Character Connes amenability of dual Banach algebras

We study the notion of character Connes amenability of dual Banach algebras and show that if A is an Arens regular Banach algebra, then A^** is character Connes amenable if and only if A is character amenable, which will resolve positively Runde’s problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension of Banach algebras. These help us to give examples of dual Banach algebras which are not Connes amenable.