Previous |  Up |  Next


Banach modules; module derivation; module amenability; inverse semigroup
Let $S$ be an inverse semigroup with the set of idempotents $E$ and $S/\approx$ be an appropriate group homomorphic image of $S$. In this paper we find a one-to-one correspondence between two cohomology groups of the group algebra $\ell ^1(S)$ and the semigroup algebra $ {\ell ^{1}}(S/\approx )$ with coefficients in the same space. As a consequence, we prove that $S$ is amenable if and only if $S/\approx $ is amenable. This could be considered as the same result of Duncan and Namioka [5] with another method which asserts that the inverse semigroup $S$ is amenable if and only if the group homomorphic image $S/\sim $ is amenable, where $\sim $ is a congruence relation on $S$.
[1] Amini, M.: Module amenability for semigroup algebras. Semigroup Forum 69 (2004), 243–254. DOI 10.1007/s00233-004-0107-3 | MR 2081295 | Zbl 1059.43001
[2] Amini, M., Bodaghi, A., Bagha, D. Ebrahimi: Module amenability of the second dual and module topological center of semigroup algebras. Semigroup Forum 80 (2010), 302–312. DOI 10.1007/s00233-010-9211-8 | MR 2601766
[3] Amioni, M.: Corrigendum, Module amenability for semigroup algebras. Semigroup Forum 72 (2006), 493. MR 2228544
[4] Dale, H. G.: Banach Algebra and Automatic Continuity. Oxford university Press, 2000.
[5] Duncan, J., Namioka, I.: Amenability of inverse semigroups and their semigroup algebra. Proc. Roy. Soc. Edinburgh Sect. A 80 (3–4) (1978), 309–321. MR 0516230
[6] Howie, J. M.: An Introduction to Semigroup Theory. London Academic Press, 1976. MR 0466355 | Zbl 0355.20056
[7] Johnson, B. E.: Cohomology in Banach algebras. Mem. Amer. Math. Soc. 127 (1972), iii+96 pp. MR 0374934 | Zbl 0256.18014
[8] Moslehian, M. S., Motlagh, A. N.: Some notes on $(\sigma ,\tau )$-amenability of Banach algebras. Stud. Univ. Babeş-Bolyai Math. 53 (3) (2008), 57–68. MR 2487108 | Zbl 1199.46111
[9] Munn, W. D.: A class of irreducible matrix representations of an arbitrary inverse semigroup. Proc. Glasgow Math. Assoc. 5 (1961), 41–48. MR 0153762 | Zbl 0113.02403
[10] Paterson, A. L. T.: Groupoids, Inverse Semigroups, and Their Operator Algebras. Birkhäuser, Boston, 1999. MR 1724106 | Zbl 0913.22001
[11] Rezavand, R., Amini, M., Sattari, M. H., Bagh, D. Ebrahimi: Module Arens regularity for semigroup algebras. Semigroup Forum 77 (2008), 300–305. DOI 10.1007/s00233-008-9075-3 | MR 2443440
[12] Wilde, C., Argabright, L.: Invariant means and factor semigroup. Proc. Amer. Math. Soc. 18 (1967), 226–228. DOI 10.1090/S0002-9939-1967-0215064-9 | MR 0215064
Partner of
EuDML logo