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Hardcover ISBN:  9781470450830 
Product Code:  SURV/243 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470454401 
Product Code:  SURV/243.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470450830 
eBook ISBN:  9781470454401 
Product Code:  SURV/243.B 
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AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 243; 2019; 560 ppMSC: Primary 46; 58; 53; 32; 17; 57; 22
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 \(\mathrm{JB}^*\)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 Sociedád Matematica Española.ReadershipGraduate students interested in complex analysis and the theory of Banach spaces.

Table of Contents

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 BanachGrassmann 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

JB$^*$triples in dual Banach spaces

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


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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 \(\mathrm{JB}^*\)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.
Graduate students interested in complex analysis and the theory of 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 BanachGrassmann 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

JB$^*$triples in dual Banach spaces

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