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Conditional expectation and stochastic integrals in non-commutative Lp spaces

Published online by Cambridge University Press:  24 October 2008

Stanisław Goldstein
Stanisław Goldstein
Affiliation:
Institute of Mathematics, Łódź University, ul. Banacha 22, 90-238 Łódź, Poland
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Extract

The aim of the paper is to propose a general scheme for the consideration of non-commutative stochastic integrals. The role of a probability space is played by a couple (, φ0), where is a von Neumann algebra and φ0 is a faithful normal state on . Our processes live in the algebra of all measurable operators associated with the crossed product of by the modular automorphism group The algebra contains all the (Haagerup's) Lp spaces over . The measure topology of the algebra has the nice feature of inducing the Lp norm topology on the Lp spaces, which makes it particularly suitable for defining stochastic integrals. The commutative theory fits smoothly into the scheme, although there exists no canonical way of embedding the algebra of (commutative) random variables into . In fact, for any commutative stochastic process we have a family of different non-commutative stochastic processes corresponding to the process. This arbitrariness seems to be quite natural in the non-commutative context. An appropriate example can be found at the end of the paper (Section 6, C4).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 2 , September 1991 , pp. 365 - 383
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Accardi, L. and Parthasarathy, K. R.. A martingale characterization of canonical commutation and anticommutation relations. J. Fund. Anal. 77 (1988), 211–231.CrossRefGoogle Scholar
[2]Barnett, C., Streater, R. F. and Wilde, I. F.. The Itô—Clifford integral. J. Funct. Anal. 48 (1982), 172–212.CrossRefGoogle Scholar
[3]Barnett, C., Streater, R. F. and Wilde, I. F.. The Itô—Clifford integral. IV: A Radon-Nikodym theorem and bracket processes. J. Operator Theory 11 (1984), 255–271.Google Scholar
[4]Barnett, C., Streater, R. F. and Wilde, I. F.. Stochastic integrals in an arbitrary gauge space. Math. Proc. Cambridge Philos. Soc. 94 (1983), 541–551.CrossRefGoogle Scholar
[5]Barnett, C., Streater, R. F. and Wilde, I. F.. Quasi-free quantum stochastic integrals for the CAR and CCR. J. Funct. Anal. 52 (1983), 19–47.CrossRefGoogle Scholar
[6]Bartle, R. G.. A general bilinear vector integral. Studia Math. 15 (1956), 337–352.CrossRefGoogle Scholar
[7]Bergh, J. and Löfström, J.. Interpolation Spaces: An Introduction (Springer-Verlag, 1976).CrossRefGoogle Scholar
[8]Cecchini, C. and Petz, D.. Norm convergence of generalized martingales in Lp-spaces over von Neumann algebras. Ada Sci. Math. (Szeged) 48 (1985), 55–63.Google Scholar
[9]Goldstein, S.. Conditional Expectations in Lp-spaces over von Neumann Algebras, Quantum Probability and Applications II. Lecture Notes in Math. vol. 1136 (Springer-Verlag, 1985), pp. 233–239.Google Scholar
[10]Goldstein, S.. Norm convergence of martingales in Lp-spaces over von Neumann algebras. Rev. Roumaine Math. Pures Appl. 32 (1987), 531–541.Google Scholar
[11]Haagerup, U.. Lp-spaces associated with an arbitrary von Neumann algebra. In Algèbres d'opérateurs et leur Application en Physique Mathématique, colloques internationaux du CNRS no. 274 (CNRS, 1979), pp. 175–184.Google Scholar
[12]Haagerup, U.. Operator valued weights in von Neumann algebras I. J. Funct. Anal. 32 (1979), 175–206.CrossRefGoogle Scholar
[13]Hiai, F. and Tsukada, M.. Conditional expectations and martingales in noncommutative Lp-spaces. J. Operator Theory 18 (1987), 265–288.Google Scholar
[14]Hudson, R. L. and Lindsay, J. M.. A non-commutative martingale representation theorem for non-Fock quantum Brownian motion. J. Funct. Anal. 61 (1985), 202–221.CrossRefGoogle Scholar
[15]Karatzas, I. and Shreve, S. E.. Brownian Motion and Stochastic Calculus (Springer-Verlag, 1988).CrossRefGoogle Scholar
[16]Kosaki, H.. Applications of the complex interpolation method to a von Neumann algebra (Noncommutative Lp-spaces). J. Funct. Anal. 56 (1984), 29–78.CrossRefGoogle Scholar
[17]Lance, E. C.. Martingale convergence in von Neumann algebras. Math. Proc. Cambridge Philos. Soc. 84 (1978), 47–56.CrossRefGoogle Scholar
[18]Lindsay, J. M. and Wilde, I. F.. On non-Fock boson stochastic integrals. J. Funct. Anal. 65 (1986), 76–82.CrossRefGoogle Scholar
[19] I.Segal, E.. A non-commutative extension of abstract integration. Ann. of Math. 57 (1953), 401–457.CrossRefGoogle Scholar
[20]Takesaki, M.. Conditional expectation in von Neumann algebra. J. Funct. Anal. 9 (1972), 306–321.CrossRefGoogle Scholar
[21]Takesaki, M.. Duality for crossed products and the structure of von Neumann algebras of type III. Ada Math. 131 (1973), 249–308.Google Scholar
[22]Terp, M.. Lp-spaces associated with von Neumann algebras. Notes, Københavns Universitet, Matematisk Institut, Rapport No. 3a/3b (1981).Google Scholar