Mathematics – Operator Algebras
Scientific paper
1996-12-17
Mem. Amer. Math. Soc. 139 (1999), no. 663
Mathematics
Operator Algebras
84 pages, 11 figures, AMS-LaTeX v1.2b, full-resolution figures available at ftp://ftp.math.uiowa.edu/pub/jorgen/PermRepCuntzAl
Scientific paper
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.
Bratteli Ola
Jorgensen Palle E. T.
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