Harmonic analysis and fractal limit-measures induced by representations of a certain \(C^*\)-algebra
type: Item
instance of: scholarly article
title: Harmonic analysis and fractal limit-measures induced by representations of a certain \(C^*\)-algebra
zbMATH Open document ID: 0829.43022
author: Palle Jorgensen
author: Steen Pedersen
published in: Journal of Functional Analysis
publication date: 1994-12-20T00:00:00Z
review text: If \(\Gamma\) is a rank \(d\) lattice in \(\mathbb{R}^d\), the corresponding fundamental domains \(\Omega \subset \mathbb{R}^d\) are well-known examples of the class of measurable subsets of \(\mathbb{R}^d\), such that \(L^2(\Omega)\) has an orthogonal basis of frequencies \(e_\lambda (x) = e^{i2 \pi \lambda x}\), indexed by elements \(\lambda\) of \(\Lambda\) in \(\mathbb{R}^d\). The main result of the paper is Theorem 2.1 giving some iteration of this spectral pair \((\Omega, \Lambda)\) to another one \((\Omega_1, \Lambda_1)\), which gives a model of the so called Cuntz \(C^*\)-algebra \({\mathcal O}_n\). It is used later to produce some fractal limit-measures \(\mu\) of the corresponding probability measures. The authors show also that such spectral pairs can be reconstructed from some representation of the Cuntz algebra \({\mathcal O}_n\) on \(L^2(\mu)\). The results were previously announced by the authors in [Bull. Am. Math. Soc., New Ser. 29, 228-234 (1993; Zbl 0785.28005)].
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 43A99
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 704496
zbMATH Keywords: orthogonal basis of frequencies
zbMATH Keywords: Cuntz \(C^*\)-algebra \({\mathcal O}_ n\)
zbMATH Keywords: fractal limit-measures
MaRDI profile type: Publication
full work available at URL: jfan.1994.1118
OpenAlex ID: W1994773938
DOI: 10.1006/JFAN.1994.1118
description: scientific article; zbMATH DE number 704496