Positivity

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A note on Huijsmans-de Pagter problem in ordered Banach algebras

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Abstract

We give an example of a positive element a in some ordered Banach algebra \(\mathcal A\) such that its spectrum is equal to \(\{1\}\) and it is not greater than or equal to the unit element of \(\mathcal A\).

Keywords

Ordered Banach algebra Spectrum Positive element 

Mathematics Subject Classification

47B60 47A10 46H05 

Notes

Acknowledgements

The author acknowledges the financial support from the Slovenian Research Agency (Research Core Funding No. P1-0222).

References

  1. 1.
    Aliprantis, C.D., Tourky, R.: Cones and Duality. Graduate Studies in Mathematics 84. American Mathematical Society, Providence (2007)CrossRefMATHGoogle Scholar
  2. 2.
    Drnovšek, R.: On positive unipotent operators on Banach lattices. Proc. Am. Math. Soc. 135, 3833–3836 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Halmos, P.R.: A Hilbert Space Problem Book. Graduate Texts in Mathematics, 19. Encyclopedia of Mathematics and its Applications, 17, 2nd edn. Springer, New York (1982)CrossRefMATHGoogle Scholar
  4. 4.
    Herzog, G., Schmoeger, C.: An example on ordered Banach algebras. Proc. Am. Math. Soc. 135, 3949–3954 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Mouton, S.: A spectral problem in ordered Banach algebras. Bull. Aust. Math. Soc. 67, 131–144 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Mouton, S., Raubenheimer, H.: Spectral theory in ordered Banach algebras. Positivity 21, 755–786 (2017)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Zhang, X.-D.: Some aspects of the spectral theory of positive operators. Acta Appl. Math. 27, 135–142 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Zhang, X.-D.: On spectral properties of positive operators. Indag. Math. (N.S.) 4, 111–127 (1993)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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