  The classical Mazur map is a uniform homeomorphism between the unit spheres
of $L_p$ spaces, and the version for noncommutative $L_p$ spaces has the same
property.
  Odell and Schlumprecht used two types of generalized Mazur maps to prove that
the unit sphere of a Banach space $X$ with an unconditional basis is uniformly
homeomorphic to the unit sphere of a Hilbert space if and only if $X$ does not
contain $\ell_\infty^n$'s uniformly. We prove a noncommutative version of this
result, yielding uniform homeomorphisms between spheres of unitarily invariant
ideals, and along the way we study noncommutative versions of the
aforementioned generalized Mazur maps: one based on the $p$-convexification
procedure, and one based on the minimization of quantum relative entropy. The
main result provides new examples of Banach spaces whose unit spheres are
uniformly homeomorphic to the unit sphere of a Hilbert space (in fact, spaces
with the property (H) of Kasparov and Yu).
