On separation theorems for subadditive and superadditive functionals

Zbigniew Gajda; Zygfryd Kominek

Studia Mathematica (1991)

  • Volume: 100, Issue: 1, page 25-38
  • ISSN: 0039-3223

Abstract

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We generalize the well known separation theorems for subadditive and superadditive functionals to some classes of not necessarily Abelian semigroups. We also consider the problem of supporting subadditive functionals by additive ones in the not necessarily commutative case. Our results are motivated by similar extensions of the Hyers stability theorem for the Cauchy functional equation. In this context the so-called weakly commutative and amenable semigroups appear naturally. The relations between these two classes of semigroups are discussed at the end of the paper.

How to cite

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Gajda, Zbigniew, and Kominek, Zygfryd. "On separation theorems for subadditive and superadditive functionals." Studia Mathematica 100.1 (1991): 25-38. <http://eudml.org/doc/215871>.

@article{Gajda1991,
abstract = {We generalize the well known separation theorems for subadditive and superadditive functionals to some classes of not necessarily Abelian semigroups. We also consider the problem of supporting subadditive functionals by additive ones in the not necessarily commutative case. Our results are motivated by similar extensions of the Hyers stability theorem for the Cauchy functional equation. In this context the so-called weakly commutative and amenable semigroups appear naturally. The relations between these two classes of semigroups are discussed at the end of the paper.},
author = {Gajda, Zbigniew, Kominek, Zygfryd},
journal = {Studia Mathematica},
keywords = {separation; semigroup; subadditive functional; superadditive functional; Hyers-Ulam stability; Cauchy equation},
language = {eng},
number = {1},
pages = {25-38},
title = {On separation theorems for subadditive and superadditive functionals},
url = {http://eudml.org/doc/215871},
volume = {100},
year = {1991},
}

TY - JOUR
AU - Gajda, Zbigniew
AU - Kominek, Zygfryd
TI - On separation theorems for subadditive and superadditive functionals
JO - Studia Mathematica
PY - 1991
VL - 100
IS - 1
SP - 25
EP - 38
AB - We generalize the well known separation theorems for subadditive and superadditive functionals to some classes of not necessarily Abelian semigroups. We also consider the problem of supporting subadditive functionals by additive ones in the not necessarily commutative case. Our results are motivated by similar extensions of the Hyers stability theorem for the Cauchy functional equation. In this context the so-called weakly commutative and amenable semigroups appear naturally. The relations between these two classes of semigroups are discussed at the end of the paper.
LA - eng
KW - separation; semigroup; subadditive functional; superadditive functional; Hyers-Ulam stability; Cauchy equation
UR - http://eudml.org/doc/215871
ER -

References

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  1. [1] A. Chaljub-Simon und P. Volkmann, Bemerkungen zu einem Satz von Rodé, manuscript. Zbl0712.39029
  2. [2] M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509-544. Zbl0078.29402
  3. [3] G. L. Forti, Remark 11, Report of Meeting, the 22nd Internat Symposium on Functional Equations, Aequationes Math. 29 (1985), 90-91. 
  4. [4] G. L. Forti, The stability of homomorphisms and amenability with applications to functional equations, Università degli Studi di Milano, Quaderno 24, 1986. 
  5. [5] Z. Gajda, Invariant means and representations of semigroups in the theory of functional equations, submitted. 
  6. [6] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. I, Springer, Berlin 1963. Zbl0115.10603
  7. [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. Zbl0061.26403
  8. [8] R. Kaufman, Interpolation of additive functionals, Studia Math. 27 (1966), 269-272. Zbl0143.36302
  9. [9] H. König, On the abstract Hahn-Banach theorem due to Rodé, Aequationes Math. 34 (1987), 89-95. Zbl0636.46005
  10. [10] P. Kranz, Additive functionals on abelian semigroups, Prace Mat. (Comment. Math.) 16 (1972), 239-246. Zbl0262.20087
  11. [11] J. Rätz, On approximately additive mappings, in: General Inequalities 2, Internat. Ser. Numer. Math. 47, Birkhäuser, Basel 1980, 233-251. Zbl0433.39014
  12. [12] G. Rodé, Eine abstrakte Version des Satzes von Hahn-Banach, Arch. Math. (Basel) 31 (1978), 474-481. Zbl0402.46003
  13. [13] L. Székelyhidi, Remark 17, Report of Meeting, the 22nd Internat. Symposium on Functional Equations, Aequationes Math. 29 (1985), 95-96. 
  14. [14] J. Tabor, Remark 18, ibid., 96. 

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