Toeplitz-Berezin quantization and non-commutative differential geometry

Harald Upmeier

Banach Center Publications (1997)

  • Volume: 38, Issue: 1, page 385-400
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Upmeier, Harald. "Toeplitz-Berezin quantization and non-commutative differential geometry." Banach Center Publications 38.1 (1997): 385-400. <http://eudml.org/doc/208643>.

@article{Upmeier1997,
abstract = {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.},
author = {Upmeier, Harald},
journal = {Banach Center Publications},
keywords = {non-commutative geometry; domains of holomorphy; quantization in multi-variable complex geometry; symmetric domains; tube domains; -algebras; functional calculi; pseudodifferential type; partial differential equations; group representations},
language = {eng},
number = {1},
pages = {385-400},
title = {Toeplitz-Berezin quantization and non-commutative differential geometry},
url = {http://eudml.org/doc/208643},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Upmeier, Harald
TI - Toeplitz-Berezin quantization and non-commutative differential geometry
JO - Banach Center Publications
PY - 1997
VL - 38
IS - 1
SP - 385
EP - 400
AB - 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.
LA - eng
KW - non-commutative geometry; domains of holomorphy; quantization in multi-variable complex geometry; symmetric domains; tube domains; -algebras; functional calculi; pseudodifferential type; partial differential equations; group representations
UR - http://eudml.org/doc/208643
ER -

References

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  2. [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. Zbl0439.47038
  3. [BLU] D. Borthwick, A. Lesniewski and H. Upmeier, Non-perturbative deformation quantization of Cartan domains, J. Funct. Anal 113 (1993), 153-176. Zbl0794.46051
  4. [FK] J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Univ. Press, 1994. Zbl0841.43002
  5. [G1] S. Gindikin, Fourier transform and Hardy spaces of ∂ ¯ -cohomology in tube domains, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 1139-1143. Zbl0771.32001
  6. [G2] V. Guillemin, Toeplitz operators in n-dimensions, Integral Equations Operator Theory 7 (1984), 145-205. 
  7. [K1] S. Kaneyuki, Pseudo-Hermitian symmetric spaces and symmetric domains over non-degenerate cones, Hokkaido Math. J. 20 (1991), 213-239. Zbl0751.32017
  8. [L1] O. Loos, Bounded Symmetric Domains and Jordan Pairs, Univ. of California, Irvine, 1979. 
  9. [S1] W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969), 61-80. Zbl0219.32013
  10. [S2] W. Schmid, On the characters of discrete series (the hermitian symmetric case), ibid. 30 (1975), 47-144. 
  11. [U1] H. Upmeier, Jordan C*-Algebras and Symmetric Banach Manifolds, North-Holland, 1985. 
  12. [U2] H. Upmeier, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math. 108 (1986), 1-25. Zbl0603.46055
  13. [U3] H. Upmeier, Toeplitz operators on bounded symmetric domains, Trans. Amer. Math. Soc. 280 (1983), 221-237. Zbl0527.47019
  14. [U4] H. Upmeier, Toeplitz C*-algebras on bounded symmetric domains, Ann. of Math. 119 (1984), 549-576. Zbl0549.46031
  15. [U5] H. Upmeier, Multivariable Toeplitz Operators and Index Theory, Birkhäuser, 1996. 
  16. [U6] A. & J. Unterberger, Quantification et analyse pseudo-différentielle, Ann. Sci. École Norm. Sup. 21 (1988), 133-158. Zbl0646.58025
  17. [U7] H. Upmeier, Weyl quantization of symmetric spaces (I): Hyperbolic matrix domains, J. Funct. Anal. 96 (1991), 297-330. Zbl0736.47014
  18. [UU] A. Unterberger and H. Upmeier, The Berezin transform and invariant differential operators, Comm. Math. Phys. 164 (1994), 563-597. Zbl0843.32019

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