Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Lie $ \sp*$-triple homomorphisms into von Neumann algebras


Author: C. Robert Miers
Journal: Proc. Amer. Math. Soc. 58 (1976), 169-172
MSC: Primary 46L10
DOI: https://doi.org/10.1090/S0002-9939-1976-0410406-9
MathSciNet review: 0410406
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ and $ N$ be associative $ \ast $-algebras. A Lie $ \ast $-triple homomorphism of $ M$ into $ N$ is a $ \ast $-linear map $ \phi :M \to N$ such that

$\displaystyle \phi [[A,B],C] = [[\phi (A),\phi (B)],\phi (C)].$

(Here $ M$ and $ N$ are considered as Lie $ \ast $-algebras with $ [X,Y] = XY - YX.)$ In this note we prove that if $ N$ is a von Neumann algebra with no central abelian projections and if $ \phi $ is onto, there exists a central projection $ D$ in $ N$ such that $ D\phi $ is a Lie $ \ast $-homomorphism of $ [M,M]$, and $ (I - D)\phi $ is a Lie $ \ast $-antihomomorphism of $ [M,M]$.

References [Enhancements On Off] (What's this?)

  • [1] J. Dixmier, Les algèbres d'opérateurs dans l'espace Hilbertien, Cahiers Scientifiques, fasc. 25, Gauthier-Villars, Paris, 1969.
  • [2] N. Jacobson and C. E. Rickart, Jordan homomorphisms of rings, Trans. Amer. Math. Soc. 69 (1950), 479-502. MR 12, 387. MR 0038335 (12:387h)
  • [3] C. R. Miers, Lie homomorphisms of operator algebras, Pacific J. Math. 38 (1971), 717-737. MR 46 #7918. MR 0308804 (46:7918)
  • [4] C. Pearcy and D. Topping, Commutators in certain II$ _{1}$-factors, J. Functional Analysis 3 (1969), 69-78. MR 39 #789. MR 0239432 (39:789)
  • [5] H. Sunouchi, Infinite Lie rings, Tôhoku Math. J. (2) 8 (1956), 291-307. MR 0101262 (21:75)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46L10

Retrieve articles in all journals with MSC: 46L10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0410406-9
Keywords: Lie $ \ast $-triple homomorphism, von Neumann algebra
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society