  A Dirichlet operator algebra is a nonself-adjoint operator algebra
$\mathcal{A}$ with the property that $\mathcal{A} + \mathcal{A}^*$ is
norm-dense in the C$^*$-envelope of $\mathcal{A}.$ We show that, under certain
restrictions, $\mathcal{A}$ has a family of completely contractive
representations $\{\pi_i\}$ with the property that the invariant subspaces of
$\pi_i(\mathcal{A})$ are totally ordered, and such that, for all $a \in
\mathcal{A}, \ ||a|| = \sup_i ||\pi_i(a)||.$ The class of Dirichlet algebras
includes strongly maximal triangular AF algebras, certain semicrossed product
algebras, and gauge-invariant subalgebras of Cuntz C$^*$-algebras.
  The main tool is the duality theory for essentially principal etale
groupoids.
