The positive contractive part of a Noncommutative $L^p$-space is a complete Jordan invariant

Let $1\leq p \leq +\infty$. We show that the positive part of the closed unit ball of a non-commmutative $L^p$-space, as a metric space, is a complete Jordan $^*$-invariant for the underlying von Neumann algebra.
Comment: 8 pages