  We consider the focusing mass-critical NLS $iu_t + \Delta u = - |u|^{4/d} u$
in high dimensions $d \geq 4$, with initial data $u(0) = u_0$ having finite
mass $M(u_0) = \int_{\R^d} |u_0(x)|^2 dx < \infty$. It is well known that this
problem admits unique (but not global) strong solutions in the Strichartz class
$C^0_{t,\loc} L^2_x \cap L^2_{t,\loc} L^{2d/(d-2)}_x$, and also admits global
(but not unique) weak solutions in $L^\infty_t L^2_x$. In this paper we
introduce an intermediate class of solution, which we call a
\emph{semi-Strichartz class solution}, for which one does have global existence
and uniqueness in dimensions $d \geq 4$. In dimensions $d \geq 5$ and assuming
spherical symmetry, we also show the equivalence of the Strichartz class and
the strong solution class (and also of the semi-Strichartz class and the
semi-strong solution class), thus establishing ``unconditional'' uniqueness
results in the strong and semi-strong classes. With these assumptions we also
characterise these solutions in terms of the continuity properties of the mass
function $t \mapsto M(u(t))$.
