Norms of idempotent Schur multipliers

Let D be a masa in B(H) where H is a separable Hilbert space. We find real numbers \eta_0 < \eta_1 < \eta_2 < ... < \eta_6 so that for every bounded, normal D-bimodule map {\Phi} on B(H) either ||\Phi|| > \eta_6, or ||\Phi|| = \eta_k for some k <= 6. When D is totally atomic, these maps are the idempotent Schur multipliers and we characterise those with norm \eta_k for 0 <= k <= 6. We also show that the Schur idempotents which keep only the diagonal and superdiagonal of an n x n matrix, or of an n x (n+1) matrix, both have norm 2/(n+1) cot(pi/(n+1)), and we consider the average norm of a random idempotent Schur multiplier as a function of dimension. Many of our arguments are framed in the combinatorial language of bipartite graphs.