Positivity pp 73-96 | Cite as

Results in f-algebras

  • K. Boulabiar
  • G. Buskes
  • A. Triki
Part of the Trends in Mathematics book series (TM)


We wrote a survey [18] on lattice ordered algebras five years ago. Why do we return to f-algebras once more? We hasten to say that there is only little overlap between the current paper and that previous survey.We have three purposes for the present paper. In our previous survey we remarked that one aspect that we did not discuss, while of some historical importance to the topic, is the theory of averaging operators. That theory has its roots in the nineteenth century and predates the rise of vector lattices. Positivity is a crucial tool in averaging, and positivity has been a fertile ground for the study of averaging-like operators. The fruits of positivity in averaging have recently (see [24]) started to appear in probability theory (to which averaging operators are close kin) and statistics. In the first section of our paper, we survey the literature for our selection of old theorems on averaging operators, at the same time providing some new perspectives and results as well.


Vector Lattice Riesz Space Weighted Composition Operator Lattice Homomorphism Maximal Ring 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    F.W. Anderson, Lattice-ordered rings of quotients. Canad. J. Math. 17 (1965), 434–448.MATHMathSciNetGoogle Scholar
  2. [2]
    W. Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. J. 32 (1983), 199–215.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    J. Araujo, E. Beckenstein and L. Narici, Biseparating maps and homeomorphic real-compactifications, J. Math. Ana. Appl. 12 (1995), 258–265.CrossRefMathSciNetGoogle Scholar
  4. [4]
    J. Arbault, Nouvelles propriétés des transformations de Reynolds, C. R. Acad. Sci. Paris 239 (1954), 858–860MathSciNetGoogle Scholar
  5. [5]
    Y. Azouzi, K. Boulabiar, and G. Buskes, The de Schipper formula and squares of vector lattices, Indag. Math. 17, (2006), 479–496.CrossRefMathSciNetGoogle Scholar
  6. [6]
    B. Banachewski, Maximal rings of quotients of semi-simple commutative rings, Archiv Math. 16 (1965), 414–420.CrossRefGoogle Scholar
  7. [7]
    S.J. Bernau and C.B. Huijsmans, The order bidual of almost f-algebras and d-algebras, Trans. Amer. Math. Soc. 347 (1995), 4259–4274.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    F. Beukers, C.B. Huijsmans and B. de Pagter, Unital embedding and complexification of f-algebras, Math. Z. 183 (1983), 131–143.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    A. Bigard and K. Keimel, Sur les endomorphismes conservant les polaires d’un groupe réticulé archimédien, Bull. Soc. Math. France 97 (1969), 381–153.MATHMathSciNetGoogle Scholar
  10. [10]
    M. Billik, Idempotent Reynolds operators, J. Math. Anal. Appl. 18 (1967), 486–496.CrossRefMathSciNetGoogle Scholar
  11. [11]
    G. Birkhoff, Moyennes des fonctions bornées, Colloque d’Algèbre et de Théorie des Nombres, pp. 143–153, Centre National de la Recherche Scientifique, Paris, 1949.Google Scholar
  12. [12]
    K. Boulabiar, G. Buskes, and M. Henriksen, A Generalization of a Theorem on Biseparating Maps, J. Math. Ana. Appl. 280 (2003), 334–339.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    K. Boulabiar, Order bounded separating linear maps on Φ-algebras, Houston J. Math. 30 (2004), 1143–1155.MATHMathSciNetGoogle Scholar
  14. [14]
    K. Boulabiar and G. Buskes, A note on bijective disjointness preserving operators, Positivity IV-theory and applications, pp. 29–33, Tech. Univ. Dresden, Dresden, 2006.Google Scholar
  15. [15]
