Permanent Weak Module Amenability of Semigroup Algebras

Open access

Abstract

We employ the fact that L1(G) is n-weakly amenable for each n ≥ 1 to show that for an inverse semigroup S with the set of idempotents E, ℓ1(S) is n- weakly module amenable as an ℓ1(E)-module with trivial left action. We study module amenability and weak module amenability of the module projective tensor products of Banach algebras.

Abstract

We employ the fact that L1(G) is n-weakly amenable for each n ≥ 1 to show that for an inverse semigroup S with the set of idempotents E, ℓ1(S) is n- weakly module amenable as an ℓ1(E)-module with trivial left action. We study module amenability and weak module amenability of the module projective tensor products of Banach algebras.

References

  • 1. Amini, M. - Module amenability for semigroup algebras, Semigroup Forum., 69 (2004), 243-254.

  • 2. Amini, M.; Bodaghi, A.; Ebrahimi Bagha, D. - Module amenability of the second dual and module topological center of semigroup algebras, Semigroup Forum, 80 (2010), 302-312.

  • 3. Amini, M.; Ebrahimi Bagha, D. - Weak module amenability of semigroup algebras, Semigroup Forum, 71 (2005), 18-26.

  • 4. Bodaghi, A. - Module amenability of the projective module tensor product, Malays. J. Math. Sci., 5 (2011), 257-265.

  • 5. Bodaghi, A. - Semigroup algebras and their weak module amenability, J. Appl. Func. Anal., 7 (2012), 332-338.

  • 6. Bodaghi, A. - Module contractibility for semigroup algebras, Math. Sci. Journal., 7 (2012), 5-18.

  • 7. Bodaghi, A. - Module amenability and tensor product of semigroup algebras, J. Math. Ext., 4 (2010), 97-106.

  • 8. Bodaghi, A.; Amini, M.; Babaee, R. - Module derivations into iterated duals of Banach algebras, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci., 12 (2011), 277-284.

  • 9. Bowling, S.; Duncan, J. - First order cohomology of Banach semigroup algebras, Semigroup Forum, 56 (1998), 130-145.

  • 10. Choi, Y.; Ghahramani, F.; Zhang, Y. - Approximate and pseudo-amenability of various classes of Banach algebras, J. Funct. Anal., 256 (2009), 3158-3191.

  • 11. Dales, H.G. - Banach Algebras and Automatic Continuity, London Mathematical Society Monographs. New Series, 24, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 2000.

  • 12. Dales, H.G.; Ghahramani, F.; Grønbæk, N. - Derivations into iterated duals of Banach algebras, Studia Math., 128 (1998), 19-54.

  • 13. Duncan, J.; Namioka, I. - Amenability of inverse semigroups and their semigroup algebras, Proc. Roy. Soc. Edinburgh Sect. A, 80 (1978), 309-321.

  • 14. Duncan, J.; Paterson, A.L.T. - Amenability for discrete convolution semigroup algebras, Math. Scand., 66 (1990), 141-146.

  • 15. Grønbæk, N. - A characterization of weakly amenable Banach algebras, Studia Math., 94 (1989), 149-162.

  • 16. Howie, J.M. - Fundamentals of Semigroup Theory, London Mathematical Society Monographs. New Series, 12, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995.

  • 17. Johnson, B.E. - Cohomology in Banach algebras, Memoirs of the American Math- ematical Society, No. 127, American Mathematical Society, Providence, R.I., 1972.

  • 18. Johnson, B.E. - Weak amenability of group algebras, Bull. London Math. Soc., 23 (1991), 281-284.

  • 19. Johnson, B.E.; Ringrose, J.R. - Derivations of operator algebras and discrete group algebras, Bull. London Math. Soc., 1 (1969), 70-74.

  • 20. Losert, V. - The derivation problem for group algebras, Ann. of Math., 168 (2008), 221-246.

  • 21. Pourmahmood-Aghababa, H. - (Super) module amenability, module topological centre and semigroup algebras, Semigroup Forum, 81 (2010), 344-356.

  • 22. Rezavand, R.; Amini, M.; Sattari, M.H.; Ebrahimi Bagha, D. - Module Arens regularity for semigroup algebras, Semigroup Forum, 77 (2008), 300-305.

1. Amini, M. - Module amenability for semigroup algebras, Semigroup Forum., 69 (2004), 243-254.

2. Amini, M.; Bodaghi, A.; Ebrahimi Bagha, D. - Module amenability of the second dual and module topological center of semigroup algebras, Semigroup Forum, 80 (2010), 302-312.

3. Amini, M.; Ebrahimi Bagha, D. - Weak module amenability of semigroup algebras, Semigroup Forum, 71 (2005), 18-26.

4. Bodaghi, A. - Module amenability of the projective module tensor product, Malays. J. Math. Sci., 5 (2011), 257-265.

5. Bodaghi, A. - Semigroup algebras and their weak module amenability, J. Appl. Func. Anal., 7 (2012), 332-338.

6. Bodaghi, A. - Module contractibility for semigroup algebras, Math. Sci. Journal., 7 (2012), 5-18.

7. Bodaghi, A. - Module amenability and tensor product of semigroup algebras, J. Math. Ext., 4 (2010), 97-106.

8. Bodaghi, A.; Amini, M.; Babaee, R. - Module derivations into iterated duals of Banach algebras, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci., 12 (2011), 277-284.

9. Bowling, S.; Duncan, J. - First order cohomology of Banach semigroup algebras, Semigroup Forum, 56 (1998), 130-145.

10. Choi, Y.; Ghahramani, F.; Zhang, Y. - Approximate and pseudo-amenability of various classes of Banach algebras, J. Funct. Anal., 256 (2009), 3158-3191.

11. Dales, H.G. - Banach Algebras and Automatic Continuity, London Mathematical Society Monographs. New Series, 24, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 2000.

12. Dales, H.G.; Ghahramani, F.; Grønbæk, N. - Derivations into iterated duals of Banach algebras, Studia Math., 128 (1998), 19-54.

13. Duncan, J.; Namioka, I. - Amenability of inverse semigroups and their semigroup algebras, Proc. Roy. Soc. Edinburgh Sect. A, 80 (1978), 309-321.

14. Duncan, J.; Paterson, A.L.T. - Amenability for discrete convolution semigroup algebras, Math. Scand., 66 (1990), 141-146.

15. Grønbæk, N. - A characterization of weakly amenable Banach algebras, Studia Math., 94 (1989), 149-162.

16. Howie, J.M. - Fundamentals of Semigroup Theory, London Mathematical Society Monographs. New Series, 12, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995.

17. Johnson, B.E. - Cohomology in Banach algebras, Memoirs of the American Math- ematical Society, No. 127, American Mathematical Society, Providence, R.I., 1972.

18. Johnson, B.E. - Weak amenability of group algebras, Bull. London Math. Soc., 23 (1991), 281-284.

19. Johnson, B.E.; Ringrose, J.R. - Derivations of operator algebras and discrete group algebras, Bull. London Math. Soc., 1 (1969), 70-74.

20. Losert, V. - The derivation problem for group algebras, Ann. of Math., 168 (2008), 221-246.

21. Pourmahmood-Aghababa, H. - (Super) module amenability, module topological centre and semigroup algebras, Semigroup Forum, 81 (2010), 344-356.

22. Rezavand, R.; Amini, M.; Sattari, M.H.; Ebrahimi Bagha, D. - Module Arens regularity for semigroup algebras, Semigroup Forum, 77 (2008), 300-305.

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