C. R., Commutativity preserving mapping of von Neumann algebras, Canad. J. Math., 1993, 45(4):695-708.
[6] Brešar M., Miers C. R., Strong commutativity preserving maps of semiprime rings, Canad. Math. Bull., 1994, 37(4):457-460.
[7] De Filippis. V., Scudo G., Strong commutativity and Engel condition preserving maps in prime and semiprime rings, Linear and Multilinear Algebra, 2013, 61(7):917-938.
[8] Gong L., Qi X. F., Shao J., et al., Strong (skew) ξ-Lie commutativity preserving maps on algebras, Cogent Math., 2015, 2(1):1003175.
[9] Lanski C., An Engel condition with derivation for left ideals, Proc. Amer. Math. Soc., 1997, 125(2):339-345.
[10] Lee T. K., Wong T. L., Nonadditive strong commutativity preserving maps, Commu. Algebra, 2012, 40(6):2213-2218.
[11] Lin J. S., Liu C. K., Strong commutativity preserving maps on Lie ideals, Linear Algebra Appl., 2008, 428(7):1601-1609.
[12] Liu C. K., Strong commutativity preserving maps on subsets of matrices that are not closed under addition, Linear Algebra Appl., 2014, 458(10):280-290.
[13] Liu M. Y., Hou J. C., Strong 3-commutativity preserving maps on standard operator algebra, Acta Mathematica Sinica, English Series, 2017, 33(12):1659-1670.
[14] Liu M. Y., Hou J. C., Strong k-commutativity preserving map on real or complex 2×2 matrices, J. Taiyuan Univ. Tech., 2017, 48(2):254-259.
[15] Molnár L., Šemrl P., Non-linear commutativity preserving maps on self-adjont operators, Q. J. Math., 2005, 56(4):589-595.
[16] Qi X. F., Hou J. C., Nonlinear strong commutativity preserving maps on prime rings, Comm. Algebra, 2010, 38(8):2790-2796.
[17] Qi X. F., Strong 2-commutativity preserving maps on prime rings, Publ. Math. Debrecen, 2016, 88(1-2):119-129.
[18] S lowik R., Continuous maps on triangular matrices that preserve commutativity, Linear and Multilinear Algebra, 2016, 64(9):1716-1730.
国家自然科学基金资助项目（11671006）；山西省优秀青年基金项目（201701D211001）
/
| 〈 |
|
〉 |