    G. Buskes and A. van Rooij, Small vector lattices, Math. Proc. Cambr. Philos. Soc. 105 (1989), 523–536.MATHGoogle Scholar
  16. [16]
    G. Buskes and A. van Rooij, Almost f-algebras: commutativity and Cauchy-Schwarz inequality, Positivity 4 (2000), 227–231.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    G. Buskes and A. van Rooij, Squares of vector lattices, Rocky Mountain J. Math. 31 (2001), 45–56.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    K. Boulabiar, G. Buskes, and A. Triki, Recent results in lattice ordered algebras, Contemporary Math. 328 (2003), 99–133.MathSciNetGoogle Scholar
  19. [19]
    B. Brainerd, On the structure of averaging operators, J. Math. Anal. Appl. 5 (1962), 135–144.CrossRefMathSciNetGoogle Scholar
  20. [20]
    P.F. Conrad and J.E. Diem, The ring of polar preserving endomorphisms of an Abelian lattice-ordered group, Illinois J. Math. 15 (1971), 222–240.MATHMathSciNetGoogle Scholar
  21. [21]
    M.L. Dubreil-Jacotin, Etude algébrique des transformations de Reynolds, Colloque d’alg`ebre supérieure, pp. 9–27, Bruxelles, 1956.Google Scholar
  22. [22]
    Y. Friedman and B. Russo, Contractive projections on C 0(K), Proc. Amer. Math. Soc. 273 (1982), 57–73.MATHMathSciNetGoogle Scholar
  23. [23]
    L. Gillman and M. Jerison, Rings of Continuous Functions, Springer Verlag, Berlin-Heidelberg-New York, 1976.MATHGoogle Scholar
  24. [24]
    J.J. Grobler, Bivariate and marginal function spaces. Positivity IV-theory and applications, pp. 63–71, Tech. Univ. Dresden, Dresden, 2006.Google Scholar
  25. [25]
    F. Hadded, Contractive projections and Seever’s identity in complex f-algebras, Comment. Math. Univ. Carolin. 44 (2003), 203–215.MATHMathSciNetGoogle Scholar
  26. [26]
    A.W. Hager, Isomorphism with a C (Y) of the maximal ring of quotients of C (X), Fund. Math. 66 (1969), 7–13.MATHMathSciNetGoogle Scholar
  27. [27]
    D.R. Hart, Some properties of disjointness preserving operators, Indag. Math. 88 (1985), 183–197.MathSciNetGoogle Scholar
  28. [28]
    C.B. Huijsmans, The order bidual of lattice ordered algebras II, J. Operator Theory 22 (1989), 277–290.MATHMathSciNetGoogle Scholar
  29. [29]
    C.B. Huijsmans and B. de Pagter, The order bidual of lattice ordered algebras, J. Funct. Anal. 59 (1984), 41–64.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    C.B. Huijsmans and B. de Pagter, Averaging operators and positive contractive projections, J. Math. Anal. Appl. 113 (1986), 163–184.MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    C.B. Huijsmans and B. de Pagter, Invertible disjointness preserving operators, Proc. Edinburgh Math. Soc. 37 (1993), 125–132.CrossRefGoogle Scholar
  32. [32]
    K. Jarosz, Automatic continuity of separating linear isomorphisms, Bull. Canadian Math. Soc. 33 (1990), 139–144MATHMathSciNetGoogle Scholar
  33. [33]
    J.S. Jeang and N.C. Wong, Weighted composition of C0 (X)’s, J. Math. Ana. Appl. 201 (1996), 981–993.MATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    E. de Jonge and A. van Rooij, Introduction to Riesz spaces, Mathematical Centre Tracts, No. 78, Mathematisch Centrum, Amsterdam, 1977.Google Scholar
  35. [35]
    J. Kampé de Fériet, Sur un problème d’algèbre abstraite posé par la définition de la moyenne dans la théorie de la turbulence, Ann. Soc. Sci. Bruxelles Sér I. 63 (1949), 156–172.Google Scholar
  36. [36]
    J.L. Kelley, Averaging operators on C (X), Illinois J. Math. 2 (1958), 214–223.MathSciNetGoogle Scholar
  37. [37]
    J. Lambek, Lectures on Rings and Modules, Blaisdell, Toronto, 1966.MATHGoogle Scholar
  38. [38]
    W.A.J. Luxemburg and A.R. Schep, A Radon-Nikodym type theorem for positive operators and a dual, Indag. Math. 40 (1978), 357–375.MathSciNetGoogle Scholar
