On the real rank of C^*-algebras of nilpotent locally compact groups - Archbold, Robert J.; Kaniuth, Eberhard (MATHEMATICA SCANDINAVICA, 2012)
If G is an almost connected, nilpotent, locally compact group then the real rank of the C^*-algebra C^* (G) is given by RR (C^* (G)) = rank (G/[G,G]) = rank (G_0/[G_0,G_0]), where G_0 is the connected component of the identity element. In particular, for the continuous Heisenberg group G_3, RR C^* (G_3))=2.