  We derive the cubic defocusing GP hierarchy in ${\mathbb R}^3$ from a bosonic
$N$-particle Schr\"odinger equation as $N\rightarrow\infty$, in the strong
topology corresponding to the space ${\mathcal H}_\xi^1$ introduced in
\cite{chpa}. In particular, we thereby eliminate the requirement of regularity
${\mathcal H}_\xi^{1+}$ for the initial data used in \cite{CPBBGKY}. Moreover,
the marginal density matrices obtained in this strong limit are allowed to be
of infinite rank. This contrasts previous results where weak-* limits were
derived, and subsequently enhanced to strong limits based on the condition that
the limiting density matrices have finite rank. Furthermore, we prove that
positive semidefiniteness of marginal density matrices is preserved over time,
which we combine with results in \cite{CPHE}, to obtain the global
well-posedness of solutions.
