Let 𝔻n be the open unit polydisc in ℂn,
$n \ges 1$
, and let H2(𝔻n) be the Hardy space over 𝔻n. For
$n\ges 3$
, we show that if θ ∈ H∞(𝔻n) 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
H2(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.