La monotonia di rappresentazioni del gruppo libero

Let Γ be a free nonabelian group on finitely many generators. Let Ω be the boundary of Γ, letC(Ω) be theC ^*-algebra of continuous functions on Ω, and let λ be the natural action of Γ onC(Ω). Aboundary representation is a representation of the crossed productC ^*-algebra Γ×_λ C(Ω). Given a unitary representation π of Γ onH, aboundary realization of π is an isometric Γ-inclusion ofH into the space of a boundary representation whose image is cyclic for that boundary representation. If the Γ-inclusion is bijective, we call, the realizationperfect. We prove below that if π admits an imperfect boundary realization, then there exists a nonzero vectorv ₀∈H satisfying $$\sum\limits_{|x| = n} {|\left\langle {v,\pi (x)v_0 } \right\rangle |^2 \leqslant |v|^2 } for each v \in {\mathcal{H}} (GVB)$$ If π is irreducible and weakly contained in the regular representation, and if no suchv ₀ exists, it follows that π satisfiesmonotony: up to equivalence, there exists exactly one realization of π, and that realization is perfect.