The Choquet–Deny Equation in a Banach Space

Jaworski, Wojciech ; Neufang, Matthias

Canadian journal of mathematics, Tome 59 (2007) no. 4, pp. 795-827

Résumé

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.

Détail

DOI : 10.4153/CJM-2007-034-4

Mots-clés : 22D12, 22D20, 43A05, 60B15, 60J50

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