Fake books I've seen have the name of the chords, such as "C" [major], "Cmin[or]", F7 etc. etc. and the melody. You then have all you need to improvise -- you can play the melody in the right hand, and block chords in the left; you can use arpeggios in the left; you can improvise a countermelody in the left; you can play the melody in the left and chords or a descant in the right... whatever suits your fancy.