Transitivity of inner automorphisms in infinite dimensional Cartan factors

Abstract

Abstract. 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. [15]. 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.

A Holomorphic Characterization of Operator Algebras

“…The same is true for finite dimensional Cartan factors of type 4 by [26]. Using these facts and the fact that the set of inner automorphisms (hence isometries) of any Cartan factor acts transitively on the set of minimal tripotents (and hence on finite rank tripotents of the same rank, [14]), one can show that if the completely symmetric part of a finite dimensional Cartan factor of type 2, 3 or 4 is not zero, then it must contain any grid which spans the Cartan factor and hence is completely isometric to a TRO. (See Problems 3 and 5 at the end of this paper.)…”