The posted chord names don't make sense to me, unless there is more music that we are not aware of. OP may have private harmonic considerations, e.g., something in their ears that we can't hear, which may mitigate the usefulness of the rest of this answer. We'll see....
How Else Might These Chords Be Named?
Seeing F9, I am expecting that chord to be going somewhere, but where? This would usually be heading to some type of Bb chord (there are a few other slightly more exotic usual suspects), but there is no Bb present among the other chords; maybe that is yet to come. Further, F9 contains an Eb, which is also not among any of the presented notes. An Fadd9 could make sense here, since the notes of that chord are F, C, and G, i.e., the root, fifth, and ninth of Fadd9. There is some ambiguity since there is no third here.
The first chord is plausibly Am7 with A, C, and G, the root, third, and seventh of that chord. This is all you need to convincingly voice a seventh chord; in fact seventh chord voicings containing only the root, third, and seventh are so useful and common that they have a name: shell voicings. Let's grant that the first chord is Am7, then.
Calling the second chord CMaj7 seems problematic. The notes here are B, C, and G. If it were a CMaj7, the third is missing; this in itself is not a deal-breaker. The major 7th is in the bass of the chord, which is a little unusual, but not unheard of, and again not a deal-breaker. I suppose that it would be fine to call this CMaj7, but thinking of the chords that way makes the chords seem more static than they are when I play through the actual voicings as written. Instead, I am tempted to call the second chord a G, or maybe better, a G11. The written voicing has the root, third, and eleventh of this chord.
The third chord contains E, C, and G, so calling it C seems natural.
Given the above, I would be tempted to write the progression as Am7 G11 C Fadd9. That is, in roman numerals: vi7 V I IV. Now the IV chord, the Fadd9 might go back to the I chord, which seems kind of natural to me since I think that C sounds like the tonic here.
It is more common to find minor chords with 11ths than it is to find major chords with 11ths; this is because the third and fourth of a major chord are only a half-step apart, i.e., the 11th is a minor ninth above the major third. This sound isn't always appealing, but sometimes it works very nicely. In the posted music the top two notes C and G are maintained throughout, making them pedal points. Pedal points are usually in the bass, but they don't have to be. This actually helps mitigate the dissonance of the G11 chord by placing that minor 9th in a larger context; pedal point passages often temporarily admit some dissonance. Viewed as a pedal point, the C is first the minor third of Am7, then the 11th of G11 (a dissonance), then the root of the C, and finally the fifth of Fadd9.
So I would, at least as a start, think of these chords as Am7 G11 C Fadd9, and if I were looking for other notes to flesh things out I might use the notes of those chords to guide my thinking.
Incidentally, when I played the posted music I was immediately reminded of this tune by The Shins that was popular several years ago.