Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-19T13:10:38.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Modular structure of algebras of unbounded operators

Published online by Cambridge University Press:  24 October 2008

Atsushi Inoue
Atsushi Inoue
Affiliation:
Department of Applied Mathematics, Fukuoka University, Fukuoka, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

A systematic analysis of standard systems and modular systems for which one can develop TomitaTakesaki theory for algebras of unbounded operators is presented. Such systems arise in the Wightman quantum field theory. The connection between such systems and the theory of local nets of von Neumann algebras initiated by Araki and HaagKastler is discussed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 2 , March 1992 , pp. 369 - 386
Copyright
Copyright © Cambridge Philosophical Society 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1Antoine, J.-P. and Karwowski, W.. Partial *-algebras of closable operators in Hilbert space. Res. Inst. Math. Sci. 21 (1985), 205236.Google Scholar
2Antoine, J.-P., Inoue, A. and Trapani, C.. Partial *-algebras of closable operators I. The basic theory and the abelian case. Res. Inst. Math. Sci. 26 (1990), 359395.Google Scholar
3Antoine, J.-P., Inoue, A. and Trapani, C., Partial *-algebras of closable operators II. States and representations of partial *-algebras. Res. Inst. Math. Sci., to appear.Google Scholar
4Araki, H.. Einfhrung in die axiomatisehe Quantenfeldtheorie. Lecture Notes, ETH Zrich (1962).Google Scholar
5Araki, H.. Von Neumann algebras of local observables for free scalar field. J. Math. Phys. 5 (1964), 113.Google Scholar
6Araki, H.. Some properties of modular conjugation operator of von Neumann algebras and a non-commutative RadomNikodym theorem with a chain rule. Pacific J. Math. 50 (1974), 309354.Google Scholar
7Bisognano, J. J. and Wichmann, E. H.. On the duality condition for a Hermitian scalar field. J. Math. Phys. 16 (1975), 9851007.Google Scholar
8Bisognano, J. J. and Wichmann, E. H.. On the duality condition for quantum fields. J.Math. Phys. 17 (1976), 303321.Google Scholar
9Borchers, H. J.. Algebraic aspects of Wightman field theory. In Statistical Mechanics and Field theory, Haifa Summer School (1972), pp. 31791.Google Scholar
10Borchers, H. J. and Zimmermann, W.. On the self-adjointness of field operators. Nuovo Cimento 31 (1963), 10471059.Google Scholar
11Borchers, H. J. and Yngvason, J.. Positivity of Wightman functionals and the existence of local nets. Comm. Math. Phys. 127 (1990), 607615.Google Scholar
12Bratteli, O. and Robinson, D. W.. Operator algebras and quantum statistical Mechanics, vols. 1, 2 (Springer-Verlag, 1979, 1981).Google Scholar
13Driessler, W. and Frhlich, J.. The reconstruction of local observable algebras from the Euclidean Green's functions of relativistic quantum field theory. Ann. Inst. H. Poincar Phys. Thor. 27 (1977), 221236.Google Scholar
14Driessler, W., Summers, S. J. and Wichmann, E. H.. On the connection between quantum fields and von Neumann algebras of local operators. Comm. Math. Phys. 105 (1986), 4984.Google Scholar
15Glimm, J. and Jaffe, A.. Quantum Physics (Springer-Verlag, 1981).CrossRefGoogle Scholar
16Gudder, S. P. and Scruggs, W.. Unbounded representations of *-algebras. Pacific J. Math. 70 (1977), 369382.Google Scholar
17Gudder, S. P. and Hudson, R. L.. A noncommutative probability theory. Trans. Amer.Math. Soc. 245 (1978), 141.Google Scholar
18Haag, R. and Kastler, D.. An algebraic approach to quantum field theory. J. Math. Phys. 5 (1964), 848861.Google Scholar
19Hislop, P. D. and Longo, R.. Modular structure of the local algebras associated with the free massless scalar field theory. Comm. Math. Phys. 84 (1982), 7185.Google Scholar
20Inoue, A.. An unbounded generalization of the TomitaTakesaki theory. Res. Inst. Math. Sci. 22 (1986), 725765.Google Scholar
21Inoue, A.. An unbounded generalization of the TomitaTakesaki theory II. Res. Inst. Math. Sci. 23 (1987), 673726.Google Scholar
22Inoue, A.. Extension of unbounded left Hilbert algebras to partial *-algebras. J. Math. Phys. 32 (1991), 323331.Google Scholar
23Inoue, A., Ueda, H. and Yamauchi, T.. Commutants and bicommutants of algebras of unbounded operators. J. Math. Phys. 28 (1987), 17.Google Scholar
24Jost, R.. The General Theory of Quantized Fields (American Mathematical Society, 1963).Google Scholar
25Katavolos, K. and Koch, I.. Extension of TomitaTakesaki theory to the unbounded algebra of the canonical commutation relation. Rep. Math. Phys. 16 (1979), 335352.Google Scholar
26Lassner, G.. Topological algebras of operators. Rep. Math. Phys. 3 (1972), 279293.Google Scholar
27Lassner, G.. Algebras of unbounded operators and quantum dynamics. Phys. 124 (1984), 471480.Google Scholar
28Lassner, G. and Timmermann, W.. Normal states on algebras of unbounded operators. Rep. Math. Phys. 3 (1972), 295305.Google Scholar
29Powers, R. T.. Self-adjoint algebras of unbounded operators. Comm. Math. Phys. 21 (1971), 85124.Google Scholar
30Powers, R. T.. Algebras of unbounded operators. Proc. Sympos. Pure Math. 38 (1982), 389406.Google Scholar
31Rieffel, M. A. and Van Daele, . A bounded operator approach to TomitaTakesaki theory. Pacific J. Math. 69 (1977), 187221.Google Scholar
32Schmdgen, K.. On trace representation of linear functionals on unbounded operator algebras. Comm. Math, Phys. 63 (1978), 113130.CrossRefGoogle Scholar
33Schmdgen, K.. Unbounded Operator Algebras and Representation theory (Akadamie-Verlag Berlin, 1990).Google Scholar
34Sherman, T.. Positive linear functionals on *-algebras of unbounded operators. J. Math. Anal. Appl. 22 (1968), 285318.Google Scholar
35Streater, R. F. and Wightman, A. S.. PCT, Spin and Statistics, and all that (Benjamin, 1964).Google Scholar
36Takesaki, M.. Tomita's Theory of Modular Hilbert Algebras and its Applications. Lecture Notes in Maths, vol. 128 (Springer-Verlag, 1970).Google Scholar
37Woronowicz, S. L.. The quantum moment problem I, II. Rep.Math. Phys. 1 (1970), 135145.Google Scholar
Woronowicz, S. L.. The quantum moment problem I, II. Rep. Math. Phys. 1 (1971), 175183.Google Scholar