In this work, we propose a method to investigate the factorization property of a adjontable Markov operator between two algebraic probability spaces without using the dilation theory. Assuming the existence of an anti-unitary operator on Hilbert space related to Stinespring representations of our Markov operator, which satisfy some particular modular relations, we prove that it admits a factorization. The method is tested on the two typologies of maps which we know admits a factorization, the Markov operators between commutative probability spaces and adjontable homomorphism. Subsequently, we apply these methods to particular adjontable Markov operator between matrix algebra which fixes the diagonal.

中文翻译：

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可伴马尔可夫算子的因式分解

在这项工作中，我们提出了一种在不使用膨胀理论的情况下研究两个代数概率空间之间的可伴马尔可夫算子分解性质的方法。假设在希尔伯特空间上存在一个与马尔可夫算子的 Stinespring 表示相关的反酉算子，它满足一些特定的模关系，我们证明它允许分解。该方法在我们知道允许分解的两种映射类型上进行测试，即交换概率空间之间的马尔可夫算子和可伴同态。随后，我们将这些方法应用于固定对角线的矩阵代数之间的特定可附马尔可夫算子。