The Choquet–Deny Equation in a Banach Space

Abstract

Abstract. 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 µ 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 L 2 (G). Other general properties of the Choquet-Deny equation in a Banach space are also discussed.

Harmonic operators: the dual perspective

“…A completely different type of quantization was recently carried out by W. Jaworski and the first author ( [15]). Starting point is the the result by Ghahramani [10] that there is a natural isometric representation θ of the measure algebra M(G) on B(L 2 (G)), such that for μ ∈ M(G) and φ ∈ L ∞ (G) -viewed as a multiplication operator on L 2 (G) -we have θ(μ)φ = μ * φ.…”

“…Starting point is the the result by Ghahramani [10] that there is a natural isometric representation θ of the measure algebra M(G) on B(L 2 (G)), such that for μ ∈ M(G) and φ ∈ L ∞ (G) -viewed as a multiplication operator on L 2 (G) -we have θ(μ)φ = μ * φ. Hence, the authors of [15] define an operator T ∈ B(L 2 (G)) to be μ-harmonic for a probability measure μ if θ(μ)(T) = T. The collection of all μ-harmonic operators is denoted byH μ . Like H μ , the spaceH μ is a von Neumann algebra, but with a product usually different from the one in B(L 2 (G)); in fact,H μ can be described as the crossed product of H μ with G, where the action of G on H μ is given by left translation ( [15,Proposition 6.3]).…”

“…In the present paper, we extend Chu's and Lau's notion of σ -harmonicity from VN(G) to B(L 2 (G)) in a way that parallels the extension of μ-harmonicity from L ∞ (G) to B(L 2 (G)) in [15].…”

“…The aim of this paper is to explore the connections between this setting and the two quantizations from [2] and [15]. For instance, one of our main results is thatunder very mild hypotheses which are always satisfied if G is amenable or the free group in two generators -H σ is a von Neumann subalgebra of B(L 2 (G)), namely the von Neumann subalgebra of B(L 2 (G)) generated by H σ and L ∞ (G).…”

“…First, we fix our notation and terminology and review a few basic facts about harmonic functions (see [15]) and harmonic functionals in VN(G) (as introduced and studied in [2]). We also recall results from [22] on the representation of M cb (A(G)) on B(L 2 (G)).…”