Maximal Abelian von Neumann algebras and Toeplitz operators with separately radial symbols
type: Item
instance of: scholarly article
title: Maximal Abelian von Neumann algebras and Toeplitz operators with separately radial symbols
zbMATH Open document ID: 1195.47015
author: Hansong Huang
published in: Integral Equations and Operator Theory
publication date: 2009-11-02T00:00:00Z
review text: The author studies the algebras generated by Toeplitz operators \(T_a\) with symbol \(a\) acting on the standard weighted Bergman spaces \(A^2_{\alpha}(\mathbb{D})\), \(\alpha > -1\), over the unit disk \(\mathbb(D)\) on the complex plane. Given a pencil \(\mathcal{P}\) of hyperbolic geodesics, denote by \(\mathcal{A}(\mathbb{D})\) the set of \(L^{\infty}\)-symbols which are constant on the corresponding cycles. It is well-known that the \(C^*\)-algebra \(\mathcal{T}_\alpha(\mathcal{A}(\mathbb{D}))\) generated by all Toeplitz operators with symbols \(a \in \mathcal{A}(\mathbb{D})\) is abelian on each weighted Bergman space \(A^2_{\alpha}(\mathbb{D})\). The author shows that the \(w^*\)-closure of \(\mathcal{T}_\alpha(\mathcal{A}(\mathbb{D}))\) is maximal abelian. That is, any operator in the commutator of \(\mathcal{T}_\alpha(\mathcal{A}(\mathbb{D}))\) lies in the \(w^*\)-closure of \(\mathcal{T}_\alpha(\mathcal{A}(\mathbb{D}))\).
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47C15
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 47L80
zbMATH DE Number: 5624099
zbMATH Keywords: weighted Bergman space
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: abelian von Neumann algebra
zbMATH Keywords: radial
zbMATH Keywords: separately radial
MaRDI profile type: Publication
full work available at URL: s00020-009-1695-9
OpenAlex ID: W2078632146
DOI: 10.1007/S00020-009-1695-9
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description: scientific article; zbMATH DE number 5624099