On $C^*$-algebras of exponential solvable Lie groups and their real ranks

TL;DR

This paper establishes a formula for the real rank of the $C^*$-algebra of exponential solvable Lie groups, linking it to the dimension of the Lie algebra quotient by its commutator.

Contribution

It proves that for exponential solvable Lie groups, the real rank of their $C^*$-algebra equals the dimension of the Lie algebra modulo its commutator, and discusses stable rank and ideal estimates.

Findings

Real rank of $C^*(G)$ equals $ ext{dim}(rak{g}/[rak{g},rak{g}])$ for exponential solvable Lie groups.

Provides a proof outline for the stable rank formula of $C^*(G)$.

Offers estimates on the ideal generated by projections in $C^*(G)$.

Abstract

For any solvable Lie group whose exponential map $\exp_G\colon{\mathfrak g}\to G$ is bijective, we prove that the real rank of $C^*(G)$ is equal to $\dim({\mathfrak g}/[{\mathfrak g},{\mathfrak g}])$. We also indicate a proof of a similar formula for the stable rank of $C^*(G)$, as well as some estimates on the ideal generated by the projections in $C^*(G)$.