  39. [39]
    W.A.J. Luxemburg and A.C. Zaanen, The linear modulus of an order bounded linear transformation I, Indag. Math. 33 (1971), 422–434.MathSciNetGoogle Scholar
  40. [40]
    H.P. Lotz, Über das spektrum positiver operatoren, Math. Z. 108 (1968), 15–32.MATHCrossRefMathSciNetGoogle Scholar
  41. [41]
    J. Martinez, The maximal ring of quotients f-ring, Algebra Univ. 33 (1995), 355–369.MATHCrossRefGoogle Scholar
  42. [42]
    I. Molinaro, Détermination d’une R-transformation de Reynolds, C. R. Acad. Sci. Paris 244 (1957), 2890–2893MATHGoogle Scholar
  43. [43]
    S.-T.C. Moy, Characterizations of conditional expectations as a transformation on function spaces, Pacific J. Math. 4 (1954), 47–63.MATHMathSciNetGoogle Scholar
  44. [44]
    A. Neeb, Positive Reynolds Operators on C 0(X), Ph.D. Thesis, Darmstadt 1996.Google Scholar
  45. [45]
    B. de Pagter, f-Algebras and Orthomorphisms, Ph.D. Thesis, Leiden, 1981.Google Scholar
  46. [46]
    B. de Pagter, The space of extended orthomorphisms on a vector lattice, Pacific J. Math. 112 (1984), 193–210.MATHMathSciNetGoogle Scholar
  47. [47]
    J. Quinn, Intermidiate Riesz spaces, Pacific J. Math. 56 (1975), 225–263.MATHMathSciNetGoogle Scholar
  48. [48]
    O. Reynolds, On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Philos. Trans. Roy. Soc. Ser. A 186 (1895), 123–164.CrossRefGoogle Scholar
  49. [49]
    G.C. Rota, On the representation of averaging operators, Rend. Sem. Mat. Univ. Padova 30 (1960), 52–64.MathSciNetMATHGoogle Scholar
  50. [50]
    G.C. Rota, Reynolds operators, Proc. Sympos. Appl. Math. 16 (1963), 70–83.Google Scholar
  51. [51]
    E. Scheffold, Über Reynoldsoperatoren und “Mittelwert bildende” Operatoren auf halbeinfachen F-Banachverbandsalgebren, Math. Nachr. 162 (1993), 329–337.MATHMathSciNetGoogle Scholar
  52. [52]
    G.L. Seever, Non-negative projections on C 0(X), Pacific J. Math. 17 (1966), 159–166.MATHMathSciNetGoogle Scholar
  53. [53]
    J. Sopka, On the characterization of Reynolds operators on the algebra of all continuous functions on a compact Hausdorff space, Ph.D. Thesis, Harvard-Cambridge, 1950.Google Scholar
  54. [54]
    A. Triki, Extensions of positive projections and averaging operators, J. Math. Anal. Appl. 153 (1990), 486–496.MATHCrossRefMathSciNetGoogle Scholar
  55. [55]
    A. Triki, A note on averaging operators, Contemporay Math. 232 (1999), 345–348.MathSciNetGoogle Scholar
  56. [56]
    Y. Utumi, On quotient rings, Osaka Math. J. 8 (1956), 1–18.MathSciNetMATHGoogle Scholar
  57. [57]
    Walker, Russell C., The Stone-Čech compactification. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83, Springer-Verlag, New York — Berlin, 1974.Google Scholar
  58. [58]
    A.W. Wickstead, The injective hull of an Archimedean f-algebra, Compositio Math. 62 (1987), 329–342.MATHMathSciNetGoogle Scholar
  59. [59]
    D.E. Wulbert, Averaging projections, Illinois J. Math. 13 (1969), 689–693.MATHMathSciNetGoogle Scholar
  60. [60]
    A.C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam, 1983.MATHGoogle Scholar

Copyright information

© Birkhäuser Verlag AG2007 2007

Authors and Affiliations

  • K. Boulabiar
    • 1
  • G. Buskes
    • 2
  • A. Triki
    • 3
  1. 1.Département du cycle agrégatif Institut Préparatoire aux Etudes Scientifiques et TechniquesUniversité du 7 Novembre à CarthageLa MarsaTunisia
  2. 2.Department of MathematicsUniversity of MississippiUniversityUSA
  3. 3.Department of Mathematics Faculté des Sciences de TunisUniversité des Sciences, des Techniques et de Médecine de Tunis (Tunis II)TunisTunisia

Personalised recommendations