We present a stable uniqueness theorem for non-unital C*-algebras. Generalized tracial rank one is defined for stably projectionless simple C*-algebras. Let A and B be two stably projectionless separable simple amenable C*-algebras with gTR(A)≤1and gTR(B)≤1.
Nearby in the stack
Suppose also that
KK(A,D)=KK(B,D)={0}
for all C*-algebras
D.
Then
A≅B
if and only if they have the same tracial cones with scales. We also show that every separable simple C*-algebra,
A
with finite nuclear dimension which satisfies the UCT with non-zero traces must have
gTR(A)≤1
if
K0(A)
is torsion. In the next part of this research, we show similar results without the restriction on