Let $T\colon L^p({\mathcal M})\to N})$ be a bounded operator between two noncommutative $L^p$-spaces, $1\leq p<\infty$. We say that $T$ is $\ell^1$-bounded (resp. $\ell^1$-contractive) if $T\otimes I_{\ell^1}$ extends to contractive) map from $L^p({\mathcal M};\ell^1)$ into N};\ell^1)$. show Yeadon's factorization theorem for $L^p$-isometries, p\not=2 <\infty$, applies an isometry L^2({\mathcal...

In [18] and [19] we have studied compact embeddings of weighted function spaces on Rn, id : Hs q (w(x),Rn) −→ Lp(R), s > 0, 1 < q ≤ p < ∞, s − n/q + n/p > 0, with, for example, w(x) = 〈x〉α, α > 0, or w(x) = logβ〈x〉, β > 0, and 〈x〉 = (2 + |x|2)1/2. We have determined the behaviour of their entropy numbers ek(id). Now we are interested in the limiting case 1/q = 1/p + s/n. Let w(x) = logβ〈x〉, β >...