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Jordan Structures in Bounded Symmetric Domains

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Geometric Function Theory in Higher Dimension

Part of the book series: Springer INdAM Series ((SINDAMS,volume 26))

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Abstract

We discuss how Jordan algebraic structures arise from the geometry of bounded symmetric domains and their useful role in the study of holomorphic functions on these domains.

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References

  1. Abate, M.: Horospheres and iterates of holomorphic maps. Math. Z. 198, 225–238 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abate, M., Raissy, J.: Wolff-Denjoy theorems in non-smooth convex domains. Ann. Mat. Pura Appl. 193, 1503–1518 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arosio, L., Bracci, F.: Canonical models for holomorphic iteration. Trans. Am. Math. Soc. 368, 3305–3339 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bassanelli, G.: On horospheres and holomorphic endomorfisms of the Siegel disc. Rend. Sen. Mat. Univ. Padova 70, 147–165 (1983)

    MathSciNet  MATH  Google Scholar 

  5. Budzynska, M.: The Denjoy-Wolff theorem in \(\mathbb {C}^n\). Nonlinear Anal. 75, 22–29 (2012)

    Google Scholar 

  6. Budzyńska, M., Kuczumow, T., Reich, S.: Theorems of Denjoy-Wolff type. Ann. Mat. Pura Appl. 192, 621–648 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cartan, H.: Les fonctions de deux variables complexes et le problème de la représentation analytique. J. Math. Pures et Appl. 10, 1–114 (1931)

    MATH  Google Scholar 

  8. Cartan, É.: Sur les domaines bornés homogènes de l’espace de n variables complexes. Abh. Math. Semin. Univ. Hamburg 11, 116–162 (1935)

    Article  MATH  Google Scholar 

  9. Chu, C.-H.: Jordan Structures in Geometry and Analysis. Cambridge Tracts in Mathematics, vol. 190. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  10. Chu, C.-H.: Iteration of holomorphic maps on Lie balls. Adv. Math. 264, 114–154 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chu, C.-H., Mellon, P.: Iteration of compact holomorphic maps on a Hilbert ball. Proc. Am. Math. Soc. 125, 1771–1777 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chu, C.-H., Rigby, M.: Iteration of self-maps on a product of Hilbert balls. J. Math. Anal. Appl. 411, 773–786 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chu, C.-H., Rigby, M.: Horoballs and iteration of holomorphic maps on bounded symmetric domains. Adv. Math. 311, 338–377 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Earle, C.J., Hamilton, R.S.: A fixed point theorem for holomorphic mappings. Proc. Symp. Pure Math. 16, 61–65 (1969)

    Article  MATH  Google Scholar 

  15. Franzoni, T., Vesentini, E.: Holomorphic Maps and Invariant Distance. Mathematics Studies, vol. 40. North-Holland, Amsterdam (1980)

    Google Scholar 

  16. Harish-Chandra: Representations of semi-simple Lie groups VI. Am. J. Math. 78, 564–628 (1956)

    Google Scholar 

  17. Hervé, M.: Iteration des transformations analytiques dans le bicercle-unité. Ann. Sci. Ecole Norm. Sup. 71, 1–28 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hervé, M.: Quelques propriétés des applications analytiques d’une boule a m dimensions dans elle-meme. J. Math. Pures et Appl. 42, 117–147 (1963)

    MathSciNet  MATH  Google Scholar 

  19. Jacobson, N.: Structure and representations of Jordan algebras. Amer. Math. Soc. Colloq. Publ. 39 (1968)

    Google Scholar 

  20. Kantor, I.L.: Classification of irreducible transitive differential groups. Dokl. Akad. Nauk SSSR 158, 1271–1274 (1964)

    MathSciNet  Google Scholar 

  21. Kantor, I.L.: Transitive differential groups and invariant connections on homogeneous spaces. Trudy Sem. Vecktor. Tenzor. Anal. 13, 310–398 (1966)

