On Connes amenable-like properties of dual Banach algebras based on w * w^{*} -character space

Abstract

Abstract In this paper, the concept of left ϕ-approximate Connes amenability for a dual Banach algebra 𝒜 {\mathcal{A}} is studied, where ϕ is a weak * {{}^{*}} -continuous multiplicative linear functional on 𝒜 {\mathcal{A}} . Some characterizations for module extension dual Banach algebras and matrix algebras are given. Moreover, the notion of approximate left ϕ-Connes biprojectivity for dual Banach algebras is introduced and some concrete examples regarding this new notion are indicated.