A description of the Stone space of Banach lattice \(C(K,E)\) (Q998168)

Let \(K\) be a Stonean space and let \(E\) be a Banach lattice with compact order intervals. The author gives a description of the Stone space of \(C(K,E)\), the space of \(E\)-valued continuous functions on \(K\). Let \(Z(E)\) denote the center of \(E\), equipped with the strong operator topology. The author notes that \(C_b(K, Z(E))\) is isometric and Riesz isomorphic to \(Z(C(K,E))\). The former space is isometric and Riesz isomorphic to \(C_b(K\odot H_I(E))\), where \(H_I(E) = \{T:Z(E) \rightarrow R\) is a lattice homomorphism, \(T(I)=1\}\).