Universal Kernel Models for Iterated Completely Positive Maps

Abstract: We study how iterated and composed completely positive maps act on operator-valued kernels. Each kernel is realized inside a single Hilbert space where composition corresponds to applying bounded creation operators to feature vectors. This model yields a direct formula for every iterated kernel and allows pointwise limits, contractive behavior, and kernel domination to be read as standard operator facts. The main results include an explicit limit kernel for unital maps, a Stein-type decomposition, a Radon-Nikodym representation under subunitality, and an almost-sure growth law for random compositions. The construction keeps all iterates in one space, making their comparison and asymptotic analysis transparent.