Transitivity of inner automorphisms in infinite dimensional Cartan factors

Let C be a Cartan-factor having arbitrary dimension dimC. It is shown that the group Inn(C) of inner automorphisms of C acts transitively on the manifold Ur(C) of tripotents with finite rank r in C. This extends results by Loos (Bounded Symmetric Domains and Jordan Pairs. Mathematical Lectures. University of California, Irvine, 1977) valid in finite dimensions, and similar findings by Isidro et al. (Math Z 233(4):741–754, 2000; Acta Sci Math (Szeged) 66(3–4), 2000; Expo Math 20(2):97–116, 2002; Q J Math 57(4):505–525, 2006). Hence, the results presented here close a significant gap concerning the transitivity property of the general infinite-dimensional case. The proofs given here are based on new methods, independent of those used for the finite-dimensional cases.