On stably finiteness for $C^*$-algebras of exponential solvable Lie groups

We study the link between stably finiteness and stably projectionless-ness for $C^*$-algebras of solvable Lie groups. We show that these two properties are equivalent if the dimension of the group is not divisible by $4$; otherwise, they are not necessarily equivalent. To provide examples proving the last assertion, we study exponential solvable Lie groups that have nonempty finite open sets in their unitary dual.