Harmonic analysis and fractal limit-measures induced by representations of a certain \(C^*\)-algebra
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)].
Cuntz \(C^*\)-algebra \({\mathcal O}_ n\)
scientific article; zbMATH DE number 704496