We continue our study of the outer Pinsker factor for probability measure-preserving actions of sofic groups. Using the notion of local and doubly empirical convergence developed by Austin we prove that in many cases the outer Pinsker factor of a product action is the product of the outer Pinsker factors. Our results are parallel to those of Seward for Rokhlin entropy. We use these Pinsker product formulas to show that if $X$ is a compact group, and $G$ is a sofic group with $G\curvearrowright X$ by automorphisms, then the outer Pinsker factor of $G\curvearrowright (X,m_{X})$ is given as a quotient by a $G$-invariant, closed, normal subgroup of $X$. We use our results to show that if $G$ is sofic and $f\in M_{n}(\mathbb Z(G))$ is invertible as a convolution operator $\ell^{2}(G)^{\oplus n}\to \ell^{2}(G)^{\oplus n},$ then the action of $G$ on the Pontryagin dual of $\mathbb Z(G)^{\oplus n}/\mathbb Z(G)^{\oplus n}f$ has completely positive measure-theoretic entropy with respect to the Haar measure.

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相对熵和苏菲克群的平斯克乘积公式

我们继续研究外层 Pinsker 因子，用于保持 sofic 群的概率度量保留行为。使用由 Austin 开发的局部和双重经验收敛的概念，我们证明在许多情况下，产品动作的外部 Pinsker 因子是外部 Pinsker 因子的乘积。对于 Rokhlin 熵，我们的结果与 Seward 的结果相似。我们使用这些 Pinsker 乘积公式来证明，如果 $X$ 是一个紧群，并且 $G$ 是一个具有 $G\curvearrowright X$ 的自同构的苏菲群，那么 $G\curvearrowright (X,m_ {X})$ 由 $X$ 的 $G$ 不变的封闭正规子群的商给出。我们使用我们的结果来证明，如果 $G$ 是软的并且 $f\in M_{n}(\mathbb Z(G))$ 作为卷积算子是可逆的 $\ell^{2}(G)^{\ oplus n}\to \ell^{2}(G)^{\oplus n},