  Let $X$ be a left introverted subspace of dual of a Banach algebra. We study
$Z_t(X^*),$ the topological center of Banach algebra $X^*$. We fined the
topological center of $(X\cA)^*$, when $\cA$ has a bounded right approximate
identity and $\cA\subseteq X^*.$ So we introduce a new notation of amenability
for a dual Banach algebra $\cal A$. A dual Banach algebra $\cal A$ is weakly
Connes-amenable if the first $weak^*-$continuous cohomology group of $\cal A$
with coefficients in $\cal A$ is zero; i.e., $H^1_{w^*}(\cal A, \cal A)=\{o\}$.
We study the weak Connes-amenability of some dual Banach algebras.
