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 Mat. Zametki, 2000, Volume 68, Issue 6, Pages 803–818 (Mi mz1003)

Hilbert Module Realization of the Square of White Noise and Finite Difference Algebras

L. Accardia, M. Skeideb

a Università degli Studi Roma Tre, Department of Mathematics
b Brandenburgische Technische Universität

Abstract: We develop an approach to the representations theory of the algebra of the square of white noise based on the construction of Hilbert modules. We find the unique Fock representation and show that the representation space is the usual symmetric Fock space. Although we started with one degree of freedom we end up with countably many degrees of freedom. Surprisingly, our representation turns out to have a close relation to Feinsilver's finite difference algebra. In fact, there exists a holomorphic image of the finite difference algebra in the algebra of square of white noise. Our representation restricted to this image is the Boukas representation on the finite difference Fock space. Thus we extend the Boukas representation to a bigger algebra, which is generated by creators, annihilators, and number operators.

DOI: https://doi.org/10.4213/mzm1003

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English version:
Mathematical Notes, 2000, 68:6, 683–694

Bibliographic databases:

UDC: 517

Citation: L. Accardi, M. Skeide, “Hilbert Module Realization of the Square of White Noise and Finite Difference Algebras”, Mat. Zametki, 68:6 (2000), 803–818; Math. Notes, 68:6 (2000), 683–694

Citation in format AMSBIB
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• https://doi.org/10.4213/mzm1003
• http://mi.mathnet.ru/eng/mz/v68/i6/p803

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Liebscher, V, “Units for the time-ordered Fock module”, Infinite Dimensional Analysis Quantum Probability and Related Topics, 4:4 (2001), 545
2. Accardi, L, “Renormalized squares of white noise and other non-Gaussian noises as Levy processes on real Lie algebras”, Communications in Mathematical Physics, 228:1 (2002), 123
3. Accardi, L, “Fock representation of the renormalized higher powers of White noise and the centreless Virasoro (or Witt)-Zamolodchikov-omega(infinity)*-Lie algebra”, Journal of Physics A-Mathematical and Theoretical, 41:30 (2008), 304001
4. Accardi L., Dhahri A., Skeide M., “Extending the Set of Quadratic Exponential Vectors”, Quantum Probability and Infinite Dimensional Analysis, Qp-Pq Quantum Probability and White Noise Analysis, 25, eds. Ouerdiane H., Barhoumi A., World Scientific Publ Co Pte Ltd, 2010, 262–266
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