On a generalized uniform zero-two law for positive contractions of non-commutative $L_1$-spaces and its vector-valued extension

Abstract

First, Ornstein and Sucheston proved that for a given positive contraction $T:L_1\to L_1$ there exists $m\in N$ such that $\Vert T^{m+1}-T^m\Vert <2$ then $$\underset{n\to \infty }{\lim }\Vert T^{n+1}-T^n\Vert =0.$$ 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.

18 pages