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E-grāmata: Jordan Triple Systems in Complex and Functional Analysis

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Jordan Triple Systems in Complex and Functional AnalysisPalielināt
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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 (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.
Introduction vii
Part 1 Prom Bounded Domains to Symmetric Banach Manifolds
1(146)
Chapter 1 Analytic manifolds and their automorphism groups
3(16)
1.1 Analytic manifolds and analytic germs
3(4)
1.2 Vector fields and one-parameter groups
7(9)
1.3 Restrictions and extensions of vector fields
16(3)
Chapter 2 Uniform manifolds and their automorphism groups
19(18)
2.1 Locally uniform manifolds and their automorphisms
19(2)
2.2 Groups of locally uniform transformations
21(16)
Chapter 3 The semigroup Oc(X) of holomorphic contractions
37(16)
3.1 Cartan's uniqueness theorem. Algebraic version
37(2)
3.2 The set O(D) of selfmaps as a topological semigroup
39(4)
3.3 Cartan's uniqueness theorem. Topological version
43(5)
3.4 The semigroup Oc(X) of holomorphic contractions in X
48(5)
Chapter 4 Manifolds with a compatible invariant metric
53(22)
4.1 Invariant locally compatible metrics in a manifold
53(3)
4.2 Cartan's uniqueness theorem in manifolds
56(3)
4.3 Aut(X, d) as a group of locally uniform transformations
59(6)
4.4 Aut(X, d) as a group of analytic transformations
65(10)
Chapter 5 Manifolds with a compatible tangent norm
75(18)
5.1 Tangent norm and integrated metrics in a manifold
75(3)
5.2 Normed manifolds
78(7)
5.3 Infinitesimal version of Cartan's uniqueness theorem
85(4)
5.4 Construction of compatible invariant norms
89(4)
Chapter 6 Symmetric normed manifolds
93(8)
6.1 Symmetric normed manifolds
93(2)
6.2 The adjoint action Ad(sa) on the Lie algebra g(X)
95(2)
6.3 Homogeneity of symmetric normed manifolds
97(4)
Chapter 7 J*-triples and their related Lie algebras
101(16)
7.1 The category of Jordan-Banach triple systems
101(4)
7.2 The subcategory of J*-triples
105(1)
7.3 Binary Banach-Lie algebras
106(3)
7.4 Lie algebras associated with a J*-triple
109(8)
Chapter 8 The J*-triple associated with a symmetric manifold
117(10)
8.1 The canonical chart of a symmetric manifold
117(4)
8.2 The J*-triple associated to a symmetric normed manifold
121(1)
8.3 Construction of the canonical functor
122(5)
Chapter 9 The symmetric manifold associated with a J*-triple
127(20)
9.1 The quotient manifold G/H
128(2)
9.2 The canonical action of G on G/H
130(2)
9.3 The immersion E → G/H of E into the manifold G/H
132(1)
9.4 The symmetric manifold associated with a J*-triple
133(5)
9.5 Examples: J*-algebras and classical Cartan factors
138(9)
Part 2 Finite Rank J*-triples and JH*-triples
147(60)
Chapter 10 Algebraic study of J*-triples
149(26)
10.1 Tripotents in a J*-triple. Peirce decomposition
149(3)
10.2 Compatibility in a J*-triple
152(1)
10.3 Orthogonality in a J*-triple
153(6)
10.4 Orthogonal families. Joint Peirce decomposition
159(3)
10.5 Ideals and inner ideals in a J*-triple
162(10)
10.6 Nilpotency and Jacobson radical of a J*-triple
172(3)
Chapter 11 Atomic J*-triples and JH*-triples
175(32)
11.1 Finite rank J*-triples. Construction of the trace form
175(2)
11.2 Finite rank J*-triples. J*-isomorphic classification
177(15)
11.3 Hilbertian triple systems or JH*-triples
192(11)
11.4 Hermitian symmetric manifolds
203(4)
Part 3 From Symmetric Banach Manifolds to JB*-Triples
207(190)
Chapter 12 Spectral properties and bounded J*-triples
209(10)
12.1 Spectral properties of J*-triples
209(2)
12.2 Anisotropy and the spectral seminorm in a J*-triple
211(2)
12.3 Bounded J*-triples and the Caratheodory metrics
213(3)
12.4 Abelian J*-triples
216(3)
Chapter 13 The Riemann mapping theorem for JB*-triples
219(18)
13.1 Monogeneous J*-triples. The Jordan representation
219(1)
13.2 The manifold associated to a monogeneous J*-subtriple
219(7)
13.3 The manifold of a monogeneous J*-triple, revisited
226(3)
13.4 Bounded J*-triples and the Riemann mapping theorem
229(8)
Chapter 14 The category of JB*-triples
237(20)
14.1 Gelfand theory for abelian J*-triples
237(8)
14.2 JB*-triples. Uniqueness of the structure
245(3)
14.3 Products and quotients of JB*-triples
248(4)
14.4 The projection theorem
252(1)
14.5 Ultraproducts of JB*-triples
253(2)
14.6 The bidual of a JB*-triple
255(2)
Chapter 15 Automorphisms of bounded symmetric domains
257(26)
15.1 Jordan families and J*-triples
257(1)
15.2 Integration of certain vector fields
