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Hilbert norms for graded algebras


Authors: Joachim Kupsch and Oleg G. Smolyanov
Journal: Proc. Amer. Math. Soc. 128 (2000), 1647-1653
MSC (2000): Primary 16W50, 16W55; Secondary 46C05, 46H25
DOI: https://doi.org/10.1090/S0002-9939-99-05569-0
Published electronically: November 24, 1999
MathSciNet review: 1707524
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper presents a solution to a problem from superanalysis about the existence of Hilbert-Banach superalgebras. Two main results are derived:

1) There exist Hilbert norms on some graded algebras (infinite-dimensional superalgebras included) with respect to which the multiplication is continuous.

2) Such norms cannot be chosen to be submultiplicative and equal to one on the unit of the algebra.


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  • [AM93] A. Arai and I. Mitoma, Comparison and nuclearity of spaces of differential forms on topological vector spaces, J. Funct. Anal. 111 (1993), 278-294. MR 94a:46051
  • [Ber66] F. A. Berezin, The method of second quantization, Academic Press, New York, 1966. MR 34:8738
  • [Ber87] F. A. Berezin, Introduction to Superanalysis, Reidel, Dordrecht, 1987. MR 89b:58006
  • [BSZ92] J. C. Baez, I. E. Segal, and Z. Zhou, Introduction to Algebraic and Constructive Quantum Field Theory, Princeton University Press, Princeton, 1992. MR 93m:81002
  • [DeW84] B. DeWitt, Supermanifolds, CUP, Cambridge, 1984. MR 87b:58007
  • [HK95] Z. Haba and J. Kupsch, Supersymmetry in euclidean quantum field theory, Fortschr. Phys. 43 (1995), 41-66. MR 96d:81109
  • [HKPS93] T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit, White noise, Kluwer, Dordrecht, 1993. MR 95f:60046
  • [Jan88] Y. Le Jan, On the Fock space representation of functionals of the occupation number field and their renormalization, J. Funct. Anal. 80 (1988), 88-108. MR 89k:60100
  • [JP81] A. Jadczyk and K. Pilch, Superspaces and supersymmetries, Commun. Math. Phys. 78 (1981), 373-390. MR 82e:58002
  • [Khr88] A. Yu. Khrennikov, Functional superanalysis, Russian Math. Surveys 43 (1988), 103-137. MR 90f:58019a
  • [Kré78] P. Krée, Méthodes fonctionelles en analyse de dimension infinie et holomorphie anticommutative , Séminaire P. Lelong et H. Skoda (Analyse) Année 1976/77, Lect. Notes Math. 694, Springer, Berlin, 1978, pp. 134-171. MR 80m:81045
  • [KS93] Yu G. Kondratev and L. Streit, Spaces of white noise distributions: constructions, descriptions, applications. I, Rep. Math. Phys. 33 (1993), 341-366. MR 95m:60095
  • [Kup90] J. Kupsch, A probabilistic formulation of bosonic and fermionic integration, Rev. Math. Phys. 2 (1990), 457-477. MR 93a:81100
  • [Mar59] J. L. Martin, Generalized classical dynamics, and the "classical analogue" of a Fermi oscillator, Proc. Roy. Soc. (London) A251 (1959), 536-542. MR 22:525
  • [Mey93] P. A. Meyer, Quantum probability for probabilists, Lect. Notes in Math. 1538, Springer, Berlin, 1993. MR 94k:81152
  • [Rog80] A. Rogers, A global theory of supermanifolds, J. Math. Phys. 21 (1980), 1352-1365. MR 82d:58001
  • [Rog86] A. Rogers, Graded manifolds, supermanifolds and infinite-dimensional Grassmann algebras, Commun. Math. Phys. 105 (1986), 375-384. MR 87f:58009
  • [SS88] O. G. Smolyanov and E. T. Shavgulidze, The Fourier transform and pseudodifferential operators in superanalysis, Soviet Math. Dokl. 37 (1988), 476-481. MR 89h:58030
  • [VV85] V. S. Vladimirov and I. V. Volovich, On the definition of the integral in superspace, Soviet Math. Dokl. 32 (1985), 817-819. MR 87g:58017
  • [Wat84] S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Institute, Bombay, and Springer-Verlag, Berlin, 1984. MR 86b:60113

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Additional Information

Joachim Kupsch
Affiliation: Fachbereich Physik der Universität Kaiserslautern, D-67663 Kaiserslautern, Germany
Email: kupsch@physik.uni-kl.de

Oleg G. Smolyanov
Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, 119899 Moscow, Russia
Email: smolian@nw.math.msu.su

DOI: https://doi.org/10.1090/S0002-9939-99-05569-0
Received by editor(s): February 20, 1998
Received by editor(s) in revised form: August 3, 1998
Published electronically: November 24, 1999
Additional Notes: The second author was supported in part by the Deutsche Forschungsgemeinschaft (DFG) and by the Russian Fund of Fundamental Research
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

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