Representations of Dirichlet Operator Algebras

A Dirichlet operator algebra is a nonself-adjoint operator algebra $\mathcal{A}$ with the property that $\mathcal{A}+{\mathcal{A}}^{\ast }$ is norm-dense in the C${}^{\ast }$-envelope of $\mathcal{A}\mathrm{.}$ 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${}^{\ast }$-algebras. The main tool is the duality theory for essentially principal etale groupoids.