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The Choquet–Deny Equation in a Banach Space
Published online by Cambridge University Press: 20 November 2018
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.
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- Research Article
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- Copyright © Canadian Mathematical Society 2007
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