Let 𝒢 be a (separable)
Hausdorff space and let K be a continuous nonnegative-definite kernel (covariance)
from 𝒢×𝒢 to C. The well known theorem of Kolmogorov states that in the case 𝒢 is
the set of integers there is a continuous mapping (stochastic process) x(⋅) from 𝒢 into
a (separable) Hilbert space 𝒦 such that K(s,t) = (x(s),x(t)). The theorem is also
known for any separable Hausdorff space. The purpose of this paper is to replace the
complex numbers C by the algebra B(ℋ,ℋ) of bounded linear operators from a
Hilbert space into itself. The factorization is then K(t,s) = X(t)∗X(s) with X a
continuous map from 𝒢 to B(ℋ,𝒦) for a suitable Hilbert space 𝒦. If 𝒢 is separable
we may take 𝒦 = ℋ.