    MathSciNet  Google Scholar 

  22. Kaup, W.: Algebraic characterization of symmetric complex Banach manifolds. Math. Ann. 228, 39–64 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kaup, W.: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z. 183, 503–529 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kaup, W.: Hermitian Jordan triple systems and their automorphisms of bounded symmetric domains. In: González, S. (ed.) Non-associative Algebra and Its Applications, Oviedo, 1993, pp. 204–214. Kluwer, Dordrecht (1994)

    Chapter  Google Scholar 

  25. Kaup, W.: On a Schwarz lemma for bounded symmetric domains. Math. Nachr. 197, 51–60 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kaup, W., Sauter, J.: Boundary structure of bounded symmetric domains. Manuscripta Math. 101, 351–360 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Koecher, M.: Imbedding of Jordan algebras into Lie algebras I. Bull. Am. J. Math. 89, 787–816 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  28. Koecher, M.: Jordan Algebras and Their Applications. University of Minnesota, 1962, Lecture Notes in Mathematics, vol. 1710. Springer, Heidelberg (1999)

    Google Scholar 

  29. Kapeluszny, J., Kuczmow, T., Reich, S.: The Denjoy-Wolff theorem in the open unit ball of a strictly convex Banach space. Adv. Math. 143, 111–123 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Loos, O.: Bounded symmetric domains and Jordan pairs (Mathematical Lectures). University of California, Irvine (1977)

    Google Scholar 

  31. McCrimmon, K.: Jordan algebras and their applications. Bull. Am. Math. Soc. 84, 612–627 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  32. McCrimmon, K.: A Taste of Jordan Algebras, Universitext. Springer, Heidelberg (2004)

    MATH  Google Scholar 

  33. Mellon, P.: Holomorphic invariance on bounded symmetric domains. J. Reine Angew. Math. 523, 199–223 (2000)

    MathSciNet  MATH  Google Scholar 

  34. Mellon, P.: Dejoy-Wolff theory for finite-dimensional bounded symmetric domains. Ann. Mate. 195, 845–855 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Meyberg, K.: Jordan-Tripelsysteme und die Koecher-Konstruktion von Lie-Algebren. Math. Z. 115, 58–78 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nirenberg, L., Webster, S., Yang, P.: Local boundary regularity of holomorphic mappings. Comm. Pure Appl. Math. 33, 305–338 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  37. Reich, S., Shoikhet, D.: The Denjoy-Wolff Theorem. In: Proceedings of Workshop on Fixed Point Theory (Kazimierz Dolny, 1997). Ann. Univ. Mariae Curie-Sk lodowska Sect. A51, pp. 219–240 (1997)

    Google Scholar 

  38. Satake, I.: Algebraic Structures of Symmetric Domains. Princeton University Press, Princeton (1980)

    MATH  Google Scholar 

  39. Stachura, A.: Iterates of holomorphic self-maps on the unit ball of a Hilbert space. Proc. Am. Math. Soc. 93, 88–90 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  40. Tits, J.: Une classe d’algèbres de Lie en relation avec les algèbres de Jordan. Indag. Math. 24, 530–535 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  41. Upmeier, H.: Über die Automorphismengruppe von Banach-Mannigfaltigkeiten mit invarianter Metrik. Math. Ann. 223, 279–288 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  42. Upmeier, H.: Jordan Algebras in Analysis, Operator Theory and Quantum Mechanics. CBMS. American Mathematical Society, Providence (1987)

    MATH  Google Scholar 

  43. Vigué, J.P.: Le groupe des automorphismes analytiques d’un domaine borné d’un espace de Banach complexe. Application aux domaines bornés symétriques. Ann. Sc. Ec. Norm. Sup. 9, 203–282 (1976)

    MATH  Google Scholar 

  44. Vinberg, E.B.: The theory of convex homogeneous cones. Trudy Moskov. Mat. Obsc. 12, 303–358 (1963)

    MathSciNet  MATH  Google Scholar 

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Chu, CH. (2017). Jordan Structures in Bounded Symmetric Domains. In: Bracci, F. (eds) Geometric Function Theory in Higher Dimension. Springer INdAM Series, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-73126-1_4

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