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.
['Behrouz Shojaee', 'Massoud Amini']
Character Connes-amenability of dual Banach algebras