Bounded Symmetric Domains in Banach Spaces

This timely book exposes succinctly recent advances in the geometric and analytic theory of bounded symmetric domains. A unique feature is the unified treatment of both finite and infinite dimensional symmetric domains, using Jordan theory in tandem with Lie theory. The highlights include a generalized Riemann mapping theorem, which realizes a bounded symmetric domain as the open unit ball of a complex Banach space with a Jordan structure. Far-reaching applications of this realization in complex geometry and function theory are discussed.This monograph is intended as a convenient reference for researchers and graduate students in geometric analysis, infinite dimensional holomorphy as well as functional analysis and operator theory.Contents: Introduction:Holomorphic Maps in Banach SpacesBanach ManifoldsSymmetric Banach ManifoldsJordan and Lie Algebraic Structures:Jordan AlgebrasJordan Triple SystemsLie Algebras and Tits–Kantor–Koecher ConstructionJordan and Lie Structures in Banach SpacesCartan FactorsBounded Symmetric Domains:Algebraic Structures of Symmetric ManifoldsRealisation of Bounded Symmetric DomainsRank of a Bounded Symmetric DomainBoundary StructuresInvariant Metrics, Schwarz Lemma and DynamicsSiegel DomainsHolomorphic Homogeneous Regular DomainsClassificationFunction Theory :The Class SBloch Constant and Bloch MapsBanach Spaces of Bloch FunctionsComposition Operators
Readership: Graduate students and researchers in diverse mathematical fields including complex geometry, function theory, functional analysis and operator theory. It would also appeal to algebraists who are interested in the applications of Jordan and Lie algebras. Bounded Symmetric Domain;Hermitian Symmetric Space;Siegel Domain;Holomorphic Homogeneous Regular Manifold;Bloch Function;Jordan Algebra;Jordan Triple;Tits–Kantor–Koecher Lie Algebra;JB*-Triple;Banach Space0Key Features:The latest book which discusses recent advances in geometric analysis on bound symmetric domains of both finite and infinite dimensionsA unique feature is the use of Jordan theory in tandem with Lie theory to treat geometric and analytic topics such as rank and boundary structures, dynamics, Siegel domains and symmetric cones, classification, as well as function theoryIt contains a chapter on Jordan and Lie structures in Banach spaces, which are used to present a self-contained proof of a Riemann mapping theorem asserting that every bounded symmetric domain can be realized as the open unit ball of a complex Banach space with a Jordan structure