An abstract approximation framework for the solution of operator algebraic Riccati equations is developed. The approach taken is based upon a formulation of the Riccati equation as an abstract nonlinear operator equation on the space of Hilbert-Schmidt operators. Hilbert-Schmidt norm convergence of solutions to generic finite dimensional Galerkin approximations to the Riccati equation to the generic finite dimensional Galerkin approximations to the Riccati equation to the solution of the original infinite dimensional problem is argued. The application of the general theory is illustrated via an operator Riccati equation arising in the linear-quadratic design of an optimal feedback control law for a one dimensional heat/diffusion equation. Numerical results demonstrating the convergence of the associated Hilbert-Schmidt kernels are included. Keywords: Operator algebraic Riccati equation; Hilbert Schmidt operator; Nonlinear operator equation; Galerkin approximation; Linear quadratic regulator. (JHD).