Character Connes-amenability of dual Banach algebras

Abstract

The concept of left character Connes-amenability for a dual Banach algebra $${\mathcal {A}}$$ is introduced. We obtain a cohomological characterization of left character Connes-amenability as well as the relation between left $$\varphi $$ -Connes-amenability and existence of left $$\varphi $$ -normal virtual diagonals for a $$\omega ^{*}$$ -continuous character $$\varphi $$ . We prove that left character amenability of $${\mathcal {A}}$$ is equivalent to left character Connes-amenability of $${\mathcal {A}}^{**}$$ when $${\mathcal {A}}$$ is Arens regular. Moreover for a locally compact group G, we show that M(G) is left character Connes-amenable. In addition by means of some examples we show that for the new notion, the corresponding class of dual Banach algebras is larger than Connes-amenable dual Banach algebras.