Centralizing mappings on von Neumann algebras
HTML articles powered by AMS MathViewer
- by Matej Brešar PDF
- Proc. Amer. Math. Soc. 111 (1991), 501-510 Request permission
Abstract:
Let $R$ be a ring with center $Z(R)$. A mapping $F$ of $R$ into itself is called centralizing if $F(x)x - xF(x) \in Z(R)$ for all $x \in R$. The main result of this paper states that every additive centralizing mapping $F$ on a von Neumann algebra $R$ is of the form $F(x) = cx + \zeta (x),x \in R$, where $c \in Z(R)$ and $\zeta$ is an additive mapping from $R$ into $Z(R)$. We also consider $\alpha$-derivations and some related mappings, which are centralizing on rings and Banach algebrasReferences
- Ram Awtar, Lie and Jordan structure in prime rings with derivations, Proc. Amer. Math. Soc. 41 (1973), 67–74. MR 318233, DOI 10.1090/S0002-9939-1973-0318233-5
- H. E. Bell and W. S. Martindale III, Centralizing mappings of semiprime rings, Canad. Math. Bull. 30 (1987), no. 1, 92–101. MR 879877, DOI 10.4153/CMB-1987-014-x
- H. E. Bell and W. S. Martindale III, Semiderivations and commutativity in prime rings, Canad. Math. Bull. 31 (1988), no. 4, 500–508. MR 971579, DOI 10.4153/CMB-1988-072-9 M. Brešar, Centralizing mappings and derivations in prime rings, preprint.
- M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc. 110 (1990), no. 1, 7–16. MR 1028284, DOI 10.1090/S0002-9939-1990-1028284-3
- M. Brešar and J. Vukman, On some additive mappings in rings with involution, Aequationes Math. 38 (1989), no. 2-3, 178–185. MR 1018911, DOI 10.1007/BF01840003
- L. O. Chung and Jiang Luh, On semicommuting automorphisms of rings, Canad. Math. Bull. 21 (1978), no. 1, 13–16. MR 480607, DOI 10.4153/CMB-1978-003-4
- I. N. Herstein, Rings with involution, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1976. MR 0442017
- Yasuyuki Hirano, Arif Kaya, and Hisao Tominaga, On a theorem of Mayne, Math. J. Okayama Univ. 25 (1983), no. 2, 125–132. MR 719626 R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras, vol. 1, Academic Press, London, 1983; vol. 2. Academic Press, London, 1986.
- Irving Kaplansky, Rings of operators, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0244778
- Charles Lanski, Differential identities, Lie ideals, and Posner’s theorems, Pacific J. Math. 134 (1988), no. 2, 275–297. MR 961236, DOI 10.2140/pjm.1988.134.275
- Wallace S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576–584. MR 238897, DOI 10.1016/0021-8693(69)90029-5
- Martin Mathieu, Elementary operators on prime $C^*$-algebras. I, Math. Ann. 284 (1989), no. 2, 223–244. MR 1000108, DOI 10.1007/BF01442873
- Joseph H. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19 (1976), no. 1, 113–115. MR 419499, DOI 10.4153/CMB-1976-017-1
- Joseph H. Mayne, Ideals and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 86 (1982), no. 2, 211–212. MR 667275, DOI 10.1090/S0002-9939-1982-0667275-4
- Joseph H. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull. 27 (1984), no. 1, 122–126. MR 725261, DOI 10.4153/CMB-1984-018-2
- C. Robert Miers, Centralizing mappings of operator algebras, J. Algebra 59 (1979), no. 1, 56–64. MR 541670, DOI 10.1016/0021-8693(79)90152-2
- Edward C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100. MR 95863, DOI 10.1090/S0002-9939-1957-0095863-0
- A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), 166–170. MR 233207, DOI 10.1090/S0002-9939-1969-0233207-X
- Marc P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. (2) 128 (1988), no. 3, 435–460. MR 970607, DOI 10.2307/1971432
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 111 (1991), 501-510
- MSC: Primary 46L57; Secondary 16E50, 46L10, 46L40
- DOI: https://doi.org/10.1090/S0002-9939-1991-1028283-2
- MathSciNet review: 1028283