Lie ideals in operator algebras

Let A be a Banach algebra for which the group of invertible elements is connected. A subspace C subset of or equal to A is a Lie ideal in A if and only if it is invariant under inner antomorphisins. 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.