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International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 79268, 34 pages
http://dx.doi.org/10.1155/IJMMS/2006/79268

Unbounded C*-seminorms, biweights, and *-representations of partial *-algebras: A review

Dipartimento di Matematica ed Applicazioni, Università di Palermo, Palermo 90123, Italy

Received 1 March 2006; Revised 30 June 2006; Accepted 18 July 2006

Copyright © 2006 Camillo Trapani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Alcántara-Bode and J. Yngvason, “Algebraic quantum field theory and noncommutative moment problems. I,” Annales de l'Institut Henri Poincaré. Physique Théorique, vol. 48, no. 2, pp. 147–159, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. J.-P. Antoine, A. Inoue, and C. Trapani, “Partial -algebras of closable operators. I. The basic theory and the abelian case,” Publications of Research Institute for Mathematical Sciences. Kyoto University, vol. 26, no. 2, pp. 359–395, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. J.-P. Antoine, A. Inoue, and C. Trapani, “Partial -algebras of closable operators. II. States and representations of partial -algebras,” Publications of Research Institute for Mathematical Sciences. Kyoto University, vol. 27, no. 3, pp. 399–430, 1991. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J.-P. Antoine, A. Inoue, and C. Trapani, “On the regularity of the partial O-algebras generated by a closed symmetric operator,” Publications of Research Institute for Mathematical Sciences. Kyoto University, vol. 28, no. 5, pp. 757–774, 1992. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J.-P. Antoine, A. Inoue, and C. Trapani, “Partial -algebras of closable operators: a review,” Reviews in Mathematical Physics, vol. 8, no. 1, pp. 1–42, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J.-P. Antoine, A. Inoue, and C. Trapani, “Biweights on partial -algebras,” Journal of Mathematical Analysis and Applications, vol. 242, no. 2, pp. 164–190, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J.-P. Antoine, A. Inoue, and C. Trapani, Partial -Algebras and Their Operator Realizations, Kluwer Academic, Dordrecht, 2002.
  8. J.-P. Antoine and W. Karwowski, “Partial -algebras of closed linear operators in Hilbert space,” Publications of Research Institute for Mathematical Sciences. Kyoto University, vol. 21, no. 1, pp. 205–236, 1985, Erratum/Addendum, 22 (1986), no. 3, 507–511. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J.-P. Antoine and C. Trapani, “A note on Banach partial -algebras,” Mediterranean Journal of Mathematics, vol. 3, no. 1, pp. 67–86, 2006. View at Google Scholar · View at MathSciNet
  10. J.-P. Antoine and C. Trapani, “Banach partial -algebras: basic structure and representations,” preprint Palermo/Louvain-la-Neuve, 2005.
  11. F. Bagarello, A. Inoue, and C. Trapani, “Unbounded C-seminorms and -representations of partial -algebras,” Zeitschrift für Analysis und ihre Anwendungen, vol. 20, no. 2, pp. 295–314, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. F. Bagarello, A. Inoue, and C. Trapani, “Derivations of quasi -algebras,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 21, pp. 1077–1096, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. F. Bagarello, A. Inoue, and C. Trapani, “Exponentiating derivations of quasi -algebras: possible approaches and applications,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 17, pp. 2805–2820, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. F. Bagarello and C. Trapani, “States and representations of CQ-algebras,” Annales de l'Institut Henri Poincaré. Physique Théorique, vol. 61, no. 1, pp. 103–133, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. F. Bagarello and C. Trapani, “CQ-algebras: structure properties,” Publications of Research Institute for Mathematical Sciences. Kyoto University, vol. 32, no. 1, pp. 85–116, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. F. Bagarello and C. Trapani, “LP-spaces as quasi -algebras,” Journal of Mathematical Analysis and Applications, vol. 197, no. 3, pp. 810–824, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. J. Bhatt, M. Fragoulopoulou, and A. Inoue, “Existence of spectral well-behaved -representations,” Journal of Mathematical Analysis and Applications, vol. 317, no. 2, pp. 475–495, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  18. S. J. Bhatt, A. Inoue, and K.-D. Kürsten, “Well-behaved unbounded operator representations and unbounded C-seminorms,” Journal of the Mathematical Society of Japan, vol. 56, no. 2, pp. 417–445, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. J. Bhatt, A. Inoue, and H. Ogi, “On C-spectral algebras,” Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento, no. 56, pp. 207–213, 1998, International Workshop on Operator Theory (Cefalù, 1997). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. J. Bhatt, A. Inoue, and H. Ogi, “Admissibility of weights on non-normed -algebras,” Transactions of the American Mathematical Society, vol. 351, no. 11, pp. 4629–4656, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. S. J. Bhatt, A. Inoue, and H. Ogi, “Unbounded C-seminorms on -algebras,” in Proceedings of the Second ISAAC Congress, Vol. 2 (Fukuoka, 1999), vol. 8 of Int. Soc. Anal. Appl. Comput., pp. 863–870, Kluwer Academic, Dordrecht, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. J. Bhatt, A. Inoue, and H. Ogi, “Unbounded C-seminorms and unbounded C-spectral algebras,” Journal of Operator Theory, vol. 45, no. 1, pp. 53–80, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, New York, 1973. View at Zentralblatt MATH · View at MathSciNet
