Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

MODULE DERIVATIONS ON COMMUTATIVE BANACH MODULES

  • Amini, Massoud (Department of Mathematics Faculty of Mathematical Sciences Tarbiat Modares University) ;
  • Bodaghi, Abasalt (Department of Mathematics Garmsar Branch Islamic Azad University) ;
  • Shojaee, Behrouz (Department of Mathematics Karaj Branch Islamic Azad University)
  • Received : 2020.01.21
  • Accepted : 2020.03.24
  • Published : 2020.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, the commutative module amenable Banach algebras are characterized. The hereditary and permanence properties of module amenability and the relations between module amenability of a Banach algebra and its ideals are explored. Analogous to the classical case of amenability, it is shown that the projective tensor product and direct sum of module amenable Banach algebras are again module amenable. By an application of Ryll-Nardzewski fixed point theorem, it is shown that for an inverse semigroup S, every module derivation of 𝑙1(S) into a reflexive module is inner.

Keywords

Acknowledgement

The authors express their sincere thanks to the reviewer for the careful and detailed reading of the manuscript and very helpful suggestions that improved the manuscript substantially.

References

  1. M. Amini, Module amenability for semigroup algebras, Semigroup Forum 69 (2004), no. 2, 243-254. https://doi.org/10.1007/s00233-004-0107-3
  2. M. Amini and D. E. Bagha, Weak module amenability for semigroup algebras, Semigroup Forum 71 (2005), no. 1, 18-26. https://doi.org/10.1007/s00233-004-0166-5
  3. M. Amini, A. Bodaghi, and D. Ebrahimi Bagha, Module amenability of the second dual and module topological center of semigroup algebras, Semigroup Forum 80 (2010), no. 2, 302-312. https://doi.org/10.1007/s00233-010-9211-8
  4. G. Asgari, A. Bodaghi, and D. Ebrahimi Bagha, Module amenability and module Arens regularity of weighted semigroup algebras, Commun. Korean Math. Soc. 34 (2019), no. 3, 743-755. https://doi.org/10.4134/CKMS.c170320
  5. A. Bodaghi, Module (${\phi},\;{\psi}$)-amenability of Banach algebras, Arch. Math. (Brno) 46 (2010), no. 4, 227-235.
  6. A. Bodaghi, Module amenability of the projective module tensor product, Malays. J. Math. Sci. 5 (2011), no. 2, 257-265.
  7. A. Bodaghi, Generalized notion of weak module amenability, Hacet. J. Math. Stat. 43 (2014), no. 1, 85-95.
  8. A. Bodaghi and M. Amini, Module character amenability of Banach algebras, Arch. Math. (Basel) 99 (2012), no. 4, 353-365. https://doi.org/10.1007/s00013-012-0430-y
  9. A. Bodaghi and M. Amini, Module biprojective and module biflat Banach algebras, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 75 (2013), no. 3, 25-36.
  10. A. Bodaghi, M. Amini, and R. Babaee, Module derivations into iterated duals of Banach algebras, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 12 (2011), no. 4, 277-284.
  11. A. Bodaghi, M. Amini, and A. Jabbari, Permanent weak module amenability of semigroup algebras, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 63 (2017), no. 2, 287-296.
  12. A. Bodaghi, H. Ebrahimi, and M. Lashkarizadeh Bami, Generalized notions of module character amenability, Filomat 31 (2017), no. 6, 1639-1654. https://doi.org/10.2298/FIL1706639B
  13. A. Bodaghi and A. Jabbari, n-weak module amenability of triangular Banach algebras, Math. Slovaca 65 (2015), no. 3, 645-666. https://doi.org/10.1515/ms-2015-0045
  14. H. G. Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs. New Series, 24, The Clarendon Press, Oxford University Press, New York, 2000.
  15. J. M. Howie, An Introduction to Semigroup Theory, Academic Press, London, 1976.
  16. H. Inceboz, B. Arslan, and A. Bodaghi, Module symmetrically amenable Banach algebras, Acta Math. Acad. Paedagog. Nyhazi. (N.S.) 33 (2017), no. 2, 233-245.
  17. B. E. Johnson, Cohomology in Banach Algebras, American Mathematical Society, Providence, RI, 1972.
  18. W. D. Munn, A class of irreducible matrix representations of an arbitrary inverse semigroup, Proc. Glasgow Math. Assoc. 5 (1961), 41-48. https://doi.org/10.1017/S2040618500034286
  19. I. Namioka and E. Asplund, A geometric proof of Ryll-Nardzewski's fixed point theorem, Bull. Amer. Math. Soc. 73 (1967), 443-445. https://doi.org/10.1090/S0002-9904-1967-11779-8
  20. H. Pourmahmood-Aghababa, (Super) module amenability, module topological centre and semigroup algebras, Semigroup Forum 81 (2010), no. 2, 344-356. https://doi.org/10.1007/s00233-010-9231-4
  21. H. Pourmahmood-Aghababa, A note on two equivalence relations on inverse semigroups, Semigroup Forum 84 (2012), no. 1, 200-202. https://doi.org/10.1007/s00233-011-9365-z
  22. H. Pourmahmood-Aghababa and A. Bodaghi, Module approximate amenability of Banach algebras, Bull. Iranian Math. Soc. 39 (2013), no. 6, 1137-1158.
  23. R. Rezavand, M. Amini, M. H. Sattari, and D. E. Bagha, Module Arens regularity for semigroup algebras, Semigroup Forum 77 (2008), no. 2, 300-305. https://doi.org/10.1007/s00233-008-9075-3
  24. M. A. Rieffel, Induced Banach representations of Banach algebras and locally compact groups, J. Functional Analysis 1 (1967), 443-491. https://doi.org/10.1016/0022-1236(67)90012-2
  25. V. Runde, Lectures on amenability, Lecture Notes in Mathematics, 1774, Springer-Verlag, Berlin, 2002. https://doi.org/10.1007/b82937