Abstract: Let \(B\) be a complete Boolean algebra, \(Q(B)\) the Stone compact of \(B\), and let \(C_\infty (Q(B))\) be the commutative unital algebra of all continuous functions \(x: Q(B) \to [-\infty, +\infty]\), assuming possibly the values \(\pm\infty\) on nowhere-dense subsets of \(Q(B)\). We consider the Orlicz-Kantorovich spaces \({(L_{\Phi}(B,m), \|\cdot\|_{\Phi})\subset C_\infty (Q(B))}\) with the Luxembourg norm associated with an Orlicz function \(\Phi\) and a vector-valued measure \(m\), with values in the algebra of real-valued measurable functions. It is shown, that in the case when \(\Phi\) satisfies the \((\Delta_2)\)-condition, the norm \(\|\cdot\|_{\Phi}\) is order continuous, that is, \(\|x_n\|_{\Phi}\downarrow \mathbf{0}\) for every sequence \(\{x_n\}\subset L_{\Phi}(B,m)\) with \(x_n \downarrow \mathbf{0}\). Moreover, in this case, the norm \(\|\cdot\|_{\Phi}\) is strictly monotone, that is, the conditions \(|x|\lneqq |y|\), \(x, y \in L_{\Phi}(B,m)\), imply \(\|x\|_{\Phi} \lneqq \|y\|_{\Phi}\). In addition, for positive elements \(x, y \in L_{\Phi}(B,m)\), the equality \(\|x+y\|_{\Phi}=\|x-y\|_{\Phi}\) is valid if and only if \(x\cdot y = 0\). Using these properties of the Luxembourg norm, we prove that for any positive linear isometry \(V: L_{\Phi}(B,m) \to L_{\Phi}(B,m)\) there exists an injective normal homomorphisms \(T : C_\infty (Q(B)) \to C_\infty (Q(B))\) and a positive element \(y \in L_{\Phi}(B,m)\) such that \(V(x ) =y\cdot T(x)\) for all \(x\in L_{\Phi}(B,m)\).
Keywords: the Banach-Kantorovich space, the Orlicz function, vector-valued measure, positive isometry, normal homomorphism.
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