  In this article, we present a method for approximating affine processes on
the cone of positive Hilbert-Schmidt operators using matrix-valued affine
processes. By leveraging results from the theory on affine processes with
values in the cone of symmetric and positive semi-definite matrices, we
construct sequences of finite-rank operator-valued affine processes that
converge weakly to the target processes and provide convergence rates for their
Laplace transforms using Galerkin approximations of the associated
operator-valued generalized Riccati equations. This article not only offers a
practical approximation scheme for operator-valued affine processes with error
bounds that hold uniformly in time, but also provides a novel existence proof
for this class of affine processes with c\`adl\`ag paths, including affine
pure-jump processes with infinite variation and state-dependent jump
intensities. In addition to the theoretical significance, the results of this
paper provide useful tools for analyzing and approximating infinite-dimensional
affine stochastic covariance models that were recently introduced in
mathematical finance.
