Inner ideals of finitary simple Lie algebras

Inner ideals of infinite dimensional finitary simple Lie algebras over a field of characteristic zero are described in geometric terms. We also study when these inner ideals are principal or minimal, and characterize those elements which are von Neumann regular. As a consequence we prove that any finitary central simple Lie algebra over a field of characteristic zero satisfies the descending chain condition on principal inner ideals. We also characterize when these algebras are Artinian, proving in particular that a finitary simple Lie algebra over an algebraically closed field of characteristic zero is Artinian if and only if it is finite dimensional. Because it is useful for our approach, we provide a characterization of the trace of a finite rank operator on a vector space over a division algebra which is intrinsic in the sense that it avoids imbeddings into finite matrices