On the weakly completely continuous operators and factorization

Abstract

In this paper, we establish some relationships between Left and right weakly completely continuous operators and topological centers of module actions and relationships between the factorization and the kinds of amenability. We define the locally topological center of the left and right module actions and investigate some of its properties. Also, we want to examine some conditions that under those the duality of a Banach algebra is strongly Connes-amenable.
Finally, we generalize the concept of the weakly strongly connes amenable to even dual in higher orders.

Keywords

• Arens regularity
• topological centers
• module actions
• n-th dual
• weakly completely continuous
• weakly strongly Connes-amenable

References

1. Arens, {\it The adjoint of a bilinear operation}, Proc. Amer. Math. Soc. {\bf 2} (1951), 839-848.
2. F. Bonsall, J. Duncan, {\it Complete normed algebras}, Springer-Verlag, Berlin 1973.
3. John B. Conway, {\it A Course in Functional Analysis}, Springer-Verlag, New York 1985.
4. H. G. Dales, {\it Banach algebra and automatic continuity}, Oxford 2000.
5. H. G. Dales, A. Rodrigues-Palacios, M.V. Velasco, {\it The second transpose of a derivation}, J. London. Math. Soc. {\bf2} 64 (2001) 707-721.
6. J. Duncan and S. A. Hosseiniun,{\it The second dual of a Banach algebra}, Proc. Roy. Soc. Edinburg Sect. A {\bf 84}(1979) 309-325.
7. } F. Ghahramani, R.J. Loy and G.A. Willis, {\it Amenability and weak amenability of second conjugate Banach algebras}, Proc. Amer. Math. Soc. {\bf 129} (1996), 1489-1497.
8. B. E. Johoson, {\it Cohomology in Banach algebra}, Mem. Amer. Math. Soc. {\bf 127}, 1972.

9. A. T. Lau and A. \"{U}lger, {\it Topological center of certain dual algebras}, Trans. Amer. Math. Soc. {\bf 348} (1996), 1191-1212.
10. S. Mohamadzadih and H. R. E. Vishki, {\it Arens regularity of module actions and the second adjoint of a derivation}, Bulletin of the Australian Mathematical Society {\bf77} (2008), 465-476.
11. V. Runde, {\it Amenability for dual Banach algebras}. Studia Math. {\bf 148} (2001), 47-66.
12. A. $\ddot{U}$lger, Arens regularity sometimes implies the RNP, Pacific J. Math. {\bf 143} (1990), 377-399.