Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-20T08:16:10.362Z Has data issue: false hasContentIssue false
  • English
  • Français

Closed Lie Ideals in Operator Algebras

Published online by Cambridge University Press:  20 November 2018

C. Robert Miers
C. Robert Miers*
Affiliation:
University of Victoria, Victoria, British Columbia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If M is an associative algebra with product xy, M can be made into a Lie algebra by endowing M with a new multiplication [x, y] = xy – yx. The Poincare-Birkoff-Witt Theorem, in part, shows that every Lie algebra is (Lie) isomorphic to a Lie subalgebra of such an associative algebra M. A Lie ideal in M is a linear subspace U ⊆ M such that [x, u] ∊ U for all x £ M, u ∊ U. In [9], as a step in characterizing Lie mappings between von Neumann algebras, Lie ideals which are closed in the ultra-weak topology, and closed under the adjoint operation are characterized when If is a von Neumann algebra. However the restrictions of ultra-weak closure and adjoint closure seemed unnatural, and in this paper we characterize those uniformly closed linear subspaces which can occur as Lie ideals in von Neumann algebras.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 5 , 01 October 1981 , pp. 1271 - 1278
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Akemann, C. A. and Pedersen, G. K., Ideal perturbations of elements in C*-algebras, Math. Scand. 41 (1977), 117–139.Google Scholar
2. Anderson, J., Commutators of compact operators, J. Reine und Ang. Math. 291 (1977), 128–132.Google Scholar
3. Calkin, J. W., Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. Math. 42 (1941), 839–873.Google Scholar
4. Civin, P. and Yood, B., Lie and Jordan structures in Banach algebras, Pacific J. of Math. 15 (1965), 775–797.Google Scholar
5. Dixmier, J., Les algebres d'opérateurs dans l'espace Hilbertien ﹛Algebres de von Neumann), (Gauthier-Villars, Paris, 1969).Google Scholar
6. Fack, T. and de la Harpe, P., Sommes de commutateurs dans les algebres de von Neumann finis continues, preprint.Google Scholar
7. Fillmore, P. A., On products of symmetries, Can. J. Math. 18 (1966), 897–900.Google Scholar
8. Herstein, I. N., Topics in ring theory, (University of Chicago Press, Chicago and London, 1969).Google Scholar
9. Miers, C. R., Lie homomorphisms of operator algebras, Pacific J. Math. 38 (1971), 717–735.Google Scholar
10. Pearcy, C. and Topping, D. M., Commutators and certain II1-factors, J. Functional Anal. 3 (1969), 69–78.Google Scholar
11. Pearcy, C. and Topping, D. M., Sums of small numbers of idempotents, Michigan Math. J. 14 (1967), 453–465.Google Scholar
12. Putnam, C. R., Commutation properties of Hilbert space operators and related topics, Ergebnisse der Mathematik 36 (Springer-Verlag, Berlin/New York, 1967).Google Scholar
13. Salinas, N., Ideals of commutators of compact operators, Acta Sci. Math. 36 (1974), 131–144.Google Scholar
14. Singer, I. M., Uniformly continuous representation of Lie groups, Annals of Math. 56 (1952), 242–247.Google Scholar
15. Sunouchi, H., Infinite Lie rings, Tôhoku Math. J. 8 (1956), 291–307.Google Scholar
16. Topping, D. M., Transcendental quasi-nilpotents in operator algebras, J. Functional Anal. 2 (1968), 342–351.Google Scholar