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

Ornstein and Sucheston first proved that for a given positive contraction T: L ₁ → L ₁ there exists m 2 N such that if ∥T ^m+1 - T ^m ∥ < 2, then lim _n→∞ ∥T ⁿ⁺¹ -T ⁿ ∥ = 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 multipara- metric 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; Φ)|the noncommutative L ₁ -spaces associated with center-valued traces.