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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
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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

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