Beurling type quotient modules over the bidisk and boundary representations

Abstract: This paper mainly concerns Beurling type quotient modules of 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 . 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 is a boundary representation of 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 -algebras, Acta Math. 123 (1969) 141–224; W. Arveson, Subalgebras of -algebras II, Acta Math. 128 (1972) 271–308]). The paper also establishes K-homology defined by Beurling type quotient modules over the bidisk. [Copyright &y& Elsevier]