Over the years there has been considerable interest in the algebraic properties of the following Riccati equation
A* X+XA-XBB*+C* C=0,
where A,B,C lies in A, a Banach algebra with identity, and the involution operation *. Conditions are sought to ensure that the Riccati equation has a stabilizing solution in A: i.e., X in A satisfies the Riccati equation and A-BB* X is stable with respect to A. In this talk this question is examined for the case that A,B,C are matrices with components in a commutative scalar Banach algebra A₀, i.e., A=A₀ⁿ. It turns out that X will be in this algebra only if the involution operation satisfies certain conditions. Applications to spatially invariant systems are discussed.