JORDAN TRIPLE SYSTEMS IN COMPLEX AND FUNCTIONAL ANALYSIS

JB$^*$-triples in dual Banach spaces
From bounded domains to symmetric Banach manifolds: Analytic manifolds and their automorphism groups
Uniform manifolds and their automorphism groups
The semigroup $\mathcal{O}_c(X)$ of holomorphic contractions
Manifolds with a compatible invariant metrics
Manifolds with a compatible tangent norm
Symmetric normed manifolds
J$^*$-triples and their related Lie algebras
The J$^*$-triple associated with a symmetric manifold
The symmetric manifold associated with a J$^*$-triple
Finite Rank J$^*$-triples and JH$^*$-triples: Algebraic study of J$^*$-triples
Atomic J$^*$-triples and JH$^*$-triples
From symmetric Banach manifolds to JB$^*$-triples: Spectral properties and bounded J$^*$-triples
The Riemann mapping theorem for JB$^*$-triples
The category of JB$^*$-triples
Automorphisms of bounded symmetric domains
Tripotents in JB$^*$-triples
Functional calculus in a JB$^*$-triple. Applications
Automorphisms of Banach-Grassmann manifolds
Symmetric Grassmann manifolds over Hilbert spaces
Affine structure of the unit ball in a JB$^*$-triple
JB$^*$-triples in dual Banach spaces or JBW$^*$-triples: Structure theory for JBW$^*$-triples and their preduals
Facial structure in JBW$^*$-triples and in JB$^*$-triples
The strong and strong* topologies in JBW$^*$-triples
Derivations of JB$^*$-triples
Some results on functional analysis
Lists of symbols and their meanings
Bibliography
Index.

This book is a systematic account of the impressive developments in the theory of symmetric manifolds achieved over the past 50 years. It contains detailed and friendly, but rigorous, proofs of the key results in the theory. Milestones are the study of the group of holomomorphic automorphisms of bounded domains in a complex Banach space (Vigué and Upmeier in the late 1970s), Kaup's theorem on the equivalence of the categories of symmetric Banach manifolds and that of hermitian Jordan triple systems, and the culminating point in the process: the Riemann mapping theorem for complex Banach spaces (Kaup, 1982). This led to the introduction of wide classes of Banach spaces known as JB*JB*-triples and JBW*JBW*-triples whose geometry has been thoroughly studied by several outstanding mathematicians in the late 1980s.

The book presents a good example of fruitful interaction between different branches of mathematics, making it attractive for mathematicians interested in various fields such as algebra, differential geometry and, of course, complex and functional analysis.