Beurling type quotient modules over the bidisk and boundary representations

This paper mainly concerns Beurling type quotient modules of H2(D2) over the bidisk. By establishing a theorem of function theory over the bidisk, it is shown that a Beurling type quotient module is essentially normal if and only if the corresponding inner function is a rational inner function having degree at most (1,1). Furthermore, we apply this result to the study of boundary representations of Toeplitz algebras over quotient modules. It is proved that the identity representation of C∗(Sz,Sw) is a boundary representation of B(Sz,Sw) in all nontrivial cases. This extends a result of Arveson to Toeplitz algebras on Beurling type quotient modules over the bidisk (cf. [W. Arveson, Subalgebras of C∗-algebras, Acta Math. 123 (1969) 141–224; W. Arveson, Subalgebras of C∗-algebras II, Acta Math. 128 (1972) 271–308]). The paper also establishes K-homology defined by Beurling type quotient modules over the bidisk.