2n-Weak module amenability of semigroup algebras

Subject Areas : Functional analysis

K. Fallahi ¹ , H. Ghahramani ²

1 - Department of Mathematics, Payam Noor University of Technology, Tehran, Iran
2 - Department of Mathematics, University of Kurdistan, P. O. Box 416, Sanandaj, Iran

Received: 2019-01-27 Accepted : 2019-04-15 Published : 2019-08-01

Keywords: Banach module, module derivation, inverse semigroup, 2n-weak module amenability, semigroup algebra,

Abstract :

‎Let $S$ be an inverse semigroup with the set of idempotents $E$‎.We prove that the semigroup algebra $\ell^{1}(S)$ is always‎‎$2n$-weakly module amenable as an $\ell^{1}(E)$-module‎, ‎for any‎‎$n\in \mathbb{N}$‎, ‎where $E$ acts on $S$ trivially from the left‎‎and by multiplication from the right‎. ‎Our proof is based on a common fixed point property for semigroups‎.

References:

[1] M. Amini, A. Bodaghi, Module amenability and weak module amenability for second dual of Banach algebras, Chamchuri. J. Math. 2 (1) (2010), 57-71.

[2] M. Amini, D. Ebrahimi Bagha, Weak module amenability for semigroup algebras, Semigroup Forum. 71 (2005), 18-26.

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

[4] M. Amini, H. Rahimi, Amenability of semigroups and their algebras modulo a group congruence, Acta. Math. Hungar. 144 (2) (2014), 407-415.

[5] A. Bodaghi, M. Amini, R. Babaee, Module derivations into iterated duals of Banach algebras, Proc. Romanian. Acad. (series A). 12 (2011), 277-284.

[6] A. Bodaghi, M. Amini, A. Jabbari, Permanent weak module amenability of semigroup algebras, Annals. Alexandru. Ioan. Cuza. Uni.-Math., DOI:10.1515/aicu-2015-0018.

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

[8] H. G. Dales, F. Ghahramani, N. Gronbek, Derivations into iterated duals of Banach algebras, Stud. Math. 128 (1998), 19-54.

[9] J. Duncan, I. Namioka, Amenability of inverse semigroups and their semigroup algebras. Proc. R. Soc. Edinb. A. 80 (1988), 309-321.

[10] D. Ebrahimi Bagha, M. Amini, Module derivation problem for inverse semigroups, Semigroup Forum. 85 (2012), 525-532.

[11] J. M. Howie, An introduction to semigroup theory, Academic Press, London, 1976.

[12] B. E. Johnson, Permanent weak amenability of group algebras of free groups, Bull. London. Math. Soc. 31 (5) (1999), 569-573.

[13] B. E. Johnson, Weak amenability of group algebras, Bull. London. Math. Soc. 23 (3) (1991), 281-284.

[14] V. Losert, On derivation and crossed homomorphisms, in: Banach Algebra 2009, in: Banach Center Pub. Inst. Math. Pol. Acad. Sci. 91 (2010), 199-217.

[15] V. Losert, The derivation problem for group algebras, Ann. Math. 168 (2008), 221-246.

[16] O. T. Mewomo, On n-weak amenability of Rees semigroup algebras, Proc. Indian. Math. Sci. 118 (4) (2008), 547-555.

[17] W. D. Munn, A class of irreducible matrix representations of an arbitrary inverse semigroup, Proc. Glasgow. Math. Assoc. 5 (1961), 41-48.

[18] H. Pourmahmood-Aghababa, A note on two equivalence relations on inverse semigroups, Semigroup Forum. 84 (2012), 200-202.

[19] H. Pourmahmood-Aghababa, (Super) Module amenability, module topological center and semigroup algebras, Semigroup Forum. 81 (2010), 344-356.

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

[21] Y. Zhang, 2m-weak amenability of group algebras, J. Math. Anal. Appl. 396 (2012), 412-416.