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 (Vigue 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 $\mathrm{JB}^*$-triples and $\mathrm{JBW}^*$-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. This book is published in cooperation with Real Sociedad Matematica Espanola.
Jose M. Isidro, University of Santiago de Compestela, Santiago de Composetla, Galicia, Spain.
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.
