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# Journal of Operator Theory

Volume 52, Issue 2, Fall 2004  pp. 325-340.

Lie ideals in operator algebras

Authors Alan Hopenwasser (1) and Vern Paulsen (2)
Author institution: (1) Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA
(2) Department of Mathematics, University of Houston, Houston, TX 77204-3476

Summary:  Let $\mathcal A$ be a Banach algebra for which the group of invertible elements is connected. A subspace $\mathcal L \subseteq \mathcal A$ 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 $C^*$-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.

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