We consider representations of Cuntz-Krieger algebras on the Hilbert space of square integrable functions on the limit set, identified with a Cantor set in the unit interval. We use these representations and the associated Perron-Frobenius and Ruelle operators to construct families of wavelets on these Cantor sets.
We identify multiresolution subspaces giving rise via Hankel transforms to
Bessel functions. They emerge as orthogonal systems derived from geometric
Hilbert-space considerations, the same way the wavelet functions from a
multiresolution scaling wavelet construction arise from a scale of Hilbert
spaces. We study the theory of representations of the $C^{\ast}$-algebra $%
O_{\nu +1}$ arising from this multiresolution analysis. A connection with
Markov chains and representations of $O_{\nu +1}$ is found. Projection valued
measures arising from the multiresolution analysis give rise to a Markov trace
for quantum groups $SO_q$.Comment: Research paper, 32 page
Abstract. We study coactions of Hopf algebras coming from compact quantum groups on the Cuntz algebra. These coactions are the natural generalization to the coalgebra setting of the canonical representation of the unitary matrix group U (d) as automorphisms of the Cuntz algebra O d .In particular we study the fixed point subalgebra under the coaction of the quantum compact groups Uq(d) on the Cuntz algebra O d by extending to any dimension d < ∞ a result of Konishi (1992).Furthermore we give a description of the fixed point subalgebra under the coaction of SUq(d) on O d in terms of generators.
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