Direct sums of Cartan factors (Q2366586)

In [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. Rend., IX. Ser. 1, No. 3, 203-213 (1990; Zbl 0747.46040)] we envisaged the orbit of the origin in the unit ball of a direct sum of two complex Banach spaces (endowed with a suitable norm), with respect to the group of holomorphic automorphisms, and we obtained some general results. As a special case we considered the class of \(p\)-norms, and we proved that the most interesting case is when \(p\) equals 2. For \(p=2\) we succeeded in giving some information about the orbit of the origin when one of the spaces is either a Hilbert space or a commutative \(C^*\)-algebra with identity. In this paper we consider the case when one of the spaces is a Cartan factor. The reason for considering Cartan factors is that, as we proved in [loc. cit.], only spaces in which the orbit of the origin in the unit ball is non-trivial can give rise to a direct sum in which such an orbit is non-trivial: and the unit ball of a Cartan factor is homogeneous. Our main result can be expressed in the following way: if \(F\) is a Cartan factor of type I, II, III or IV and \(F\) is not isometric to a Hilbert space, then, given a non-trivial complex Banach space \(G\), no point in the orbit of the origin in the unit ball of the 2-sum of \(G\) and \(F\) can have non-zero \(F\)-coordinate. In the last section we prove some results concerning duality theory for Cartan factors.