Document Type : Original Manuscript

Authors

1 Department of Mathematics, University of Semnan, P.O. Box 35195-363, Semnan, Iran.

2 Faculty of Engineering- East Guilan, University of Guilan, P.O. Box 44891-63157, Rudsar, Iran.

Abstract

In this paper we define $\varphi$-Connes module amenability of
a dual Banach algebra $\mathcal{A}$ where $\varphi$ is a bounded $w_{k^*}$-module
homomorphism from $\mathcal{A}$ to $\mathcal{A}$. We are mainly
concerned with the study of $\varphi$-module normal
virtual diagonals. We show that if $S$ is a weakly cancellative
inverse semigroup with subsemigroup $E$ of idempotents, $\chi$
is a bounded $w_{k^*}$-module homomorphism from $l^1(S)$ to $l^1(S)$ and $l^1(S)$
as a Banach module over $l^1(E)$ is $\chi$-Connes module amenable, then it has a $\chi$-module normal virtual
diagonal. In the case $\chi=id$, the converse holds

Keywords

###### ##### References
1. M. Amini, Module amenability for semigroup algebras, Semigroup Forum, 69 (2004), 243–254.
2. M. Amini, A. Bodaghi, D. Bagha and D. Ebrahimi, Module amenability of the second dual and module topological center of semigroup algebras, Semigroup
Forum, 80 (2010), 302–312.
3. M. Amini and R. Rezavand, Module operator amenability of the Fourier algebra of an inverse semigroup, Semigroup Forum, 92 (2014), 45–70.
4. M. Amini and R. Rezavand, Module operator virtual diagonals on the Fourier algebra of an inverse semigroup, Semigroup Forum, 97 (2018), 562–570.
5. M. Lashkarizadeh Bami, M. Valaei and M. Amini, Super module amenability of inverse semigroup algebras, Semigroup Forum, 86 (2013), 279–283.
6. H. G. Dales, A. T. Lau and D. Strauss, Banach algebras on semigroups and their compactifications, Mem. Amer. Math. Soc., 205 (2010), 1–165.
7. H. G. Dales, Banach algebras and automatic continuity, London Math. Soc. Monogr. ser., Clarendon Press (2000)
8. M. Daws, Connes amenability of bidual and weighted semigroup algebras, Math. Scand., 99 (2006), 217–246.
9. M. M. Day, Amenable Semigroups, Illinois J. Math., 1 (1957), 509–544.
10. E. G. Effros , Amenability and virtual diagonals for Von Neumann algebras, J. Funct. Anal., 78 (1988), 137–156.
11. A. Ghaffari and S. Javadi, φ-Connes amenability of dual Banach algebras, Bull. Iranian Math. Soc., 43 (2017), 25–39.
12. B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127
(1972).
13. W. D. Munn, A class of irreducible matrix representations of an arbitrary inverse semigroup, Proc. Glasgow Math. Assoc., 5 (1961), 41–48.
14. T. W. Palmer, Arens multiplication and a characterization of W-algebras, Proc. Amer. Math. Soc. 44 (1974), 81–87.
15. M. A. Rieffel, Induced Banach representations of Banach algebras and locally compact groups, J. Func. Analysis, 1 (1967), 443–491.
16. R. Rezavand, M. Amini, M. H. Sattari and D. Ebrahimi Bagha, Module Arens regularity for semigroup algebras, Semigroup Forum, 77 (2008), 300–305.
17. W. Rudin, Functional Analysis, McGraw Hill, New York, 1991.
18. V. Runde, Lectures on Amenability, Lecture Notes in Mathematics 1774, Springer Verlag, 2002.
19. V. Runde, Amenability for dual Banach algebras, Studia Math., 148 (2001), 47–66.
20. V. Runde, Connes amenability and normal virtual diagonals for measure algebras I, J. London Math. Soc., 67 (2003), 643–656.
21. V. Runde, Connes amenability and normal virtual diagonals for measure algebras II, Bull. Austral. Math. Soc., 68 (2003), 325–328.