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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Causal compactification of compactly causal spaces


Author: Frank Betten
Journal: Trans. Amer. Math. Soc. 355 (2003), 4699-4721
MSC (2000): Primary 53C35; Secondary 32M15
DOI: https://doi.org/10.1090/S0002-9947-03-03010-1
Published electronically: July 28, 2003
MathSciNet review: 1997579
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Abstract: We give a classification of causal compactifications of compactly causal spaces. Introduced by Ólafsson and Ørsted, for a compactly causal space $G/H$, these compactifications are given by $G$-orbits in the Bergman-Silov boundary of $G_1/K_1$, with $G \subset G_1$ and $(G_1, K_1, \theta)$ a Hermitian symmetric space of tube type. For the classical spaces an explicit construction is presented.


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  • [Be] W. Bertram, Algebraic structures of Makarevic spaces I, Transformation Groups 3 (1988), 3-32. MR 99c:32047
  • [FÓ] F. Betten and G. Ólafsson, Causal compactification and Hardy spaces for spaces of Hermitian type, Pacific J. Math. 200 (2001), 273-312. CMP 2002:05
  • [He 78] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York 1978 MR 80k:53081
  • [HÓ 96] J. Hilgert and G. Ólafsson, Causal symmetric spaces, The Geometry and Harmonic Analysis, Perspectives in Math. 18, Academic Press, New York 1996 MR 97m:43006
  • [HÓØ] J. Hilgert, G. Ólafsson, and B. Ørsted, Hardy spaces on affine symmetric spaces, J. reine angew. Math. 415 (1991), 189-218 MR 92h:22030
  • [Ka 87] S. Kaneyuki, On orbit structure of compactification of para-hermitian symmetric spaces, Japan J. Math. 13 (1987), 333-370 MR 88m:53094
  • [Ka 89] S. Kaneyuki, On the causal structures of the Silov boundary of symmetric bounded domains, in ``Prospects in Complex Geometry'', Springer LN 1468, New York 1989, pp. 127-159 MR 92h:32049
  • [Ko 93] K. Koufany, Semi-groupe de Lie associé à une algèbre de Jordan euclidienne, thesis, Université de Nancy I, 1993
  • [KØ96] K. Koufany and B. Ørsted, Function spaces on the Ol'shanskii semigroup and the Gelf'and-Gindikin program, Ann. Inst. Fourier 46 (1996), 689-722 MR 97k:22021
  • [KØ97] K. Koufany and B. Ørsted, Hardy spaces on two sheeted covering semigroups, J. Lie Theory 7 (1997), 245-267 MR 98k:22060
  • [KW 65] A. Korányi and J. A. Wolf, Realization of Hermitian symmetric spaces as generalized half planes, Ann. of Math. 81 (1965), 265-288 MR 30:4980
  • [Ma 73] B. O. Makarevic, Open symmetric orbits of reductive groups in symmetric $R$-spaces, Math. USSR Sbornik 20 (1973), 406-418 MR 50:1170
  • [Ma 79] T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), 331-357 MR 81a:53049
  • [Ma 82] T. Matsuki, Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math. J. 12 (1982), 307-320 MR 83k:53072
  • [Mo 97] V. F. Molchanov, Holomorphic discrete series for hyperboloids of Hermitian type, J. Funct. Anal. 147 (1997), 26-50 MR 99b:22026
  • [Ól 90] G. Ólafsson, Causal symmetric spaces, Mathematica Gottingensis 15, 1990
  • [ÓØ 97] G. Ólafsson and B. Ørsted, Causal compactification and Hardy spaces, Trans. AMS 351 (1999), 3771-3792 MR 99m:22005
  • [Va 84] V. S. Varadarajan, Lie Groups, Lie Algebras and Their Representations, 2nd edition, Springer, New York 1984 MR 85e:22001
  • [Wa 72] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Springer Grundlehren der mathematischen Wissenschaften 188, Berlin 1972 MR 58:16979
  • [Wo 72] J. A. Wolf, Fine Structure of Hermitian Symmetric Spaces, in ``Symmetric Spaces'', pp. 271-357, Dekker, New York 1972 MR 53:8516

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Additional Information

Frank Betten
Affiliation: Mathematisches Institut, Universität Göttingen, Bunsenstraße 3–5, D–37073 Göttingen, Germany

DOI: https://doi.org/10.1090/S0002-9947-03-03010-1
Received by editor(s): October 7, 1996
Published electronically: July 28, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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