In this paper, we study the notion of Connes amenability for a class of I I-upper triangular matrix algebra UP(I;A), where A is a dual Banach algebra with a non-zero wk-continuous character and I is a totally ordered set. For this purpose, we characterize the ϕ-Connes amenability of a dual Banach algebra A through the existence of a specified net in A^ A, where ϕ is a non-zero wk-continuous character. Using this, we show that UP(I;A) is Connes amenable if and only if I is singleton and A is Connes amenable. In addition, some examples of ϕ-Connes amenable dual Banach algebras, which is not Connes amenable are given.
