2018/09/16 by Weihua Liu, Liu, Weihua
Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Mathematical Inequalities and Applications #Operator Algebras (math.OA) #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1809.05789
openalex publication_date 2018/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a class of independence relations, which include free, Boolean and monotone independence, in operator valued probability. We show that this class of independence relations have a matricial extension property so that we can easily study their associated convolutions via Voiculescu's fully matricial function theory. Based the matricial extension property, we show that many results can be generalized to multi-variable cases. Besides free, Boolean and monotone independence convolutions, we will focus on two important convolutions, which are orthogonal and subordination additive convolutions. We show that the operator-valued subordination functions, which come from the free additive convolutions or the operator-valued free convolution powers, are reciprocal Cauchy transforms of operator-valued random variables which are uniquely determined up to Voiculescu's fully matricial function theory. In the end, we study relations between certain convolutions and transforms in C^*-operator valued probability.