Lie Ideals in Operator Algebras

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Let A be a Banach algebra for which the group of invertible elements is connected. A subspace L A is a Lie ideal in A if, and only if, it is invariant under inner automorphisms. This applies, in particular, to any canonical subalgebra of an AF C*-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 C*-algebra.