  We extend Gour et al's characterization of quantum majorization via
conditional min-entropy to the context of semifinite von Neumann algebras. Our
method relies on a connection between conditional min-entropy and operator
space projective tensor norm for injective von Neumann algebras. This approach
also connects the tracial Hahn-Banach theorem of Helton, Klep and McCullough to
noncommutative vector-valued $L_1$-space.
