Strong pseudo-Connes amenability of dual Banach algebras

In this paper, we introduce the new notion of strong pseudo-Connes amenability for dual Banach algebras. We study the relation between this new notion to the various notions of Connes amenability. Also we show that for every non-empty set $I$, $M_I(\mathbb{C})$ is strong pseudo-Connes amenable if and only if $I$ is finite. We provide some examples of certain dual Banach algebras and we study its strong pseudo-Connes amenability. In the last section, we investigate the property ultra central approximate identity for a Banach algebra $\mathcal{A}$ and its second dual $\mathcal{A}^{**}$. Also we show that for a left cancellative regular semigroup $S$, ${\ell^{1}(S)}^{**}$ has an ultra central approximat identity if and only if $S$ is a group. Finally we study this property for $\varphi$-Lau product Banach algebras and the module extension Banach algebras.