Document Type : Research Paper

Author

Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord, Iran.

Abstract

In this paper, we define a new concept of factorization for a bounded bilinear mapping $f:X\times Y\to Z$, depended on  a natural number $n$ and a cardinal number $\kappa$; which is called $n$-factorization property of level $\kappa$. Then we study the relation between $n$-factorization property of  level $\kappa$ for $X^*$ with respect to $f$ and automatically boundedness and $w^*$-$w^*$-continuity and also strong Arens irregularity. These results may help us to prove some previous  problems related to strong Arens irregularity more easier than old. These include some results proved by Neufang in ~\cite{neu1} and ~\cite{neu}.  Some applications to certain bilinear mappings on convolution algebras, on a locally compact group, are also included. Finally, some solutions related to  the Ghahramani-Lau conjecture is raised.

Keywords

###### ##### References

[1] G.R. Allan and A.M. Sinclair, Bounded approximate identities, factorization, and a convolution algebra, J. Funct. Anal., 29 (1978), pp. 308-318.

[2] G.R. Allan and A.M. Sinclair, Power factorization in Banach algebras with bounded approximate identity, Studia Math., 56 (1976), pp. 31-38.

[3] A. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc., 2 (1951), pp. 839-848.

[4] S. Barootkoob, Topological centers and factorization of certain module actions, Sahand Commun. Math. Anal., 15 (1) (2019), pp. 203-215.

[5] P. Cohen, Factorization in group algebras, Dllke Math. J., 26 (1959), pp. 199-205.

[6] H.G. Dales, Banach algebras and automatic continuity, Vol. 24 of London Mathematical Society Monographs, The Clarendon Press, Oxford, UK, 2000.

[7] F. Ghahramani and A.T.-M. Lau, Multipliers and ideals in second conjugate algebras related to locally compact groups, J. Funct. Anal., 132 (1) (1995), pp. 170-191.

[8] F. Ghahramani and J.P. McClure, Module homomorphisms of the dual modules of convolution Banach algebras, Canad. Math. Bull., 35 (2) (1992), pp. 180-185.

[9] M. Eshaghi Gordji and M. Filali, Arens regularity of module actions, Studia Math., 181 (3) (2007), pp. 237-254.

[10] N. Gronbaek, Power factorization in Banach modules over commutative radical Banach algebras, Math. Scand., 50 (1982), pp. 123-134.

[11] K. Haghnejad Azar, Arens Regularity and Factorization Property, J. Sci. Kharazmi University, 13 (2) (2013), pp. 321-336.

[12] K. Haghnejad Azar, Factorization properties and generalization of multipliers in module actions, Journal of Hyperstructures, 4 (2) (2015), pp. 142-155.

[13] K. Haghnejad Azar and Masoud Ghanji, Factorization properties and topologicalL centers of module actions and $*$-involution algebras, U.P.B. Sci. Bull., Series A, 75 (1) (2013), pp. 35-46.

[14] E. Hewitt and K. A. Ross, Abstract harmonic analysts, Volume II: Structure and analysts for compact groups, analysis on locally compart Abeltan gnmps, Springer-Verlag, Berlin, Heidelberg, and New York, 1970.

[15] H. Hofmeier and G. Wittstock, A bicommutant theorem for completely bounded module homomorphisms, Math. Ann., 308 (1) (1997), pp. 141-154.

[16] Z. Hu and M. Neufang, Decomposability of von Neumann algebras and the Mazur property of higher level, Canad. J. Math., 58 (4) (2006), pp. 768-795.

[17] B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127 (1972).

[18] A. T.-M Lau and A. Ulger, Topological centers of certain dual algebras, Trans. Amer. Math. Soc., 348 (3) (1996), pp. 1191-1212.

[19] V. Losert, M. Neufang, J. Pachl, and J. Steprans, Proof of the Ghahramani–Lau conjecture, Advanc. Math., 290 (2016), pp. 709-738.

[20] M. Neufang, On a conjecture by Ghahramani-Lau and related problems concerning topological centers, J. Funct. Anal., 224 (1) (2005), pp. 217-229.

[21] M. Neufang, On Mazur's property and property (X), J. Operat. Theory, 60 (2) (2008), pp. 301-316.

[22] M. Neufang, Solution to a conjecture by Hofmeier-Wittstock, J. Funct. Anal., 217 (1) (2004), pp. 171-180.

[23] D. Poulin, Characterization of amenability by a factorization property of the group Von Neumann algebra, arXiv:1108.3020v1 [math.OA] (2011).