On $C^{\ast }$-algebras of exponential solvable Lie groups and their real ranks

Abstract

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

arXiv comment8 pages

References in corpus (3)

• Fourier transforms of $C^{\ast }$-algebras of nilpotent Lie groups
• Coadjoint structure of Borel subgroups and their nilradicals
• On higher real and stable ranks for CCR C*-algebras