  We prove maximal ergodic theorems for spherical averages on the Heisenberg
groups acting on $L_p$ spaces over measure spaces not necessarily commutative,
that is, on noncommutative $L_p$ spaces. The scale of $p$ is optimal in the
reduced Heisenberg group case. We also obtain the corresponding individual
ergodic theorems and differential theorems in the noncommutative setting. The
results can be regarded as noncommutative analogues of Nevo-Thangavelu's
ergodic theorems. The approach of proof involves recent developments in
noncommutative $L_p$ spaces and in noncommutative harmonic analysis, in
addition to the spectral theory used in the commutative setting.
