Positive isometries of Orlicz–Kantorovich spaces

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 $x\in L_{\Phi}(B,m)$.