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Strongly and weakly almost periodic linear maps on semigroup algebras

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Abstract

Let S be a foundation locally compact topological semigroup. Two new topologies τ c and τ w are introduced on M a (S)*. We introduce τ c and τ w almost periodic functionals in M a (S)*. We study these classes and compare them with each other and with the norm almost periodic and weakly almost periodic functionals. For f∈M a (S)*, it is proved that T f ∈ℬ(M a (S),M a (S)*) is strong almost periodic if and only if f is τ c -almost periodic. Indeed, we have obtained a generalization of a well known result of Crombez for locally compact group to a more general setting of foundation topological semigroups. Finally if P(S) (the set of all probability measures in M a (S)) has the semiright invariant isometry property, it is shown that the set of τ w -almost periodic functionals has a topological left invariant mean.

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References

  1. Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic, New York (1985)

    MATH  Google Scholar 

  2. Amini, M., Medghalchi, A.R.: Fourier algebras on topological foundation *-semigroup. Semigroup Forum 68, 322–334 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baker, J., Lau, A.T., Pym, J.S.: Module homomorphisms and topological centers associated with weakly sequentially complete Banach algebras. J. Funct. Anal. 158, 186–208 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berglund, J.F., Junghenn, H.D., Milnes, P.: Analysis on Semigroups, Function Spaces, Compactifications, Representions. New York (1989)

  5. Chou, C., Lau, A.T.M., Rosenblatt, J.: Approximation of compact operators by sums of translations. Ill. J. Math. 29, 340–350 (1985)

    MATH  MathSciNet  Google Scholar 

  6. Crombez, G.: Subspaces of L ∞(G) with unique topological left invariant mean. Czechoslovak Math. J. 109, 178–182 (1984)

    MathSciNet  Google Scholar 

  7. Crombez, G., Govaerts, W.: Strongly and weakly almost periodic multipliers from L 1(G) to L ∞(G). Bull. Math. Belg. 32, 179–188 (1980)

    MATH  MathSciNet  Google Scholar 

  8. Dunford, N., Schwartz, J.: Linear Operators, Part 1. Interscience, New York (1958)

    Google Scholar 

  9. Dzinotyiweyi, H.A.M.: The Analogue of the Group Algebra for Topological Semigroups. Pitman, Boston (1984)

    MATH  Google Scholar 

  10. Edwards, R.E.: Functional Analysis. Holt, Rinehart and Winston, New York (1965)

    MATH  Google Scholar 

  11. Ghaffari, A.: Ergodic theory of amenable semigroup actions. Proc. Indian Acad. Sci. 117, 117–183 (2007)

    MathSciNet  Google Scholar 

  12. Ghaffari, A.: Convolution operators on semigroup algebras. Southeast Asian Bull. Math. 27, 1025–1036 (2004)

    MATH  MathSciNet  Google Scholar 

  13. Ghaffari, A.: Topologically left invariant mean on semigroup algebras. Proc. Indian Acad. Sci. 115, 453–459 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ghahramani, F., Lau, A.T.M.: Multipliers and ideals in second conjugate algebras related to locally compact groups. J. Funct. Anal. 132, 170–191 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Graniner, E.E.: Criteria for compactness and for discreteness of locally compact amenable groups. Proc. Am. Math. Soc. 40, 615–624 (1973)

    Article  Google Scholar 

  16. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, vol. I. Springer, Berlin (1963); vol. II. Springer, Berlin (1970)

    MATH  Google Scholar 

  17. Lashkarizadeh Bami, M.: The topological centers of LUC(S)* and M a (S)* of certain foundation semigroups. Glasg. Math. J. 42, 335–343 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lashkarizadeh Bami, M.: Isometric isomorphism between Banach algebras related to certain class of Clifford topological semigroups. Semigroup Forum 69, 219–229 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lashkarizadeh Bami, M., Samea, H.: Approximate amenability of certain semigroup algebras. Semigroup Forum 71, 312–322 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lau, A.T.: Operators which commute with convolution on subspaces of L ∞(G). Colloq. Math. 39, 351–359 (1978)

    MATH  MathSciNet  Google Scholar 

  21. Lau, A.T.: Amenability of semigroups. In: Hofman, K.H., Lawson, J.D., Pym, J.S. (eds.) The Analytical and Topological Theory of Semigroups, pp. 331–334. de Gruyter, Berlin (1999)

    Google Scholar 

  22. Lau, A.T., Pym, J.: Concerning the second dual of the group algebra of a locally compact group. J. Lond. Math. Soc. 41, 445–460 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  23. Paterson, A.L.T.: Amenability. Am. Math. Soc. Math. Survey and Monographs, vol. 29. Providence (1988)

  24. Pier, J.P.: Amenable Locally Compact Groups. Wiley, New York (1984)

    Google Scholar 

  25. Pourabbas, A., Riazi, A.: Approximate identities in spaces of all absolutely continuous measures on locally compact semigroups. Semigroup Forum 71, 312–322 (2005)

    Article  MathSciNet  Google Scholar 

  26. Riazi, A., Moosaei, M.: On the space of weakly almost periodic functionals and its amenability. Southeast Asian Bull. Math. 26, 113–119 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Riazi, A., Wong, J.C.S.: Characterizations of amenable locally compact semigroups. Pac. J. Math. 108, 479–496 (1983)

    MATH  MathSciNet  Google Scholar 

  28. Rudin, W.: Functional Analysis. McGraw-Hill, New York (1991)

    MATH  Google Scholar 

  29. Rudin, W.: Invariant means on L ∞(G). Studia Math. 44, 219–227 (1972)

    MathSciNet  Google Scholar 

  30. Sleijpen, A.L.: The dual of the space of measures with continuous translations. Semigroup Forum 22, 139–150 (1981)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics, Semnan University, Semnan, Iran

    Ali Ghaffari

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  1. Ali Ghaffari
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Correspondence to Ali Ghaffari.

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Communicated by Jimmie D. Lawson

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Ghaffari, A. Strongly and weakly almost periodic linear maps on semigroup algebras. Semigroup Forum 76, 95–106 (2008). https://doi.org/10.1007/s00233-007-9001-0

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