# Weak$^*$-continuous derivations in dual Banach algebras

## M. Eshaghi-Gordji , A. Ebadian, F. Habibian, and B. Hayati

Address:

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

Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran

Department of Mathematics, Malayer University, Malayer, Hamedan, Iran

E-mail:

madjid.eshaghi@gmail.com

habibianf72@yahoo.com

ebadian.ali@gmail.com

hayati@malayeru.ac.ir

Abstract:Let $\mathcal{A}$ be a dual Banach algebra. We investigate the first weak$^*$-continuous cohomology group of $\mathcal{A}$ with coefficients in $\mathcal{A}$. Hence, we obtain conditions on ${\mathcal{A}}$ for which \[ H^1_{w^*}(\mathcal{A}, \mathcal{A})=\lbrace 0\rbrace \,. \]

AMSclassification:primary 46H25.

Keywords:Arens product, 2-weakly amenable, derivation.