\(C^*\)-algebras of anisotropic Schrödinger operators on trees
type: Item
instance of: scholarly article
title: \(C^*\)-algebras of anisotropic Schrödinger operators on trees
zbMATH Open document ID: 1068.81022
author: Sylvain Golénia
published in: Annales Henri Poincaré
publication date: 2005-02-09T00:00:00Z
review text: Let \(\Gamma\) be a \(\nu\)-fold tree and \(\widehat\Gamma= \Gamma\cup\partial\Gamma\) be its the hyperbolic compactification. On \(\ell^2(\Gamma)\) defined is the differential operator given by \((\partial f)(x)=\sum_{y'=x}f(y)\). Let \(\mathcal{D}\) be the \(C^*\)-algebra generated by \(\partial\). By \(C(\widehat\Gamma)\) is denoted the set of complex-valued continuous functions on \(\widehat\Gamma\). Denote by \(\mathcal{C}(\widehat\Gamma)\) the \(C^*\)-algebra generated by \(\mathcal{D}\) and \(C(\widehat\Gamma)\). This algebra contains the set \(\mathbb{K}(\Gamma)\) of compact operators on \(\ell^2(\Gamma)\). In this paper the algebra \(\mathcal{C}(\widehat\Gamma)\) is studied. Following [\textit{V. Georgescu, A. Iftimovici}, Commun. Math. Phys. 228, 519--560 (2002; Zbl 1005.81026)] the quotient \(\mathcal{C}(\widehat\Gamma)/\mathbb{K}(\Gamma)\) is considered. Namely, it is proved the following Theorem. Let \(\nu>1\). There is a unique morphism \(\Phi:\mathcal{C}(\widehat\Gamma)\to \mathcal{D}\otimes C(\partial\Gamma)\) such that \(\Phi(D)=D\otimes 1\), for all \(D\in\mathcal{D}\) and \(\Phi(\varphi(Q))=1\otimes(\varphi| _{\partial\Gamma})\). This morphism is surjective and its kernel is \(\mathbb{K}(\Gamma)\). This result allowed to compute the essential spectrum of self-adjoint operators affiliated to \(\mathcal{C}(\widehat\Gamma)\). The results cover Schrödinger operators with highly anisotropic potential.
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 46L08
Mathematics Subject Classification ID: 81R15
zbMATH DE Number: 2133173
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: compact operators
zbMATH Keywords: quotient
zbMATH Keywords: essential spectrum
reviewed by: Farruh Mukhamedov
MaRDI profile type: Publication
full work available at URL: s00023-004-0192-6
OpenAlex ID: W2365225304
DOI: 10.1007/S00023-004-0192-6
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description: scientific article; zbMATH DE number 2133173