Connes amenability-like properties
We introduce and study the notions of $w^*$-approximate Connes amenability and pseudo-Connes amenability for dual Banach algebras. We prove that the dual Banach sequence algebra $\ell ^1$ is not $w^*$-approximately Connes amenable. We show that in general the concepts of pseudo-Connes amenability and Connes amenability are distinct. Moreover the relations between these new notions are also discussed.