ℓ1-contractive maps on noncommutative Lp-spaces

Abstract

Let T:Lp(M)→Lp(N) be a bounded operator between two noncommutative Lp-spaces, 1⩽p<∞. We say that T is ℓ1-bounded (respectively ℓ1-contractive) if T⊗Iℓ1 extends to a bounded (respectively contractive) map from Lp(M;ℓ1) into Lp(N;ℓ1). We show that Yeadon's factorization theorem for Lp-isometries, 1⩽p≠2<∞, applies to an isometry T:L2(M)→L2(N) if and only if T is ℓ1-contractive. We also show that a contractive operator T:Lp(M)→Lp(N) is automatically ℓ1-contractive if it satisfies one of the following two conditions: either T is 2-positive; or T is separating, that is, for any disjoint a,b∈Lp(M) (i.e.\ a∗b=ab∗=0), the images T(a),T(b) are disjoint as well.