On the uniform zero-two law for positive contractions of Jordan algebras

Following an idea of Ornstein and Sucheston, Foguel proved the so-called uniform "zero-two" law: let $T:\ L^1(X,\mathcal{F}, \mu)\to L^1(X,\mathcal{F}, \mu)$ be a positive contraction. If for some $m\in\mathbb{N}\cup\{0\}$ one has $||T^{m+1}-T^m||<2$, then
$$ \lim_{n\to\infty}|| T^{m+1}-T^m||=0. $$
In this paper we prove a non-associative version of the unform "zero-two" law for positive contractions of $L_1$-spaces associated with $JBW$-algebras.