  The band structure of the recently synthesized (3$\times$3) silicene
monolayer on (4$\times$4) Ag(111) is investigated using density functional
theory. A $k$-projection technique that includes the $k_\bot$-dependence of the
surface bands is used to separate the contributions arising from the silicene
and the substrate, allowing a consistent comparison between the calculations
and the angle-resolved photoemission experiments. Our calculations not only
reproduce the observed gap and linear dispersion across the K point of
(1$\times$1) silicene, but also demonstrate that these originate from the
$k_\bot$-dependence of Ag(111) substrate states (modified by interactions with
the silicene) and \textit{not} from a Dirac state.
