Lie Ideals in Operator Algebras

2002/11/21 by Alan Hopenwasser, Hopenwasser, Alan, Vern Paulsen +1
Mathematics · #47L40 #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:47L40

paper · pdf · doi:10.48550/arxiv.math/0211347

LaTeX; approx. 18 pages

arxiv created 2002/11/21 · arxiv updated 2009/11/30

Abstract

Let \mathcal A be a Banach algebra for which the group of invertible elements is connected. A subspace \mathcal L ⊆ \mathcal A is a Lie ideal in \mathcal A if, and only if, it is invariant under inner automorphisms. This applies, in particular, to any canonical subalgebra of an AF \ensuremathC^*-algebra. The same theorem is also proven for strongly closed subspaces of a totally atomic nest algebra whose atoms are ordered as a subset of the integers and for CSL subalgebras of such nest algebras. We also give a detailed description of the structure of a Lie ideal in any canonical triangular subalgebra of an AF \ensuremathC^*-algebra.

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