2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68655We show that the representations of the Cuntz C$^\ast$-algebras $O_n$ 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, an application in quantum information theory is obtained; namely, a structure theorem for the fixed point set of a unital quantum channel. We also include some open problems motivated by this work.15 pages, preprint versionOperator AlgebrasQuantum Physics46L45, 47A20, 46L60, 42C40, 81P15text