We present a detailed analysis of the band structure of the root 3 x root 3 R 30 degrees BiAg2/Ag/Si(111) trilayer system by means of high resolution Angle Resolved Photoemission Spectroscopy (ARPES). BiAg2/Ag/Si(111) exhibits a complex spin-polarized electronic structure due to giant spin-orbit interactions. We show that a complete set of constant energy ARPES maps, supplemented by a modified nearly free electron calculation, provides a unique insight into the structure of the spin-polarized bands and spin gaps. We also show that the complex gap structure can be continuously tuned in energy by a controlled deposition of an alkali metal.