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On quotient modules of H2(𝔻n): essential normality and boundary representations

Part of: Selfadjoint operator algebras Linear spaces and algebras of operators General theory of linear operators

Published online by Cambridge University Press:  31 January 2019

B. Krishna Das ,
Sushil Gorai  and
Jaydeb Sarkar
B. Krishna Das
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, India400076 (dasb@math.iitb.ac.in; bata436@gmail.com)
Sushil Gorai
Affiliation:
Department of Mathematics and Statistics, Indian Institute of Science Education and Research Kolkata, Mohanpur 741 246, West Bengal, India (sushil.gorai@iiserkol.ac.in)
Jaydeb Sarkar
Affiliation:
Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore560059, India (jay@isibang.ac.in; jaydeb@gmail.com)
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Abstract

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(𝔻2) 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.

Keywords

Hardy spacereproducing kernel Hilbert spacesquotient modulesessential normalityboundary representations

MSC classification

Primary: 47A13: Several-variable operator theory (spectral, Fredholm, etc.)
Secondary: 47A20: Dilations, extensions, compressions 47L25: Operator spaces (= matricially normed spaces) 47L40: Limit algebras, subalgebras of $C^*$-algebras 46L05: General theory of $C^*$-algebras 46L06: Tensor products of $C^*$-algebras
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1339 - 1359
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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