Lie triple derivations of von Neumann algebras
HTML articles powered by AMS MathViewer
 by C. Robert Miers PDF
 Proc. Amer. Math. Soc. 71 (1978), 5761 Request permission
Abstract:
A Lie triple derivation of an associative algebra M is a linear map $L:M \to M$ such that \[ L[[X,Y],Z] = [ {[L(X),Y],Z} ] + [ {[X,L(Y)],Z} ] + [ {[X,Y],L(Z)} ]\] for all $X,Y,Z \in M$. (Here $[X,Y] = XY  YX$ and [M, M] is the linear subspace of M generated by such terms.) We show that if M is a von Neumann algebra with no central abelian summands then there exists an operator $A \in M$ such that $L(X) = [A,X] + \lambda (X)$ where $\lambda :M \to {Z_M}$ is a linear map which annihilates brackets of operators in M.References

J. Dixmier, Les algèbres d’operateurs dans l’espace Hilbertien, Cahiers Scientifiques, fasc. 25, GauthierVillars, Paris, 1969.
 Richard A. Howland, Lie isomorphisms of derived rings of simple rings, Trans. Amer. Math. Soc. 145 (1969), 383–396. MR 252441, DOI 10.1090/S00029947196902524410
 N. Jacobson and C. E. Rickart, Jordan homomorphisms of rings, Trans. Amer. Math. Soc. 69 (1950), 479–502. MR 38335, DOI 10.1090/S0002994719500038335X
 William G. Lister, A structure theory of Lie triple systems, Trans. Amer. Math. Soc. 72 (1952), 217–242. MR 45702, DOI 10.1090/S00029947195200457029
 Wallace S. Martindale III, Lie derivations of primitive rings, Michigan Math. J. 11 (1964), 183–187. MR 166234
 C. Robert Miers, Derived ring isomorphisms of von Neumann algebras, Canadian J. Math. 25 (1973), 1254–1268. MR 336365, DOI 10.4153/CJM19731320
 C. Robert Miers, Lie derivations of von Neumann algebras, Duke Math. J. 40 (1973), 403–409. MR 315466
 C. Robert Miers, Lie $^*$triple homomorphisms into von Neumann algebras, Proc. Amer. Math. Soc. 58 (1976), 169–172. MR 410406, DOI 10.1090/S00029939197604104069
 C. R. Putnam, Commutation properties of Hilbert space operators and related topics, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 36, SpringerVerlag New York, Inc., New York, 1967. MR 0217618
 Shôichirô Sakai, Derivations of $W^{\ast }$algebras, Ann. of Math. (2) 83 (1966), 273–279. MR 193528, DOI 10.2307/1970432
Additional Information
 © Copyright 1978 American Mathematical Society
 Journal: Proc. Amer. Math. Soc. 71 (1978), 5761
 MSC: Primary 46L10; Secondary 17B65
 DOI: https://doi.org/10.1090/S00029939197804874809
 MathSciNet review: 0487480