Bounded Symmetric Homogeneous Domains in Infinite Dimensional Spaces

In this article, we exhibit a large class of Banach spaces whose open unit balls are bounded symmetric homogeneous domains. These Banach spaces, which we call J*-algebras, are linear spaces of operators mapping one Hilbert space into another and have a kind of Jordan tripte product structure. In particular, all Hilbert spaces and all B*--algebras are J*-algebras. Moreover, all four types of the classical Cartan domains and their infinite dimensional analogues are the open unit balls of J*-algebras, and the same holds for any finite or infinite product of these domains. Thus we have a setting in which a large number of bounded symmetric homogeneous domains may be studied simultaneously. A particular advantage of this setting is the interconnection which exists between function-theoretic problems and problems of functional analysis. This leads to a simplified discussion of both types of problems. We shall see that the open unit balls of J*-algebras are natural generalizations of the open unit disc of the complex plane. In fact, we give an explicit algebraic formula for Mobius transformations of these balls and show that the origin can be mapped to any desired operator in the ball with one of the Mobius transformations. An extremal form of the Schwarz lemma then leads immediately to the representation of each biholomorphic mapping between the open unit balls of two J*-algebras as a composition of a Mobius transformation and a linear isometry of one of the J*-algebras onto the other. Such linear isometries reduce to a multiplication by unitary operators for mappings in the identity component of the group of all biholomorphic mappings of the open unit ball of a C*-algebra with identity. However, in general, linear isometries of one J*-algebra onto another can be complicated. Still, using the mentioned Schwarz lemma and Mobius transformations, we show that all such linear isometries preserve the J*-structure. A consequence of these results is that the open unit balls of two J*-a{gebras are holomorphically equivalent if and only if the J*-algebras are isometrically isomorphic under a mapping preserving the J * structure. Another consequence is that the open unit ball of a J*-algebra is holomorphically equivalent to a product of balls if and only if the J*-algebra is isometrically isomorphic to a product of J*--algebras. The last result connects the factorization of domains with the factorization of J*-algebras and has a number of interesting applications. For example, using Cartan's classification of bounded symmetric domains in C n, we classify all J*-algebras of dimension less than 16. Moreover, we reduce the problem of classifying all finite dimensional J*-algebras to the problem of finding some J*-algebras whose open unit balls are holomorphically equivalent to the two exceptional Cartan domains in dimensions 16 and 27, respectively, when such J*-algebras exist. If there are such J*-atgebras in both cases, then every bounded symmetric