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1997 | 38 | 1 | 385-400
Tytuł artykułu

Toeplitz-Berezin quantization and non-commutative differential geometry

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this survey article we describe how the recent work in quantization in multi-variable complex geometry (domains of holomorphy, symmetric domains, tube domains, etc.) leads to interesting results and problems in C*-algebras which can be viewed as examples of the "non-commutative geometry" in the sense of A. Connes. At the same time, one obtains new functional calculi (of pseudodifferential type) with possible applications to partial differential equations and group representations.
Słowa kluczowe
Rocznik
Tom
38
Numer
1
Strony
385-400
Opis fizyczny
Daty
wydano
1997
Twórcy
  • Fachbereich Mathematik, Universität Marburg, Lahnberge, 35032 Marburg, Germany
Bibliografia
  • [AG] J. d'Atri and S. Gindikin, Siegel domain realization of pseudo-Hermitian symmetric manifolds, Geom. Dedicata 46 (1993), 91-125.
  • [B1] F. A. Berezin, A connection between the co- and contravariant symbols of operators on classical complex symmetric spaces, Soviet Math. Dokl. 19 (1978), 786-789.
  • [BLU] D. Borthwick, A. Lesniewski and H. Upmeier, Non-perturbative deformation quantization of Cartan domains, J. Funct. Anal 113 (1993), 153-176.
  • [FK] J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Univ. Press, 1994.
  • [G1] S. Gindikin, Fourier transform and Hardy spaces of $\bar{∂}$-cohomology in tube domains, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 1139-1143.
  • [G2] V. Guillemin, Toeplitz operators in n-dimensions, Integral Equations Operator Theory 7 (1984), 145-205.
  • [K1] S. Kaneyuki, Pseudo-Hermitian symmetric spaces and symmetric domains over non-degenerate cones, Hokkaido Math. J. 20 (1991), 213-239.
  • [L1] O. Loos, Bounded Symmetric Domains and Jordan Pairs, Univ. of California, Irvine, 1979.
  • [S1] W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969), 61-80.
  • [S2] W. Schmid, On the characters of discrete series (the hermitian symmetric case), ibid. 30 (1975), 47-144.
  • [U1] H. Upmeier, Jordan C*-Algebras and Symmetric Banach Manifolds, North-Holland, 1985.
  • [U2] H. Upmeier, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math. 108 (1986), 1-25.
  • [U3] H. Upmeier, Toeplitz operators on bounded symmetric domains, Trans. Amer. Math. Soc. 280 (1983), 221-237.
  • [U4] H. Upmeier, Toeplitz C*-algebras on bounded symmetric domains, Ann. of Math. 119 (1984), 549-576.
  • [U5] H. Upmeier, Multivariable Toeplitz Operators and Index Theory, Birkhäuser, 1996.
  • [U6] A. & J. Unterberger, Quantification et analyse pseudo-différentielle, Ann. Sci. École Norm. Sup. 21 (1988), 133-158.
  • [U7] H. Upmeier, Weyl quantization of symmetric spaces (I): Hyperbolic matrix domains, J. Funct. Anal. 96 (1991), 297-330.
  • [UU] A. Unterberger and H. Upmeier, The Berezin transform and invariant differential operators, Comm. Math. Phys. 164 (1994), 563-597.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv38i1p385bwm
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