  24. H. J. Borchers, “Decomposition of families of unbounded operators,” in Quantum Dynamics: Models and Mathematics (Proceedings of the Symposium, Centre for Interdisciplinary Research, Bielefeld University, Bielefeld, 1975), L. Streit, Ed., Acta Phys. Austriaca, Suppl. XVI, pp. 15–46, Springer, Vienna, 1976, RCP 25 (Strasbourg), 22 (1975), 26–53. View at Google Scholar · View at MathSciNet
  25. J. Dixmier, Les C-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964. View at Zentralblatt MATH · View at MathSciNet
  26. P. G. Dixon, “Generalized B-algebras,” Proceedings of the London Mathematical Society. Third Series, vol. 21, pp. 693–715, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. R. S. Doran and V. A. Belfi, Characterizations of C-Algebras, vol. 101 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 1986. View at Zentralblatt MATH · View at MathSciNet
  28. D. A. Dubin and M. A. Hennings, Quantum Mechanics, Algebras and Distributions, vol. 238 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990. View at Zentralblatt MATH · View at MathSciNet
  29. E. G. Effros, “A decomposition theory for representations of C-algebras,” Transactions of the American Mathematical Society, vol. 107, no. 1, pp. 83–106, 1963. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. J. M. G. Fell, “The structure of algebras of operator fields,” Acta Mathematica, vol. 106, pp. 233–280, 1961. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. M. Fragoulopoulou, “Structure of contractible locally C-algebras,” Proceedings of the American Mathematical Society, vol. 129, no. 10, pp. 2889–2896, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. S. S. Horuzhy and A. V. Voronin, “Field algebras do not leave field domains invariant,” Communications in Mathematical Physics, vol. 102, no. 4, pp. 687–692, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. A. Inoue, Tomita-Takesaki Theory in Algebras of Unbounded Operators, vol. 1699 of Lecture Notes in Mathematics, Springer, Berlin, 1998. View at Zentralblatt MATH · View at MathSciNet
  34. A. Inoue and N. Takeshita, “On structure of locally convex -algebras with normal unbounded C-norms,” Kyushu Journal of Mathematics, vol. 59, no. 1, pp. 127–144, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  35. G. Lassner, “Topological algebras and their applications in quantum statistics,” Wissenschaftliche Zeitschrift der Karl-Marx-Universität Leipzig. Mathematisch-Naturwissenschaftliche Reihe, vol. 30, no. 6, pp. 572–595, 1981. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. G. Lassner, “Algebras of unbounded operators and quantum dynamics,” Physica A: Statistical and Theoretical Physics, vol. 124, no. 1–3, pp. 471–479, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. T. W. Palmer, Banach Algebras and the General Theory of -Algebras. Vol. 2. -Algebras, vol. 79 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2001. View at Zentralblatt MATH · View at MathSciNet
  38. R. T. Powers, “Self-adjoint algebras of unbounded operators,” Communications in Mathematical Physics, vol. 21, no. 2, pp. 85–124, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. K. Schmüdgen, “The order structure of topological -algebras of unbounded operators. I,” Reports on Mathematical Physics, vol. 7, no. 2, pp. 215–227, 1975. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. K. Schmüdgen, Unbounded Operator Algebras and Representation Theory, vol. 37 of Operator Theory: Advances and Applications, Birkhüuser, Basel, 1990. View at MathSciNet
  41. K. Schmüdgen, “On well-behaved unbounded representations of -algebras,” Journal of Operator Theory, vol. 48, no. 3, suppl., pp. 487–502, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. Z. Sebestyén, “Every C-seminorm is automatically submultiplicative,” Periodica Mathematica Hungarica, vol. 10, no. 1, pp. 1–8, 1979. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. C. Trapani, “Quasi -algebras of operators and their applications,” Reviews in Mathematical Physics, vol. 7, no. 8, pp. 1303–1332, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. C. Trapani, “Some seminorms on quasi -algebras,” Studia Mathematica, vol. 158, no. 2, pp. 99–115, 2003, Erratum/Addendum, 160 (2004), no. 1, 101–101. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. C. Trapani, “Bounded elements and spectrum in Banach quasi -algebras,” Studia Mathematica, vol. 172, no. 3, pp. 249–273, 2006. View at Google Scholar · View at MathSciNet
  46. C. Trapani and F. Tschinke, “Unbounded C-seminorms and biweights on partial -algebras,” Mediterranean Journal of Mathematics, vol. 2, no. 3, pp. 301–313, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  47. J. Yngvason, “Algebraic quantum field theory and noncommutative moment problems. II,” Annales de l'Institut Henri Poincaré. Physique Théorique, vol. 48, no. 2, pp. 161–173, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  48. B. Yood, “C-seminorms,” Studia Mathematica, vol. 118, no. 1, pp. 19–26, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet