Iterated function systems and permutation representations of the Cuntz algebra

TL;DR

This paper investigates a class of representations of the Cuntz algebras related to wavelet theory, analyzing their decomposition into irreducible components and extending the results to higher dimensions and fractal structures.

Contribution

It introduces a new class of permutation-based representations of Cuntz algebras and describes their irreducible decomposition, including extensions to fractal and higher-dimensional cases.

Findings

Decomposition of representations into irreducibles described.

Extensions to L^2 spaces on higher-dimensional tori.

Representation structures linked to arithmetic and combinatorial properties.

Abstract

We study a class of representations of the Cuntz algebras O_N, N=2,3,..., acting on L^2(T) where T=R/2\pi Z. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into irreducibles, and show how the O_N-irreducibles decompose when restricted to the subalgebra UHF_N\subset O_N of gauge-invariant elements; and we show that the whole structure is accounted for by arithmetic and combinatorial properties of the integers Z. We have general results on a class of representations of O_N on Hilbert space H such that the generators S_i as operators permute the elements in some orthonormal basis for H. We then use this to extend our results from L^2(T) to L^2(T^d), d>1 ; even to L^2(\mathbf{T}) where \mathbf{T} is some fractal version of the torus which carries more of the algebraic information encoded in our representations.