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

Abstract

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.