Physical Applications of Algebras of Unbounded Operators



During the past 20 years a long series of papers concerning algebras of unbounded operators appeared in the literature, papers which, though being originally motivated by physical arguments, contain almost no physics at all. On the contrary the mathematical aspects of these algebras have been analyzed in many details and this analysis produced, up to now, the monographes [32] and [2]. Some physics appeared first in [28] and [31], in the attempt to describe systems with a very large (1024) number of degrees of freedom, following some general ideas originally proposed in the famous Haag and Kastler’s paper, [27], on QM.


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  1. [1]
    G. Alli and G. L. Sewell, New methods and structures in the theory of the multi-mode Dicke laser model, J. Math. Phys. 36, 5598 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J.-P. Antoine, A. Inoue and C. Trapani Partial *-algebras and Their Operator Realizations, Kluwer, Dordrecht, 2002CrossRefzbMATHGoogle Scholar
  3. [3]
    J.-P. Antoine and W. Karwowski, Partial *-Algebras of Closed Linear Operators in Hilbert Space, Publ.RIMS, Kyoto Univ. 21, 205–236 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    F. Bagarello, Applications of Topological *-Algebras of Unbounded Operators, J. Math. Phys., 39, 2730 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    F. Bagarello, Fixed Points in Topological *-Algebras of Unbounded Operators, Publ. RIMS, Kyoto Univ. 37, (2001), 397–418.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    F. Bagarello, G. Morchio, Dynamics of mean field spin models from basic results in abstract differential equations, J. Stat. Phys. 66, 849 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    F. Bagarello and C. Trapani, ‘Almost’ Mean Field Ising Model: an Algebraic Approach, J.Statistical Phys. 65, (1991), 469–482.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    F. Bagarello and C. Trapani, A note on the algebraic approach to the “almost” mean field Heisenberg model, II Nuovo Cimento B 108, (1993), 779–784.MathSciNetCrossRefGoogle Scholar
  9. [9]
    F. Bagarello and C. Trapani, States and representations of CQ*-algebras, Ann. Inst. H. Poincaré 61,(1994), 103–133.MathSciNetzbMATHGoogle Scholar
  10. [10]
    F. Bagarello, C. Trapani, The Heisenberg Dynamics of Spin Sistems: a Quasi*-Algebras Approach, J. Math. Phys. 37, (1996), 4219–4234.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    F. Bagarello, C. Trapani, Algebraic dynamics in 0*-algebras: a perturbative approach , J. Math. Phys. 43, (2002), 3280–3292.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    F. Bagarello, A. Inoue, C. Trapani, Derivations of quasi *-algebras, Int. Jour. Math. and Math. Sci., 21, 1077–1096 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    F. Bagarello, A. Inoue, C. Trapani, Exponentiating derivations of quasi *-algebras: possible approaches and applications, Int. Jour. Math, and Math. Sci., 17, 2805–2820 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    F. Bagarello, G. L. Sewell, New Structures in the Theory of the Laser Model II: Microscopic Dynamics and a Non-Equilibrim Entropy Principle, J. Math. Phys., 39 (1998), 2730–2747.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    F. Bagarello, C. Trapani Morphisms of Certain Banach C*-Algebras, Publ. RIMS, Kyoto Univ., 36, No. 6, 681–705, (2000)CrossRefzbMATHGoogle Scholar
  16. [16]
    F. Bagarello, A. Inoue, C. Trapani, Some classes of topological quasi *-algebras, Proc. Amer. Math. Soc, 129, 2973–2980 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    O. Bratteli and D.W. Robinson, Operator algebras and Quantum statistical mechanics I, Springer-Verlag, New York, 1987.CrossRefzbMATHGoogle Scholar
  18. [18]
    O. Bratteli and D.W. Robinson, Operator algebras and Quantum statistical mechanics 2, Springer-Verlag, New York, 1987.CrossRefzbMATHGoogle Scholar
  19. [19]
    E. Buffet, P.A. Martin, Dynamics of the Open BCS Model, J. Stat. Phys., 18, No. 6, 585–632, (1978)MathSciNetCrossRefGoogle Scholar
  20. [20]
    A.M. Chebotarev, Lectures on quantum probability, Sociedad Matemática Mexicana (2000)zbMATHGoogle Scholar
  21. [21]
    D.A. Dubin, G.L. Sewell, Time-translations in the algebraic formulation of statistical mechanics, J. Math. Phys. 11, (1970), 2990–2998.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    G.G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley, New York, (1972)zbMATHGoogle Scholar
  23. [23]
    G.G. Emch, H.J.F. Knops, Pure thermodynamical phases as extremal KMS states, J. Math. Phys. 11, (1970), 3008–3018.MathSciNetCrossRefGoogle Scholar
  24. [24]
    F. Fagnola, Quantum Markov Semigroups and Quantum Flows, Proyecciones, 18, no. 3, 1–144, (1999)MathSciNetzbMATHGoogle Scholar
  25. [25]
    R. Haag, Local Quantum Physics, Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  26. [26]
    R. Haag, N.M. Hugenholtz, M. Winnink, On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys., 5, 215–236, (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    R. Haag and D. Kastler, An Algebraic Approach to Quantum Field Theory, J.Math.Phys. 5, (1964), 848–861.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    G. Lassner, Topological algebras and their applications in Quantum Statistics, Wiss. Z. KMU-Leipzig, Math.-Naturwiss. R. 30, (1981), 572–595.MathSciNetzbMATHGoogle Scholar
  29. [29]
    G. Morchio and F. Strocchi, Mathematical structures for long range dynamics and symmetry breaking J. Math. Phys. 28, (1987), 622–635.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    D. Ruelle, Statistical Mechanics, W.A. Benjamin, New York (1969)zbMATHGoogle Scholar
  31. [31]
    M. Schröder and W. Timmermann, Invariance of domains and automorphisms in algebras of unbounded operators, in Proc. Int. Conf. on Operator Algebras and Group Representations, Romania (1980), 134–139.Google Scholar
  32. [32]
    K. Schmüdgen, Unbounded operator algebras and Representation theory, Birkhäuser, Basel, 1990CrossRefzbMATHGoogle Scholar
  33. [33]
    G.L. Sewell, Quantum Theory of Collective Phenomena, Oxford University Press, Oxford (1989)Google Scholar
  34. [34]
    G.L. Sewell, Quantum Mechanics and its Emergent Macrophysics, Princeton University Press, (2002)zbMATHGoogle Scholar
  35. [35]
    F. Strocchi, Elements of Quantum Mechanics of Infinite Systems, World Scientific, (1985)CrossRefGoogle Scholar
  36. [36]
    W. Thirring, A Course in Mathematical Physics 4: Quantum Mechanics of Large Systems, Springer-Verlag, Wien (1980)Google Scholar
  37. [37]
    W. Thirring and A. Wehrl, On the Mathematical Structure of the B.C.S.-Model, Commun.Math.Phys. 4, (1967), 303–314.CrossRefzbMATHGoogle Scholar
  38. [38]
    C. Trapani, Bounded elements and spectrum in Banach quasi *-algebras, Studia Matematica, in pressGoogle Scholar

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© Hindustan Book Agency 2007

Authors and Affiliations

  1. 1.Dipartimento di Metodi e Modelli Matematici, Facoltà di IngegneriaUniversità di PalermoPalermoItaly

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