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