Advances in Operator Theory

A Banach algebra with its applications over paths of bounded variation

Dong Hyun Cho

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Abstract

‎Let $C[0,T]$ denote the space of continuous real-valued functions on $[0,T]$‎. ‎In this paper we introduce two Banach algebras‎: ‎one of them is defined on $C[0,T]$ and the other is a space of equivalence classes of measures over paths of bounded variation on $[0,T]$‎. ‎We establish an isometric isomorphism between them and evaluate analytic Feynman integrals of the functions in the Banach algebras‎, ‎which play significant roles in the Feynman integration theories and quantum mechanics‎.

Article information

Source
Adv. Oper. Theory, Volume 3, Number 4 (2018), 794-806.

Dates
Received: 11 February 2018
Accepted: 14 May 2018
First available in Project Euclid: 8 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1528444823

Digital Object Identifier
doi:10.15352/aot.1802-1310

Mathematical Reviews number (MathSciNet)
MR3856173

Zentralblatt MATH identifier
06946378

Subjects
Primary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]
Secondary: 28C20‎ ‎60H05

Keywords
Banach algebra ‎Feynman integral‎ ‎Itô integral ‎Paley-Wiener-Zygmund integral ‎Wiener space

Citation

Cho, Dong Hyun. A Banach algebra with its applications over paths of bounded variation. Adv. Oper. Theory 3 (2018), no. 4, 794--806. doi:10.15352/aot.1802-1310. https://projecteuclid.org/euclid.aot/1528444823


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References

  • S. A. Albeverio, R. J. H\aoegh-Krohn, and S. Mazzucchi, Mathematical theory of Feynman path integrals, An introduction, 2nd edition, Lecture Notes in Math., 523, Springer-Verlag, Berlin, 2008.
  • R. H. Cameron and D. A. Storvick, Some Banach algebras of analytic Feynman integrable functionals, Analytic functions, Kozubnik 1979 (Proc. Seventh Conf., Kozubnik, 1979), pp. 18–67, Lecture Notes in Math., 798, Springer, Berlin-New York, 1980.
  • D. H. Cho, Measurable functions similar to the Itô integral and the Paley-Wiener-Zygmund integral over continuous paths, Priprint.
  • D. H. Cho, A Banach algebra similar to the Cameron-Storvick's one with its equivalent spaces, J. Funct. Spaces (accepted).
  • D. H. Cho, A Banach algebra and its equivalent space over continuous paths with a positive measure, Preprint.
  • R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Physics 20 (1948), 367–387.
  • G. Kallianpur and C. Bromley, Generalized Feynman integrals using analytic continuation in several complex variables, Stochastic analysis and applications, 217–267, Adv. Probab. Related Topics, 7, Dekker, New York, 1984.
  • I. D. Pierce, On a family of generalized Wiener spaces and applications [Ph.D. thesis], University of Nebraska-Lincoln, Lincoln, Neb, USA, 2011.
  • K. S. Ryu, The generalized analogue of Wiener measure space and its properties, Honam Math. J. 32 (2010), no. 4, 633–642.
  • K. S. Ryu, The translation theorem on the generalized analogue of Wiener space and its applications, J. Chungcheong Math. Soc. 26 (2013), no. 4, 735–742.