Iterated function systems and permutation representations of the Cuntz algebra
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_Nirreducibles decompose when restricted to the subalgebra UHF_N\subset O_N of gaugeinvariant 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.
 Publication:

arXiv eprints
 Pub Date:
 December 1996
 DOI:
 10.48550/arXiv.functan/9612002
 arXiv:
 arXiv:functan/9612002
 Bibcode:
 1996funct.an.12002B
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Operator Algebras;
 46L55;
 47C15 (Primary) 42C05;
 22D25;
 11B85 (Secondary)
 EPrint:
 84 pages, 11 figures, AMSLaTeX v1.2b, fullresolution figures available at ftp://ftp.math.uiowa.edu/pub/jorgen/PermRepCuntzAlg in eps files with the same names as the lowresolution figures included here