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

TL;DR

This paper investigates the relationship between stably finiteness and stably projectionless-ness in $C^*$-algebras of exponential solvable Lie groups, revealing a dimension-dependent equivalence and providing specific examples.

Contribution

It establishes a dimension-based criterion for the equivalence of stably finiteness and stably projectionless-ness in these $C^*$-algebras and constructs examples with particular dual properties.

Findings

Equivalence holds if the group's dimension is not divisible by 4.

Non-equivalence occurs when the dimension is divisible by 4.

Examples with nonempty finite open sets in the unitary dual are provided.

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.