Advertisement

Annali di Matematica Pura ed Applicata (1923 -)

, Volume 198, Issue 5, pp 1711–1729 | Cite as

Unbounded derivations and *-automorphisms groups of Banach quasi *-algebras

  • Maria Stella AdamoEmail author
  • Camillo Trapani
Article
  • 36 Downloads

Abstract

This paper is devoted to the study of unbounded derivations on Banach quasi *-algebras with a particular emphasis to the case when they are infinitesimal generators of one-parameter automorphisms groups. Both of them, derivations and automorphisms are considered in a weak sense, i.e., with the use of a certain families of bounded sesquilinear forms. Conditions for a weak *-derivation to be the generator of a *-automorphisms group are given.

Keywords

Banach quasi *-algebras Unbounded derivations *-Automorphisms groups and their infinitesimal generators Integrability of derivation 

Mathematics Subject Classification

Primary 46L57 Secondary 46L08, 47L60 

Notes

Acknowledgements

This work has been done in the framework of the Project “Alcuni aspetti di teoria spettrale di operatori e di algebre; frames in spazi di Hilbert rigged”, INDAM-GNAMPA 2018. The first author wishes to thank prof. M. Fragoulopoulou for her valuable suggestions and the Department of Mathematics of National and Kapodistrian University of Athens in Greece for its hospitality.

References

  1. 1.
    Adamo, M.S., Trapani, C.: Representable and continuous functionals on a Banach quasi \(^{\ast }-\)algebra. Mediterr. J. Math. 14, 157 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Antoine, J.-P., Inoue, A., Trapani, C.: Partial *-Algebras and their Operator Realizations, vol. 553. Kluwer Academic, Dordrecht (2003)zbMATHGoogle Scholar
  3. 3.
    Antoine, J.-P., Inoue, A., Trapani, C.: Spatiality of *-derivations of partial O*-algebras. J. Math. Phys. 36, 3743 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Antoine, J.-P., Inoue, A., Trapani, C.: Spatial theory of *-automorphisms on partial O*-algebras. J. Math. Phys. 35, 3059 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Antoine, J.-P., Inoue, A., Trapani, C.: O*-dynamical systems and *-derivations of unbounded operator algebras. Math. Nachr. 204, 5–28 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bagarello, F., Inoue, A., Trapani, C.: Derivations of quasi *-algebras. Int. J. Math. Math. Sci. 21, 1077–1096 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bagarello, F., Inoue, A., Trapani, C.: Representations and derivations of quasi *-algebras induced by local modifications of states. J. Math. Anal. Appl. 356, 615–623 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bagarello, F., Trapani, C.: The Heisenberg dynamics of spin systems: a quasi *-algebras approach. J. Math. Phys. 37, 4219–4234 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bagarello, F., Trapani, C.: \(\rm CQ^{\ast }-\)algebras: structure properties. Publ. RIMS, Kyoto Univ. 32, 85–116 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bratteli, O.: Derivation, Dissipation and Group Actions on C*-Algebras. Lecture Notes in Mathematics, vol. 1229. Springer, Berlin (1986)CrossRefGoogle Scholar
  11. 11.
    Bratteli, O., Robinson, D.W.: Unbounded derivations of C*-algebras. Commun. Math. Phys. 42(3), 253–268 (1975)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. Theoretical and Mathematical Physics. Springer, Berlin (2002)Google Scholar
  13. 13.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)zbMATHGoogle Scholar
  14. 14.
    Garth Dales, H.: Banach Algebras and Automatic Continuity. Oxford Science Publications, Oxford (2000)zbMATHGoogle Scholar
  15. 15.
    Fragoulopoulou, M., Trapani, C., Triolo, S.: Locally convex quasi *-algebras with sufficiently many *-representations. J. Math. Anal. Appl. 388, 1180–1193 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Trapani, C., Fragoulopoulou, M.: Locally convex quasi *-algebras and their representations. Monograph (2012) (in preparation)Google Scholar
  17. 17.
    Hille, E., Phillips, R.: Functional Analysis and Semi-Groups, vol. 31. American Mathematical Society, Providence (1996)Google Scholar
  18. 18.
    Kato, T.: Perturbation Theory for Linear Operators Classics in Mathematics, vol. 132. Springer, Berlin (1995)CrossRefGoogle Scholar
  19. 19.
    Kishimoto, A.: Dissipations and derivations. Commun. Math. Phys. 47(1), 25–32 (1976)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)zbMATHGoogle Scholar
  21. 21.
    Ringrose, J.: Automatic continuity of derivations of operator algebras. J. Lond. Math. Soc. 5, 4 (1972)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Sakai, S.: Operator Algebras in Dynamical Systems. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  23. 23.
    Thirring, W., Wehrl, A.: On the mathematical structure of the B.C.S.-model. Commun. Math. Phys. 4(5), 303–314 (1967)CrossRefGoogle Scholar
  24. 24.
    Trapani, C.: *-Representations, seminorms and structure properties of normed quasi *-algebras. Studia Math. 186, 47–75 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Trapani, C.: Bounded elements and spectrum in Banach quasi *-algebras. Studia Mathematica 172, 249–273 (2006)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Weigt, M.: Derivations of \(\tau \)-measurable operators. Oper. Theory Adv. Appl. 195, 273–286 (2009)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Weigt, M., Zarakas, I.: Unbounded derivations of GB*-algebras. Oper. Theory Adv. Appl. 247, 69–82 (2015)CrossRefGoogle Scholar
  28. 28.
    Weigt, M., Zarakas, I.: Derivations of Fréchet nuclear GB*-algebras. Bull. Aust. Math. Soc. 92, 290–301 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di PalermoPalermoItaly

Personalised recommendations