On a generalized uniform zero-two law for positive contractions of non-commutative L₁-spaces and its vector-valued extension

First, Ornstein and Sucheston proved that for a given positive contraction $T:L_1\to L_1$ there exists $m\in N$ such that $\big\|T^{m+1}-T^m\|<2$ then $$ \lim_{n\to\infty}\|T^{n+1}-T^n\|=0. $$ Such a result was labeled as "zero-two" law. In the present paper, we prove a generalized uniform "zero-two" law for multi-parametric family of positive contractions of the non-commutative $L_1$-spaces. Moreover, we also establish a vector-valued analogous of the uniform "zero-two" law for positive contractions of $L_1(M,\Phi)$-- the non-commutative $L_1$-spaces associated with center valued trace.