258(10)
15.3 The manifold associated with a JB*-triple, revisited
268(8)
15.4 Examples: Automorphisms of the unit ball in Cartan factors
276(2)
15.5 Behaviour of automorphisms at the boundary of the ball
278(5)
Chapter 16 Tripotents in JB*-triples
283(32)
16.1 Existence of non-zero tripotents in JB*-triples
283(1)
16.2 JB*-triples and their related JB*-algebras
283(4)
16.3 Peirce projectors in JB*-triples
287(3)
16.4 Order relation in the set of tripotents
290(5)
16.5 Regular tripotents and extreme points
295(4)
16.6 Examples: Tripotents in J*-algebras and Cartan factors
299(16)
Chapter 17 Functional calculus in a JB*-triple. Applications
315(10)
17.1 The odd functional calculus
315(2)
17.2 The structure group of a JB*-triple
317(3)
17.3 Invertible elements in a JB*-triple
320(2)
17.4 The structure group of the JB*-triple C(Ω)
322(3)
Chapter 18 Automorphisms of Banach-Grassmann manifolds
325(28)
18.1 Banach-Grassmann manifolds
325(4)
18.2 The group of collineations of a Grassmann manifold
329(4)
18.3 Collineations of Hilbert-Grassmann manifolds
333(3)
18.4 Correlations of Grassmann manifolds
336(3)
18.5 Holomorphic vector fields in Hilbert-Grassmann manifolds
339(7)
18.6 Groups of automorphims of Grassmann manifolds
346(7)
Chapter 19 Symmetric Grassmann manifolds over Hilbert spaces
353(22)
19.1 The dual manifolds of the classical symmetric manifolds In,m
353(1)
19.2 An alternative description of the unit ball l0
354(2)
19.3 Complete vector fields on the symmetric manifolds In,m
356(1)
19.4 Isometries of rectangular Cartan factors In,m
357(1)
19.5 The symmetric dual manifolds of IIn and IIIn
358(3)
19.6 Complete vector fields on the manifolds IIn and IIIn
361(2)
19.7 Isometries of Cartan factors IIn and IIIn
363(1)
19.8 Complex quadrics. The symmetric dual manifold of IVn
364(4)
19.9 The quadrics Qn as the symmetric dual manifold of IVn
368(4)
19.10 Isometries of Cartan factors of type IVn
372(3)
Chapter 20 Affine structure of the unit ball in a JB*-triple
375(22)
20.1 Affine structure of the unit ball in a Banach space
375(5)
20.2 Affine structure of the unit ball of a JB*-triple
380(8)
20.3 Boundaries, stable and determining subsets in JB*-triples
388(4)
20.4 The dynamic system associated to a JB*-triple
392(1)
20.5 The manifold of tripotents in a JB*-triple
393(4)
Part 4 JB*-triples in Dual Banach Spaces or JBW*-triples
397(130)
Chapter 21 JB*-triples in dual Banach spaces
399(26)
21.1 Definition and elementary properties of JBW*-triples
399(5)
21.2 Characterisation of the predual of a JBW*-triple
404(2)
21.3 Uniqueness of the predual of a JBW*-triple
406(5)
21.4 Separate w*-w*-continuity and uniqueness of the predual
411(5)
21.5 The bidual of a JB*-triple is a JBW*-triple
416(5)
21.6 Separate w*-w*-continuity of the triple product in a JBW*-triple
421(2)
21.7 Examples: Cartan factors are JBW*-triples
423(2)
Chapter 22 Structure theory for JBW*-triples and their preduals
425(42)
22.1 w*-closed ideals in a JBW*-triple
425(9)
22.2 Representations of JB*-triples
434(6)
22.3 Ideals in JB*-algebras and JB*-triples
440(1)
22.4 Normal functional on a JBW*-triple
441(6)
22.5 Atomic and non-atomic ideals of a JBW*-triple
447(9)
22.6 The Gelfand-Naimark theorem
456(4)
22.7 Normal representations of JBW*-triples
460(7)
Chapter 23 Facial structure in JBW*-triples and in JB*-triples
467(26)
23.1 Facial structure of convex sets
467(4)
23.2 Facial structure of the unit ball in Banach spaces
471(3)
23.3 Facial structure of the predual of a JBW-algebra
474(1)
23.4 Support of an element in a JBW*-triple
475(6)
23.5 Facial structure of the predual of a JBW*-triple
481(12)
Chapter 24 The strong and strong* topologies in JBW*-triples
493(18)
24.1 The strong* topology in JBW*-triples
493(8)
24.2 The strong topology in JB*-triples
501(2)
24.3 Behaviour of the strong and the strong* topologies
503(3)
24.4 Admissible topologies and holomorphic automorphisms
506(5)
Chapter 25 Derivations of JB*-triples
511(16)
25.1 Automatic continuity of derivations
511(2)
25.2 Existence of outer derivations
513(8)
25.3 Approximation by inner derivations
521(6)
Appendix A Some results on functional analysis
527(16)
A.1 Monogeneous J*-triples. The Gelfand representation
527(8)
A.2 Derivations of the Banach-Lie algebra C(E)
535(5)
A.3 Isomorphisms of the Banach algebra C(E)
540(3)
List of symbols and their meanings 543(4)
Bibliography 547(10)
Index 557
Jose M. Isidro, University of Santiago de Compestela, Santiago de Composetla, Galicia, Spain.
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