Equivariant Maps and Bimodule Projections

We construct a contractive, idempotent, MASA bimodule map on B(H), whose range is not a ternary subalgebra of B(H). Our method uses a crossed-product to reduce the existence of such an idempotent map to an analogous problem about the ranges of idempotent maps that are equivariant with respect to a group action and Hamana's theory of G-injective envelopes.