On Quotient modules of $H^2({\mathbb{D}}^n)$: Essential Normality and Boundary Representations

Let ${\mathbb{D}}^n$ be the open unit polydisc in ${\mathbb{C}}^n$, $n\ge 1$, and let $H^2({\mathbb{D}}^n)$ be the Hardy space over ${\mathbb{D}}^n$. For $n\ge 3$,weshowthatif$\theta \in H^\infty($\mathbb{D}$^n)$ is an inner function, then the $n$-tuple of commuting operators $(C_{z_1},\dots ,C_{z_n})$ on the Beurling type quotient module ${\mathcal{Q}}_{\theta }$ is not essentially normal, where $${\mathcal{Q}}_{\theta }=H^2({\mathbb{D}}^n)\mathrm{/}\theta H^2({\mathbb{D}}^n)and{C}_{z_j}=P_{{\mathcal{Q}}_{\theta }}{M}_{z_j}{\mathrm{\mid }}_{{\mathcal{Q}}_{\theta }}(j=1,\dots ,n)\mathrm{.}$$ Rudin’s quotient modules of $H^2({\mathbb{D}}^2)$ are also shown to be not essentially normal. We prove several results concerning boundary representations of $C^{\ast }$-algebras corresponding to different classes of quotient modules including doubly commuting quotient modules and homogeneous quotient modules.