, Volume 18, Issue 1, pp 119–130 | Cite as

Domination by ergodic elements in ordered Banach algebras

  • S. Mouton
  • K. Muzundu


We recall the definition and properties of an algebra cone in an ordered Banach algebra (OBA) and continue to develop spectral theory for the positive elements. An element \(a\) of a Banach algebra is called ergodic if the sequence of sums \(\sum _{k=0}^{n-1} \frac{a^k}{n}\) converges. If \(a\) and \(b\) are positive elements in an OBA such that \(0\le a\le b\) and if \(b\) is ergodic, an interesting problem is that of finding conditions under which \(a\) is also ergodic. We will show that in a semisimple OBA that has certain natural properties, the condition we need is that the spectral radius of \(b\) is a Riesz point (relative to some inessential ideal). We will also show that the results obtained for OBAs can be extended to the more general setting of commutatively ordered Banach algebras (COBAs) when adjustments corresponding to the COBA structure are made.


Ordered Banach algebra Commutatively ordered Banach algebra Positive element Spectrum Ergodic element 

Mathematics Subject Classification (2000)

46H05 47A10 47B65 06F25 


  1. 1.
    Arendt, W.: On the \(o\)-spectrum of regular operators and the spectrum of measures. Math. Z. 178, 271–287 (1981)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Aupetit, B.: A primer on spectral theory. Springer, New York (1991)CrossRefMATHGoogle Scholar
  3. 3.
    Behrendt, D., Raubenheimer, H.: On domination of inessential elements in ordered Banach algebras. Ill. J. Math. 51(3), 927–936 (2007)MATHMathSciNetGoogle Scholar
  4. 4.
    Braatvedt, G., Brits, R., Raubenheimer, H.: Gelfand-Hille type theorems in ordered Banach algebras. Positivity 13, 39–50 (2009)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Dunford, N.: Spectral theory. I. convergence to projections. Trans. Am. Math. Soc. 54, 185–217 (1943)MATHMathSciNetGoogle Scholar
  6. 6.
    Herzog, G., Schmoeger, C.: A note on a theorem of Raubenheimer and Rode. Proc. Am. Math. Soc. 131(11), 3507–3509 (2003)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Mouton, H. du T., Mouton, S.: Domination properties in ordered Banach algebras. Studia Math. 149(1), 63–73 (2002)Google Scholar
  8. 8.
    Mouton, S.: A condition for spectral continuity of positive elements. Proc. Am. Math. Soc. 137(5), 1777–1782 (2009)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Mouton, S.: A spectral problem in ordered Banach algebras. Bull. Aust. Math. Soc. 67, 131–144 (2003)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Mouton, S.: Convergence properties of positive elements in Banach algebras. Math. Proc. R. Ir. Acad. 102 A(2), 149–162 (2002)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Mouton, S.: On spectral continuity of positive elements. Studia Math. 174(1), 75–84 (2006)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Mouton, S.: On the boundary spectrum in Banach algebras. Bull. Aust. Math. Soc. 74, 239–246 (2006)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Mouton, S., Muzundu, K.: Commutatively ordered Banach algebras. Quaest. Math. 36, 1–29 (2013)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Mouton, S., Raubenheimer, H.: More spectral theory in ordered Banach algebras. Positivity 1, 305–317 (1997)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Räbiger, F., Wolff, M.P.H.: Spectral and asymptotic properties of dominated operators. J. Aust. Math. Soc. (Ser. A) 63, 16–31 (1997)CrossRefMATHGoogle Scholar
  16. 16.
    Raubenheimer, H., Rode, S.: Cones in Banach algebras. Indag. Math. 7, 489–502 (1996)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Schaefer, H.H.: Banach lattices and positive operators. Springer, New York (1974)CrossRefMATHGoogle Scholar
  18. 18.
    Zaanen, A.C.: Riesz spaces II. North-Holland, Amsterdam (1983)MATHGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of StellenboschStellenboschSouth Africa

Personalised recommendations