On positive embeddings of C(K) spaces into C(S,X) lattices

Let K and S be compact Hausdorff spaces and X a Banach lattice with λ+(X)>1, whereλ+(X)=inf⁡{max⁡{‖x−y‖,‖x+y‖}:‖x‖=1,‖y‖=1 and x,y>0}. We prove that if T is a positive isomorphism from C(K) into C(S,X) then for some natural number m there exists an upper semicontinuous set-valued mapping ψ from S to K such that⋃s∈Sψ(s)=K, and each set ψ(s) has at most m elements. This result is an extension of a Plebanek theorem, the case X=R. We also show that if ‖T‖‖T−1‖<λ+(X), then m can be taken equal to 1. This result is optimal for the classical spaces X=ℓp, with 1≤p<∞.