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 toward approximating this algebra in a Chern band. In this Letter, we prove that the GMP algebra, up to form factors, is the 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.