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.
