The Choquet-Deny equation in a Banach space

Let $G$ be a locally compact group and $\pi$ a representation of $G$ by weakly^* continuous isometries acting in a dual Banach space $E$. Given a probability measure $\mu$ on $G$ we study the Choquet-Deny equation $\pi(\mu)x=x$, $x\in E$. We prove that the solutions of this equation form the range of a projection of norm 1 and can be represented by means of a ``Poisson formula'' on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law $\mu$. The relation between the space of solutions of the Choquet-Deny equation in $E$ and the space of bounded harmonic functions can be understood in terms of a construction resembling the $W^*$-crossed product and coinciding precisely with the crossed product in the special case of the Choquet-Deny equation in the space $E=B(L^2(G))$ of bounded linear operators on $L^2(G)$. Other general properties of the Choquet-Deny equation in a Banach space are also discussed.