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Ideal Connes amenability of \(l^{1}\)-Munn algebras and its application to semigroup algebras

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Abstract

We show that for any weakly cancellative uniformly locally finite inverse semigroup S such that \(l^{1}(S)\) is ideally Connes amenable, for each \({\mathcal {D}}\)-classes D of S, E(D) is finite, where E(D) is the set of idempotents of D. This is similar to a theorem of Duncan and Paterson, that says if \(l^{1}(S)\) is amenable then E(D) is finite. Also we study ideal Connes amenability of \(l^{1}\)-Munn algebras.

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References

  1. Civin, P., Yood, B.: The second conjugate space of a Banach algebra as an algebra. Pacific J. Math. 11, 820–847 (1961)

    MathSciNet  MATH  Google Scholar 

  2. Dales, H.G., Lau, A.T.-M, Strauss, D.: Banach algebras on semigroups and on their compactifications. Mem. Am. Math. Soc. 205 (2010)

  3. Duncan, J., Hosseiniun, S.A.R.: The second dual of a Banach algebra. Proc. R. Soc. Edinburgh. Sect. A 19, 309–325 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Duncan, J., Paterson, A.L.T.: Amenability for discrete convolution semigroup algebras. Math. Scand. 66, 141–146 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Day, M.M.: Means on semigroups and groups. Bull. Am. Math. Soc. 55, 1054–1055 (1949)

    Google Scholar 

  6. Esslamzadeh, G.H.: Banach algebra structure and amenability of a class of matrix algebras with applications. J. Funct. Anal. 161, 364–383 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Esslamzadeh, G.H.: Ideals and representation of certain semigroup algebras. Semigroup Forum 69, 51–62 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Eshaghi Gordji, M., Yazdanpanah, T.: Derivations into duals of ideals of Banach algebras. Proc. Ind. Acad. Sci. (Math. Sci.) 114(4), 399–408 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hosseinzadeh, H., Jabbari, A.: Permanently weak amenability of Rees semigroup algebras. Int. J. Anal. 16(1), 117–124 (2018)

    MATH  Google Scholar 

  10. Howie, J.M.: Fundamentals of Semigroup Theory. London Mathematics Society Monographs, vol. 12. Clarendon Press, Oxford (1995)

    MATH  Google Scholar 

  11. Jabbari, A., Hosseinzadeh, H.: Approximate amenability of matrix algebras. Scientia. Ser. A Math. Sci. 21, 17–24 (2011)

    MathSciNet  MATH  Google Scholar 

  12. Johnson, B.E.: Cohomology in Banach algebras. Mem. Am. Math. Soc. 127 (1972)

  13. Johnson, B.E., Kadison, R.V., Ringrose, J.R.: Cohomology of operator algebras III. Bull. Soc. Math. France 100, 73–96 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lau, A.T.-M., Losert, V.: On the second conjugate algebra of \(L^{1}(G)\) of a locally compact group. J. Lond. Math. Soc. 37, 464–470 (1988)

    MATH  Google Scholar 

  15. Minapoor, A.: Derivations on dual triangular Banach algebras. Glob. Anal. Discrete Math. 4(1), 31–40 (2019)

    Google Scholar 

  16. Minapoor, A.: Approximate ideal Connes amenability of dual Banach algebras and ideal Connes amenability of discrete Beurling algebras. Eurasian Math. J. 11(2), 72–85 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Minapoor, A., Bodaghi, A., Ebrahimi Bagha, D.: Derivations on the tensor product of Banach algebras. J. Math. Ext. 11, 117–125 (2017)

    MATH  Google Scholar 

  18. Minapoor, A., Bodaghi, A., Ebrahimi Bagha, D.: Ideal Connes-amenability of dual Banach algebras. Mediterr. J. Math. 14, 174 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ramsden, P.: Biflatness of semigroup algebras. Semigroup Forum. 79, 515–530 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Runde, V.: Amenability for dual Banach algebras. Studia Math. 148, 47–66 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Runde, V.: Connes-amenability and normal, virtual diagonals for measure algebras (II). Bull. Austral. Math. Soc. 68, 325–328 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Shojaee, B., Esslamzadeh, G.H., Pourabbas, A.: First cohomology of \(l^{1}\)-Munn algebras and certain semigroup algebras. Bull. Iran. Math. Soc. 35(1), 211–219 (2009)

    MATH  Google Scholar 

  23. Soroushmehr, M.: Homomorphism of \(l^{1}\)-Munn algebras and application to semigroup algebras. Bull. Aust. Math. Soc. 92(1), 115–122 (2015)

    MathSciNet  MATH  Google Scholar 

  24. von Neumann, J.: Zur allgemeinen Theorie des Masses. Fund. Math. 13, 73–116 (1929)

    Article  MATH  Google Scholar 

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Acknowledgements

The author thanks the unknown referee for careful reading, constructive comments and fruitful suggestions to improve the quality of the paper. The author also would like to thank Professor Massoud Amini for his valuable discussions and useful comments.

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  1. Department of Mathematics, Ayatollah Borujerdi University, Borujerd, Iran

    Ahmad Minapoor

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  1. Ahmad Minapoor
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Correspondence to Ahmad Minapoor.

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Communicated by Anthony To-Ming Lau.

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Minapoor, A. Ideal Connes amenability of \(l^{1}\)-Munn algebras and its application to semigroup algebras. Semigroup Forum 102, 756–764 (2021). https://doi.org/10.1007/s00233-021-10175-0

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