  We associate a dimer algebra A to a Postnikov diagram D (in a disk)
corresponding to a cluster of minors in the cluster structure of the
Grassmannian Gr(k,n). We show that A is isomorphic to the endomorphism algebra
of a corresponding Cohen-Macaulay module T over the algebra B used to
categorify the cluster structure of Gr(k,n) by Jensen-King-Su. It follows that
B can be realised as the boundary algebra of A, that is, the subalgebra eAe for
an idempotent e corresponding to the boundary of the disk. The construction and
proof uses an interpretation of the diagram D, with its associated plabic graph
and dual quiver (with faces), as a dimer model with boundary. We also discuss
the general surface case, in particular computing boundary algebras associated
to the annulus.
