 \(\mathbf{G}_N^*(E)\) denotes the proposition defined by \(G_1^*\) and \(G_2^*\) for a set \(N\) of \(A^*\)-symmetric reasoners, so we can say that \(E\) is Lewis-common knowledge for the agents of \(N\) iff \(\omega \in \mathbf{G}_N^*(E)\).