Iterated function systems and permutation representations of the Cuntz algebra

Ola Bratteli, Palle E. T. Jorgensen
1999 Memoirs of the American Mathematical Society  
We study a class of representations of the Cuntz algebras ON, N = 2, 3, ... , acting on L 2 (1!') where 11' = lR/27rZ. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into irreducibles, and show how the ON-irreducibles decompose when restricted to the subalgebra UHFN C ON 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
more » ... al results on a class of representations of ON on Hilbert space 1l such that the generators Si as operators permute the elements in some orthonormal basis for 1£. We then use this to extend our results from L 2 (11') to L 2 ('ll'd), d > 1; even to L 2 (T) where Tis some fractal version of the torus which carries more of the algebraic information encoded in our representations.
doi:10.1090/memo/0663 fatcat:s4v6kt6itzdlpfxtfnc6cfgsme