On approximate Connes-amenability of enveloping dual Banach algebras

For a Banach algebra $\mathcal{A}$, we introduce various approximate virtual diagonals such as approximate WAP-virtual diagonal and approximate virtual diagonal. For the enveloping dual Banach algebra $F(\mathcal{A})$ of $\mathcal{A}$, we show that $F(\mathcal{A})$ is approximately Connes-amenable if and only if $\mathcal{A}$ has an approximate WAP-virtual diagonal. Further, for a discrete group $G$, we show that if the group algebra ${\mathrm{\ell }}^1(G)$ has an approximate WAP-virtual diagonal, then it has an approximate virtual diagonal.