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

Abstract

Ornstein and Sucheston first proved that for a given positive contraction $T:L_1\to L_1$ there exists $m\in \mathbb{N}$ such that if $\Vert T^{m+1}-T^m\Vert <2$, then ${lim\hspace{0.17em}}_{n\to \infty }\Vert T^{n+1}-T^n\Vert =0$. This result was referred to as the zero-two law. In the present article, we prove a generalized uniform zero-two law for the multiparametric family of positive contractions of noncommutative $L_1$-spaces. Moreover, we also establish a vector-valued analogue of the uniform zero-two law for positive contractions of $L_1(M,\Phi )$—the noncommutative $L_1$-spaces associated with center-valued traces.