A Closed Band-Projected Density Algebra Must be Girvin-MacDonald-Platzman

The band-projected density operators in a Landau level obey the Girvin-MacDonald-Platzman (GMP) algebra, and a large amount of effort in the study of fractional Chern insulators has been directed towards approximating this algebra in a Chern band. In this paper, we prove that the GMP algebra, up to form factors, is the $\mathit{\text{only}}$ closed algebra that projected density operators can satisfy in two and three dimensions, highlighting the central place it occupies in the study of Chern bands in general. A number of interesting corollaries follow.