On quotient modules of H 2 (𝔻 n ): essential normality and boundary representations

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Abstract Let 𝔻 n be the open unit polydisc in ℂ n , $n \ges 1$ , and let H 2 (𝔻 n ) be the Hardy space over 𝔻 n . For $n\ges 3$ , we show that if θ ∈ H ∞ (𝔻 n ) is an inner function, then the n -tuple of commuting operators $(C_{z_1}, \ldots , C_{z_n})$ on the Beurling type quotient module ${\cal Q}_{\theta }$ is not essentially normal, where $${\rm {\cal Q}}_\theta = H^2({\rm {\open D}}^n)/\theta H^2({\rm {\open D}}^n)\quad {\rm and}\quad C_{z_j} = P_{{\rm {\cal Q}}_\theta }M_{z_j}\vert_{{\rm {\cal Q}}_\theta }\quad (j = 1, \ldots ,n).$$ Rudin's quotient modules of H 2 (𝔻 2 ) are also shown to be not essentially normal. We prove several results concerning boundary representations of C *-algebras corresponding to different classes of quotient modules including doubly commuting quotient modules and homogeneous quotient modules.