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 -algebras and quantization of para-Hermitian spaces
Authors:
Gerrit van Dijk and Michael Pevzner
Journal:
Proc. Amer. Math. Soc. 136 (2008), 2253-2260
MSC (2000):
Primary 22E46, 43A85, 46B25
Posted:
February 14, 2008
MathSciNet review:
2383532
Full-text PDF
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Additional Information
Abstract: In the present note we describe a family of  -algebra structures on the set  of square integrable functions on a rank-one para-Hermitian symmetric space  .
- [A]
Warren
Ambrose, Structure theorems for a special class
of Banach algebras, Trans. Amer. Math. Soc.
57 (1945),
364–386. MR 0013235
(7,126c), http://dx.doi.org/10.1090/S0002-9947-1945-0013235-8
- [DM]
G.
Van Dijk and V.
F. Molchanov, The Berezin form for rank one para-Hermitian
symmetric spaces, J. Math. Pures Appl. (9) 77 (1998),
no. 8, 747–799 (English, with English and French summaries). MR 1646796
(99j:22012), http://dx.doi.org/10.1016/S0021-7824(98)80008-1
- [DM2]
G.
Van Dijk and V.
F. Molchanov, Tensor products of maximal degenerate series
representations of the group 𝑆𝐿(𝑛,𝐑),
J. Math. Pures Appl. (9) 78 (1999), no. 1,
99–119. MR
1671222 (2000a:22020), http://dx.doi.org/10.1016/S0021-7824(99)80011-7
- [DP]
Gerrit
van Dijk and Michael
Pevzner, Ring structures for holomorphic discrete series and
Rankin-Cohen brackets, J. Lie Theory 17 (2007),
no. 2, 283–305. MR 2325700
(2008e:11057)
- [G]
I.
S. Gradshteyn and I.
M. Ryzhik, Table of integrals, series, and products, 6th ed.,
Academic Press Inc., San Diego, CA, 2000. Translated from the Russian;
Translation edited and with a preface by Alan Jeffrey and Daniel
Zwillinger. MR
1773820 (2001c:00002)
- [KK]
Soji
Kaneyuki and Masato
Kozai, Paracomplex structures and affine symmetric spaces,
Tokyo J. Math. 8 (1985), no. 1, 81–98. MR 800077
(87c:53078), http://dx.doi.org/10.3836/tjm/1270151571
- [L]
Lynn
H. Loomis, An introduction to abstract harmonic analysis, D.
Van Nostrand Company, Inc., Toronto-New York-London, 1953. MR 0054173
(14,883c)
- [N]
M.
A. Naĭmark, Normed rings, Revised edition. Translated
from the first Russian edition by Leo F. Boron, P. Noordhoff N. V.,
Groningen, 1964. MR 0205093
(34 #4928)
- [PU]
Michael
Pevzner and André
Unterberger, Projective pseudodifferential analysis and harmonic
analysis, J. Funct. Anal. 242 (2007), no. 2,
442–485. MR 2274817
(2008h:47097), http://dx.doi.org/10.1016/j.jfa.2006.07.002
- [UU]
André
Unterberger and Julianne
Unterberger, Algebras of symbols and modular forms, J. Anal.
Math. 68 (1996), 121–143. MR 1403254
(97i:11044), http://dx.doi.org/10.1007/BF02790207
- [V]
N.
Ya. Vilenkin, Spetsialnye funktsii i teoriya predstavlenii
grupp, 2nd ed., “Nauka”, Moscow, 1991 (Russian, with
Russian summary). MR 1177172
(93d:33013)
- [A]
- W. Ambrose, Structure theorem for a special class of Banach algebras, Trans. Amer. Math. Soc. 57 (1945), pp. 364-386. MR 0013235 (7:126c)
- [DM]
- G. van Dijk, V.F. Molchanov, The Berezin form for rank-one para-Hermitian symmetric spaces, J. Math.Pures Appl. 77 (1998), no. 8, pp. 747-799. MR 1646796 (99j:22012)
- [DM2]
- G. van Dijk, V.F. Molchanov, Tensor products of maximal degenerate series representations of the group
 . J. Math. Pures Appl. 78 (1999), pp. 99-119. MR 1671222 (2000a:22020)
- [DP]
- G. van Dijk, M. Pevzner, Ring structures for holomorphic discrete series and Rankin-Cohen brackets, J. Lie Theory 17, No. 2 (2007), pp. 283-305. MR 2325700
- [G]
- I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products, sixth edition, Academic Press, San Diego, CA, 2000. MR 1773820 (2001c:00002)
- [KK]
- S. Kaneyuki, M. Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math. 8, No. 1 (1985), pp. 81-98. MR 800077 (87c:53078)
- [L]
- L.H. Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand Company, Inc., Princeton, NJ, 1953. MR 0054173 (14:883c)
- [N]
- M.A. Naimark, Normed Rings, P. Noordhoff N.V., Groningen, 1964. MR 0205093 (34:4928)
- [PU]
- M. Pevzner, A. Unterberger, Projective pseudodifferential analysis and harmonic analysis, J. Funct. Anal. 242, No. 2 (2007), pp. 442-485. MR 2274817
- [UU]
- A. Unterberger, J. Unterberger, Algebras of symbols and modular forms. J. Anal. Math. 68 (1996), pp. 121-143. MR 1403254 (97i:11044)
- [V]
- N. Ya. Vilenkin, Special functions and the theory of group representations. Nauka, Moscow, 1991. MR 1177172 (93d:33013)
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Additional Information
Gerrit van Dijk
Affiliation:
Mathematish Instituut, Universiteit Leiden, P.O. Box 9512, NL-2300 RA Leiden, The Netherlands
Email:
dijk@math.leidenuniv.nl
Michael Pevzner
Affiliation:
Laboratoire de Mathématiques, UMR CNRS 6056, Université de Reims, Campus Moulin de la Housse BP 1039, F-51687, Reims, France
Email:
pevzner@univ-reims.fr
DOI:
http://dx.doi.org/10.1090/S0002-9939-08-09219-8
PII:
S 0002-9939(08)09219-8
Keywords:
Quantization,
para-Hermitian symmetric spaces,
Hilbert algebras
Received by editor(s):
March 5, 2007
Posted:
February 14, 2008
Communicated by:
Mikhail Shubin
Article copyright:
© Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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