On C⁎-algebras of exponential solvable Lie groups and their real ranks

For any solvable Lie group whose exponential map expG⁡:g→GexpG⁡:g→G is bijective, we prove that the real rank of C⁎(G)C⁎(G) is equal to dim⁡(g/[g,g])dim⁡(g/[g,g]). We also indicate a proof of a similar formula for the stable rank of C⁎(G)C⁎(G), as well as some estimates on the ideal generated by the projections in C⁎(G)C⁎(G).