Algebraic characterization of symmetric complex Banach manifolds

Abstract

The symmetric hermitian complex manifolds (of finite dimension) have been classi fled completely by E. Cartan [4] using the classification of simple complex Lie algebras. A Jordan theoretic approach is due to Koecher [18] and more recently to Loos [25] : The symmetric bounded domains are in a one-to-one correspondence to hermitian Jordan triple systems, for which a certain trace form is positive-definite hermitian. Bounded symmetric domains in infinite dimensions have been considered by various authors [7, 9, 11, 14-16, 28, 32]. Harris for instance proved for a big class of complex Banach spaces U (including all C*-algebras) that the open unit ball of U is homogeneous and hence symmetric. In this paper we study symmetric complex Banach manifolds. A complex Banach manifold is a complex manifold (of possibly infinite dimension) M together with a fixed norm v on the tangent bundle of M. We do not require that the restriction of v to every tangent space T~, x~M, is a Hilbert norm--otherwise we would exclude many interesting examples such as the open unit ball in the Banach space L(H) of all bounded operators on a Hilbert space H with d imH= oo. Therefore our notion of hermitian Jordan triple system cannot rely on trace forms. The major part of the paper is devoted to the proof of the following Main Theorem. The category of simply connected, symmetric, complex Banach manifolds with base point is equivalent to the category of hermitian Jordan triple systems.