On the extension of positive maps to Haagerup noncommutative L ^p-spaces

Let M be a von Neumann algebra, let φ be a normal faithful state on M and let Lp(M,φ) be the associated Haagerup noncommutative Lp-spaces, for 1≤p≤∞. Let D∈L1(M,φ) be the density of φ. Given a positive map T:M→M such that φ∘T≤C1φ for some C1≥0, we study the boundedness of the Lp-extension Tp,θ:D1−θpMDθp→Lp(M,φ) which maps D1−θpxDθp to D1−θpT(x)Dθp for all x∈M. Haagerup–Junge–Xu showed that Tp,12 is always bounded and left open the question whether Tp,θ is bounded for θ≠12. We show that for any 1≤p<2 and any θ∈[0,2−1(1−p−1−−−−√)]∪[2−1(1+p−1−−−−√),1], there exists a completely positive T such that Tp,θ is unbounded. We also show that if T is 2-positive, then Tp,θ is bounded provided that p≥2 or 1≤p<2 and θ∈[1−p/2,p/2].