Beurling type quotient modules over the bidisk and boundary representations

Abstract

This paper mainly concerns Beurling type quotient modules of H 2 ( D 2 ) 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 ∗ ( S z , S w ) is a boundary representation of B ( S z , S w ) 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.