  We present a classification theorem for amenable simple stably projectionless
C*-algebras with generalized tracial rank one whose $K_0$ vanish on traces
which satisfy the Universal Coefficient Theorem. One of them is denoted by
${\cal Z}_0$ which has a unique tracial state and $K_0({\cal Z}_0)=\mathbb{Z}$
and $K_1({\cal Z}_0)=\{0\}.$
  Let $A$ and $B$ be two separable simple $C^*$-algebras satisfying the UCT and
have finite nuclear dimension. We show that $A\otimes {\cal Z}_0\cong B\otimes
{\cal Z}_0$ if and only if ${\rm Ell}(B\otimes {\cal Z}_0)={\rm Ell}(B\otimes
{\cal Z}_0).$ A class of simple separable $C^*$-algebras which are
approximately sub-homogeneous whose spectra having bounded dimension is shown
to exhaust all possible Elliott invariant for $C^*$-algebras of the form
$A\otimes {\cal Z}_0,$ where $A$ is any finite separable simple amenable
$C^*$-algebras. Suppose that $A$ and $B$ are two finite separable simple
$C^*$-algebras with finite nuclear dimension satisfying the UCT such that
traces vanishe on $K_0(A)$ and $K_0(B)$
  (but arbitrary $K_1$). One consequence of the main results in this situation
is that $A\cong B$ if and only if $A$ and $B$ have the isomorphic Elliott
invariant.
