Positive isometries of Orlicz-Kantorovich spaces

Let B be a complete Boolean algebra, Q(B) the Stone compact of B, and let C∞(Q(B)) be the commutative unital algebra of all continuous functions x:Q(B)→[-∞,+∞], assuming possibly the values ±∞ on nowhere-dense subsets of Q(B). We consider the Orlicz-Kantorovich spaces (LΦ(B,m),∥⋅∥Φ)⊂C∞(Q(B)) with the Luxembourg norm associated with an Orlicz function Φ and a vector-valued measure m, with values in the algebra of real-valued measurable functions. It is shown, that in the case when Φ satisfies the (Δ2)-condition, the norm ∥⋅∥Φ is order continuous, that is, ∥xn∥Φ↓0 for every sequence {xn}⊂LΦ(B,m) with xn↓0. Moreover, in this case, the norm ∥⋅∥Φ is strictly monotone, that is, the conditions |x|≨|y|, x,y∈LΦ(B,m), imply ∥x∥Φ≨∥y∥Φ. In addition, for positive elements x,y∈LΦ(B,m), the equality ∥x+y∥Φ=∥x-y∥Φ is valid if and only if x⋅y=0. Using these properties of the Luxembourg norm, we prove that for any positive linear isometry V:LΦ(B,m)→LΦ(B,m) there exists an injective normal homomorphisms T:C∞(Q(B))→C∞(Q(B)) and a positive element y∈LΦ(B,m) such that V(x)=y⋅T(x) for all x∈LΦ(B,m).