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The Choquet–Deny Equation in a Banach Space

Published online by Cambridge University Press:  20 November 2018

Wojciech Jaworski  and
Matthias Neufang
Wojciech Jaworski
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6 email: wjaworsk@math.carleton.ca, mneufang@math.carleton.ca
Matthias Neufang
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6 email: wjaworsk@math.carleton.ca, mneufang@math.carleton.ca
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Abstract

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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.

Keywords

22D1222D2043A0560B1560J50
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 4 , 01 August 2007 , pp. 795 - 827
Copyright
Copyright © Canadian Mathematical Society 2007

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