QUANTUM CHANNELS, WAVELETS, DILATIONS AND REPRESENTATIONS OF Oₙ

We show that the representations of the Cuntz C^*-algebras Oₙ which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finite-dimensional space. Based on this analysis, we find an application in quantum information theory; namely, a structure theorem for the fixed-point set of a unital quantum channel. We also include some open problems motivated by this work. AMS 2000 Mathematics subject classification: Primary 46L45; 47A20; 46L60; 42C40; 81P15