**Mathematical Surveys and Monographs**

Volume: 243;
2019;
560 pp;
Hardcover

MSC: Primary 46; 58; 53; 32; 17; 57; 22;

**Print ISBN: 978-1-4704-5083-0
Product Code: SURV/243**

List Price: $129.00

AMS Member Price: $103.20

MAA Member Price: $116.10

**Electronic ISBN: 978-1-4704-5440-1
Product Code: SURV/243.E**

List Price: $129.00

AMS Member Price: $103.20

MAA Member Price: $116.10

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#### Supplemental Materials

# Jordan Triple Systems in Complex and Functional Analysis

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*José M. Isidro*

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 Sociedad Matemática Española (RSME)

#### Readership

Graduate students interested in complex analysis and the theory of Banach spaces.

#### Table of Contents

# Table of Contents

## Jordan Triple Systems in Complex and Functional Analysis

- Cover Cover11
- Title page i2
- Introduction vii8
- Part 1 . From Bounded Domains to Symmetric Banach Manifolds 116
- Chapter 1. Analytic manifolds and their automorphism groups 318
- Chapter 2. Uniform manifolds and their automorphism groups 1934
- Chapter 3. The semigroup 𝒪_{𝒸}(𝒳) of holomorphic contractions 3752
- Chapter 4. Manifolds with a compatible invariant metric 5368
- Chapter 5. Manifolds with a compatible tangent norm 7590
- Chapter 6. Symmetric normed manifolds 93108
- Chapter 7. J*-triples and their related Lie algebras 101116
- Chapter 8. The J*-triple associated with a symmetric manifold 117132
- Chapter 9. The symmetric manifold associated with a J*-triple 127142

- Part 2 . Finite Rank J*-triples and JH*-triples 147162
- Chapter 10. Algebraic study of J*-triples 149164
- 10.1. Tripotents in a J*-triple. Peirce decomposition 149164
- 10.2. Compatibility in a J*-triple 152167
- 10.3. Orthogonality in a J*-triple 153168
- 10.4. Orthogonal families. Joint Peirce decomposition 159174
- 10.5. Ideals and inner ideals in a J*-triple 162177
- 10.6. Nilpotency and Jacobson radical of a J*-triple 172187

- Chapter 11. Atomic J*-triples and JH*-triples 175190

- Part 3 . \040From Symmetric Banach Manifolds to JB*-Triples 207222
- Chapter 12. Spectral properties and bounded J*-triples 209224
- Chapter 13. The Riemann mapping theorem for JB*-triples 219234
- Chapter 14. The category of JB*-triples 237252
- Chapter 15. Automorphisms of bounded symmetric domains 257272
- Chapter 16. Tripotents in JB*-triples 283298
- 16.1. Existence of non-zero tripotents in JB*-triples 283298
- 16.2. JB*-triples and their related JB*-algebras 283298
- 16.3. Peirce projectors in JB*-triples 287302
- 16.4. Order relation in the set of tripotents 290305
- 16.5. Regular tripotents and extreme points 295310
- 16.6. Examples: Tripotents in J*-algebras and Cartan factors 299314

- Chapter 17. Functional calculus in a JB*-triple. Applications 315330
- Chapter 18. Automorphisms of Banach-Grassmann manifolds 325340
- 18.1. Banach-Grassmann manifolds 325340
- 18.2. The group of collineations of a Grassmann manifold 329344
- 18.3. Collineations of Hilbert-Grassmann manifolds 333348
- 18.4. Correlations of Grassmann manifolds 336351
- 18.5. Holomorphic vector fields in Hilbert-Grassmann manifolds 339354
- 18.6. Groups of automorphims of Grassmann manifolds 346361

- Chapter 19. Symmetric Grassmann manifolds over Hilbert spaces 353368
- 19.1. The dual manifolds of the classical symmetric manifolds 𝐈_{𝐧,𝐦} 353368
- 19.2. An alternative description of the unit ball 𝔄₀ 354369
- 19.3. Complete vector fields on the symmetric manifolds 𝐈_{𝐧,𝐦} 356371
- 19.4. Isometries of rectangular Cartan factors 𝐈_{𝐧,𝐦} 357372
- 19.5. The symmetric dual manifolds of 𝐈𝐈_{𝐧} and 𝐈𝐈𝐈_{𝐧} 358373
- 19.6. Complete vector fields on the manifolds 𝐈𝐈_{𝐧} and 𝐈𝐈𝐈_{𝐧} 361376
- 19.7. Isometries of Cartan factors 𝐈𝐈_{𝐧} and 𝐈𝐈𝐈_{𝐧} 363378
- 19.8. Complex quadrics. The symmetric dual manifold of 𝐈𝐕_{𝐧} 364379
- 19.9. The quadrics \Q_{𝑛} as the symmetric dual manifold of 𝐈𝐕_{𝐧} 368383
- 19.10. Isometries of Cartan factors of type 𝐈𝐕_{𝐧} 372387

- Chapter 20. Affine structure of the unit ball in a JB*-triple 375390
- 20.1. Affine structure of the unit ball in a Banach space 375390
- 20.2. Affine structure of the unit ball of a JB*-triple 380395
- 20.3. Boundaries, stable and determining subsets in JB*-triples 388403
- 20.4. The dynamic system associated to a JB*-triple 392407
- 20.5. The manifold of tripotents in a JB*-triple 393408

- Part 4 . JB*-triples in Dual Banach Spaces or JBW*-triples 397412
- Chapter 21. JB*-triples in dual Banach spaces 399414
- 21.1. Definition and elementary properties of JBW*-triples 399414
- 21.2. Characterisation of the predual of a JBW*-triple 404419
- 21.3. Uniqueness of the predual of a JBW*-triple 406421
- 21.4. Separate 𝑤*-𝑤*-continuity and uniqueness of the predual 411426
- 21.5. The bidual of a JB*-triple is a JBW*-triple 416431
- 21.6. Separate 𝑤*-𝑤*-continuity of the triple product in a JBW*-triple 421436
- 21.7. Examples: Cartan factors are JBW*-triples 423438

- Chapter 22. Structure theory for JBW*-triples and their preduals 425440
- 22.1. 𝑤*-closed ideals in a JBW*-triple 425440
- 22.2. Representations of JB*-triples 434449
- 22.3. Ideals in JB*-algebras and JB*-triples 440455
- 22.4. Normal functionals on a JBW*-triple 441456
- 22.5. Atomic and non-atomic ideals of a JBW*-triple 447462
- 22.6. The Gelfand-Naimark theorem 456471
- 22.7. Normal representations of JBW*-triples 460475

- Chapter 23. Facial structure in JBW*-triples and in JB*-triples 467482
- Chapter 24. The strong and strong* topologies in JBW*-triples 493508
- Chapter 25. Derivations of JB*-triples 511526
- Appendix A. Some results on functional analysis 527542
- List of symbols and their meanings 543558
- Bibliography 547562
- Index 557572

- Back Cover Back Cover1577