The atomic and electronic structures of the Ag∕Si(111)√3×√3 surface are currently under debate. By employing angle-resolved valence-band spectroscopy, the surface band dispersions around the K̅ point of the Ag∕Si(111)√3×√3 surface have been investigated in detail. Contrary to a recent study, we conclude that the S₂ and S₃ surface state bands do not show any detectable split at 100 K. Thus, photoemission spectra at both room temperature and 100 K show only a single peak at the K̅ point without any direct evidence of a split. Calculated band structures for the inequivalent triangle (IET) model show a gap at the K̅ point in contrast to the honeycomb-chain-trimer (HCT) model. We find, however, that there is no real contradiction between our photoemission data and the IET model provided the energy gap of the latter model is small as indicated by a recent calculation [ Phys. Rev. B 70 245431 (2004)].