The choquet-deny equation in a banach space

Let G be a locally compact group and π a representation of G by weakly* continuous isometries acting in a dual Banach space E. Given a probability measure π on G, we study the Choquet-Deny equation π(μ)x = x, x ∈ 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 μ. 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 L2(G). Other general properties of the Choquet-Deny equation in a Banach space are also discussed.