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# The Choquet--Deny Equation in a Banach Space

Published:2007-08-01
Printed: Aug 2007
• Wojciech Jaworski
• Matthias Neufang
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## Abstract

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.
 MSC Classifications: 22D12 - Other representations of locally compact groups 22D20 - Representations of group algebras 43A05 - Measures on groups and semigroups, etc. 60B15 - Probability measures on groups or semigroups, Fourier transforms, factorization 60J50 - Boundary theory