Conditional covariance kernels in a noncommutative setting

We address a new factorization theory, and conditioning analysis, in the context of noncommutative stochastic systems. Our focus is that of transition matrices/kernels as they arise in diverse contexts, including in quantum theory. Our initial positive definite operator valued kernels [Formula: see text] are considered as generators of both a noncommutative systems; and, in different contexts, as a noncommutative covariance operators, or a covariance kernel. We address the following two questions: For the first, we study an associated system of path spaces. For the second, we consider associated stochastic process (typically Gaussian) fitting a framework introduced by Arveson [J. Funct. Anal. 66, 173 (1986)]. In the general context, our analysis of the associated stochastic processes we introduce new infinite-dimensional systems of polymorphisms.