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Extensions of the Noncommutative Integration

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Abstract

In this paper we will continue the analysis undertaken in Bagarello et al. (Rend Circ Mat Palermo (2) 55:21–28, 2006), Bongiorno et al. (Rocky Mt J Math 40(6):1745–1777, 2010), Triolo (Rend Circ Mat Palermo (2) 60(3):409–416, 2011) on the general problem of extending the noncommutative integration in a *-algebra of measurable operators. As in Aiena et al. (Filomat 28(2):263–273, 2014), Bagarello (Stud Math 172(3):289–305, 2006) and Bagarello et al. (Rend Circ Mat Palermo (2) 55:21–28, 2006), the main problem is to represent different types of partial *-algebras into a *-algebra of measurable operators in Segal’s sense, provided that these partial *-algebras posses a sufficient family of positive linear functionals (states) (Fragoulopoulou et al., J Math Anal Appl 388(2):1180–1193, 2012; Trapani and Triolo, Stud Math 184(2):133–148, 2008; Trapani and Triolo, Rend Circolo Mat Palermo 59:295–302, 2010; La Russa and Triolo, J Oper Theory, 69:2, 2013; Triolo, J Pure Appl Math, 43(6):601–617, 2012). In this paper, a new condition is given in an attempt to provide a extension of the non commutative integration.

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References

  1. Aiena, P., Trapani, C., Triolo, S.: SVEP and local spectral radius formula for unbounded operators. Filomat 28(2), 263–273 (2014)

    Article  MathSciNet  Google Scholar 

  2. Bagarello, F., Trapani, C., Triolo, S.: Quasi \(*\)-algebras of measurable operators. Stud. Math. 172(3), 289–305 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bagarello, F., Trapani, C., Triolo, S.: A note on faithful traces on a von Neumann algebra. Rend. Circ. Mat. Palermo (2) 55, 21–28 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bagarello, F., Triolo, S.: An invariant Gram-Schmidt like orthonormalization procedure and applications to coherent states. J. Math. Phys. 48, 043505 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bagarello, F., Abdollahpour, M.R., Triolo, S.: An invariant analytic orthonormalization procedure with applications. J. Math. Phys. 48, 103513 (2007). doi:10.1063/1.2800286

    Article  MathSciNet  MATH  Google Scholar 

  6. Bagarello, F., Trapani, C., Triolo, S.: Representable states on quasi-local quasi *-algebras. J. Math. Phys. 52(1), 013510 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bellomonte, G., Trapani, C., Triolo, S.: Absolutely convergent extensions of nonclosable positive linear functionals. Mediterr. J. Math. 7, 63 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bongiorno, B.: The Henstock kurzweil Integral. In: Pap, E. (ed.) Handbook of measure theory, vol. 13. Elsevier, pp. 587-614 (2002). ISBN: 978-0-444-50263-6

  9. Bongiorno, B., Trapani, C., Triolo, S.: Extensions of positive linear functionals on a topological *-algebra. Rocky Mt. J. Math. 40(6), 1745–1777 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Foran, J.: An extension of the Denjoy integral. Proc. Am. Math. Soc. 49, 359–365 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fragoulopoulou, M., Trapani, C., Triolo, S.: A locally convex quasi *-algebras with sufficiently many *-representations. J. Math. Anal. Appl. 388(2), 1180–1193 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gordon, R.A.: The integrals of Lebesgue, Denjoy, Perron, and Henstock. In: Graduate Studies in Mathematics, vol. 4. American Mathematical Society, Providence (1994)

  13. Nelson, E.: Note on non-commutative integration. J. Funct. Anal. 15, 103–116 (1974)

    Article  MATH  Google Scholar 

  14. Henstock, R.: Lectures on the Theory of Integration. World Scientific, Singapore (1988)

    Book  MATH  Google Scholar 

  15. Henstock, R.: The General Theory of Integration, Oxford Mathematical Monographs. Clarendon Press, Oxford (1991)

    MATH  Google Scholar 

  16. Henstock, R.: Theory of Integration. Butterworths, Londra (1963)

    MATH  Google Scholar 

  17. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, I. Academic Press, New York (1983)

    MATH  Google Scholar 

  18. Kurzweil, J.: Nichtalsolut Konvergente Integrale. Teubner, Leipzig (1980)

    MATH  Google Scholar 

  19. Segal, I.E.: A noncommutative extension of abstract integration. Ann. Math. 57, 401–457 (1953)

    Article  MATH  Google Scholar 

  20. Trapani, C., Triolo, S.: Locally convex quasi \(C^*\)-algebras and noncommutative integration. Studia Mathematica 228(1), 33–45 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Trapani, C., Triolo, S.: Representations of modules over a \(*\)-algebra and related seminorms. Stud. Math. 184(2), 133–148 (2008). 46L08 (46L51 47L60)

    Article  MathSciNet  MATH  Google Scholar 

  22. Trapani, C., Triolo, S.: Faithful Representations of left C*-modules. Rend. Circolo Mat. Palermo 59, 295–302 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. La Russa, C., Triolo, S.: Radon–Nikodym theorem in topological quasi *-algebras. J. Oper. Theory 69, 2 (2013)

    MATH  Google Scholar 

  24. Triolo, S.: WQ*-algebras of measurable operators. Indian J. Pure Appl. Math. 43(6), 601–617 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Triolo, S.: Possible extensions of the noncommutative integral. Rend. Circ. Mat. Palermo (2) 60(3), 409–416 (2011). doi:10.1007/s12215-011-0063-1

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

I would like to thank warmly the Referee for useful comments. This work has been supported by the Università degli Studi di Palermo and by the Gruppo Nazionale per l’ Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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  1. Dipartimento di Energia, ingegneria dell’ Informazione e modelli Matematici (DEIM), Fac. Ingegneria, Università di Palermo, 90128, Palermo, Italy

    Salvatore Triolo

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  1. Salvatore Triolo
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Correspondence to Salvatore Triolo.

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Communicated by Yuri Kondratiev.

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Triolo, S. Extensions of the Noncommutative Integration. Complex Anal. Oper. Theory 10, 1551–1564 (2016). https://doi.org/10.1007/s11785-015-0526-z

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