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First, Ornstein and Sucheston proved that for a given positive contraction T : L1 → L1 there exists m ∈ N such that ∥Tm+1 − T‖ < 2 then lim n→∞ ‖T − T‖ = 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 L1-spaces. Moreover, we also establish a vector-valued analogous of the uniform ”zero-two” law for positive contractions of L1(M,Φ)– the noncommutative L1-spaces associated with center valued trace.

On a generalized uniform zero-two law for positive contractions of noncommutative $L_{1}$ -spaces and its vector-valued extension