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Let $(B,\\nu_B)$ be the associated Poisson boundary. We show that every intermediate $G$-von Neumann algebra $\\mathcal{M}$ with \\[ \\mathcal{N} \\subseteq \\mathcal{M} \\subseteq \\mathcal{N} \\,\\bar{\\otimes}\\, L^{\\infty}(B,\\nu) \\] splits as a tensor product of the form $\\mathcal{N}\\bar{\\otimes}L^{\\infty}(C,\\nu_C)$, where $(C,\\nu_C)$ is a $(G,\\mu)$-boundary. 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Let $(B,\\nu_B)$ be the associated Poisson boundary. We show that every intermediate $G$-von Neumann algebra $\\mathcal{M}$ with \\[ \\mathcal{N} \\subseteq \\mathcal{M} \\subseteq \\mathcal{N} \\,\\bar{\\otimes}\\, L^{\\infty}(B,\\nu) \\] splits as a tensor product of the form $\\mathcal{N}\\bar{\\otimes}L^{\\infty}(C,\\nu_C)$, where $(C,\\nu_C)$ is a $(G,\\mu)$-boundary. 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