Bilocal derivations of standard operator algebras
HTML articles powered by AMS MathViewer
- by Jun Zhu and Changping Xiong
- Proc. Amer. Math. Soc. 125 (1997), 1367-1370
- DOI: https://doi.org/10.1090/S0002-9939-97-03722-2
- PDF | Request permission
Abstract:
In this paper, we shall show the following two results: (1) Let $A$ be a standard operator algebra with $I$, if $\Phi$ is a linear mapping on $A$ which satisfies that $\Phi (T)$ maps $\ker T$ into $\operatorname {ran} T$ for all $T\in A$, then $\Phi$ is of the form $\Phi (T)=TA+BT$ for some $A,B$ in $B(X)$. (2) Let $X$ be a Hilbert space, if $\Phi$ is a norm-continuous linear mapping on $B(X)$ which satisfies that $\Phi (P)$ maps $\ker P$ into $\operatorname {ran} P$ for all self-adjoint projection $P$ in $B(X)$, then $\Phi$ is of the form $\Phi (T)=TA+BT$ for some $A,B$ in $B(X)$.References
- Matej Brešar, Characterizations of derivations on some normed algebras with involution, J. Algebra 152 (1992), no. 2, 454–462. MR 1194314, DOI 10.1016/0021-8693(92)90043-L
- S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089
- Paul R. Chernoff, Representations, automorphisms, and derivations of some operator algebras, J. Functional Analysis 12 (1973), 275–289. MR 0350442, DOI 10.1016/0022-1236(73)90080-3
- Richard V. Kadison, Local derivations, J. Algebra 130 (1990), no. 2, 494–509. MR 1051316, DOI 10.1016/0021-8693(90)90095-6
- David R. Larson and Ahmed R. Sourour, Local derivations and local automorphisms of ${\scr B}(X)$, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 187–194. MR 1077437, DOI 10.1090/pspum/051.2/1077437
- Peter emrl, Additive derivations of some operator algebras, Illinois J. Math. 35 (1991), no. 2, 234–240. MR 1091440
Bibliographic Information
- Jun Zhu
- Affiliation: Department of Mathematics, Hubei Institute for Nationalities, Enshi, Hubei, 445000, People’s Republic of China
- Changping Xiong
- Affiliation: Department of Mathematics, Hubei Institute for Nationalities, Enshi, Hubei, 445000, People’s Republic of China
- Received by editor(s): June 14, 1995
- Received by editor(s) in revised form: November 8, 1995
- Additional Notes: Project supported by the Science Foundation of HBEC, People’s Republic of China
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 1367-1370
- MSC (1991): Primary 47D30, 47D25, 47B47
- DOI: https://doi.org/10.1090/S0002-9939-97-03722-2
- MathSciNet review: 1363442