Beurling type quotient modules over the bidisk and boundary representations
Beurling type quotient modules over the bidisk and boundary representations
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DOI：
10.1016/j.jfa.2009.06.031
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发表时间：
2009-11
影响因子：
1.7
通讯作者：
K. Guo;Kai Wang
中科院分区：
文献类型：
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作者：
K. Guo;Kai Wang
本文主要研究双圆盘上H2（D2）的Beurling型商模.通过建立双圆盘上函数论的一个定理，证明了Beurling型商模本质正规当且仅当对应的内函数是次数至多为（1，1）的有理内函数.此外，我们将这个结果应用于商模上Toeplitz代数的边界表示的研究。证明了在所有非平凡情形下，C ∈（Sz，Sw）的单位表示是B（Sz，Sw）的边界表示.这将Arveson的一个结果推广到了双圆盘上Beurling型商模上的Toeplitz代数（参见[W.李文，C-代数的子代数，数学学报，1999年，第123期，第141-224页; Arveson，Subalgebras of C-algebras II，Acta Math.128（1972）271-308]）.本文还建立了双圆盘上由Beurling型商模定义的K-同调。
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.