On the real rank of C*-algebras of nilpotent locally compact groups

If GG is an almost connected, nilpotent, locally compact group then the real rank of the C∗C∗-algebra C∗(G)C∗(G) is given by RR(C∗(G))=rank(G/[G,G])=rank(G0/[G0,G0])RR⁡(C∗(G))=rank⁡(G/[G,G])=rank⁡(G0/[G0,G0]), where G0G0 is the connected component of the identity element. In particular, for the continuous Heisenberg group G3G3, RRC∗(G3))=2RR⁡C∗(G3))=2.