Toeplitz algebras, subnormal tuples and rigidity on reproducing \(C[z_{1},\dots ,z_{d}]\)-modules
type: Item
instance of: scholarly article
title: Toeplitz algebras, subnormal tuples and rigidity on reproducing \(C[z_{1},\dots ,z_{d}]\)-modules
zbMATH Open document ID: 1066.47075
author: Kunyu Guo
author: Junyun Hu
author: Xianmin Xu
published in: Journal of Functional Analysis
publication date: 2004-08-16T00:00:00Z
review text: It is the aim of the paper under review to study Toeplitz algebras, subnormal tuples, rigidity on reproducing modules \(H\) on a domain \(\Omega\subset{\mathbb C}^d\) over the polynomial ring of \(d\)-complex variables \({\mathbb C}[z_1,\dots,z_d]\) (or on their submodules) in connection to Arveson's boundary representation theory. Several examples prove that the theory is consistent: the Dirichlet module and the Arveson module have the boundary property, while the Hardy module and the weighted Bergman module do not have this property (all modules considered over the unit ball \({\mathbb B}_d)\). If \(H\) has the boundary property, it is proved that there are no inner multipliers for this module (except for constants), hence such a module must be rigid. The same conclusion holds true if \(H\) is given by a positive measure on \({\mathbb C}^d\) (hence \(H\) is subnormal), supported on a set of positive Lebesgue measure. However, a subnormal module \(H\) on a bounded domain \(\Omega\) cannot have the boundary property if \(\bar\Omega\) covers the Taylor spectrum of the tuple \(\{M_{z_1},\dots,M_{z_d}\}\) of coordinate multipliers on \(H\). A \(\mathcal U\)-invariant kernel on the ball \({\mathbb B}^r_d\) (of radius \(r\)) is given by a unique power series \(f\) with nonnegative coefficients and convergence radius \(R\geq r^2\). If \(f\) is regular, then the corresponding reproducing function space \(H^f_d\) is actually a reproducing \({\mathbb C}[z_1,\dots,z_d]\)-module on \({\mathbb B}_d^{\sqrt{R}}\). The Taylor spectrum and the essential Taylor spectrum of \(\{M_{z_1},\dots,M_{z_d}\}\) are computed. An exact sequence of \(C^*\)-algebras is then associated with the Toeplitz algebra \({\mathcal T}^f_d\) on \(H^f_d\). As a consequence of the above results, a necessary condition for modules being subnormal is obtained. Finally, similar properties are studied for submodules \(M\) of Arveson's module \(H^2_d\). It is shown that there are no inner multipliers for \(M\) and…
Mathematics Subject Classification ID: 47L80
Mathematics Subject Classification ID: 47A13
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 47B32
zbMATH DE Number: 2091745
zbMATH Keywords: Toeplitz algebra
zbMATH Keywords: subnormal tuple
zbMATH Keywords: equivalence problem
MaRDI profile type: Publication
full work available at URL: j.jfa.2003.06.003
OpenAlex ID: W2069188697
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DOI: 10.1016/J.JFA.2003.06.003
description: scientific article; zbMATH DE number 2091745