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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 Voiculescus 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 Voiculescus fully matricial function theory. In the end, we study relations between certain convolutions and transforms in $C^*$-operator valued probability.
We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a $C^*$-algebra $mathcal{A}$. We define an $mathcal{A}$-valued chordal Loewner chain as a subordination chain of analytic self-maps of the $mathcal{A
We show that the limit laws of random matrices, whose entries are conditionally independent operator valued random variables having equal second moments proportional to the size of the matrices, are operator valued semicircular laws. Furthermore, we
Let $(X_1,dots,X_m)$ be self-adjoint non-commutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,dots,S_m)$ be a free semicircular family. We show t
We study analogues of classical Hilbert transforms as fourier multipliers on free groups. We prove their complete boundedness on non commutative $L^p$ spaces associated with the free group von Neumann algebras for all $1<p<infty$. This implies that t
Suppose that $X_{1}$ and $X_{2}$ are two selfadjoint random variables that are freely independent over an operator algebra $mathcal{B}$. We describe the possible operator atoms of the distribution of $X_{1}+X_{2}$ and, using linearization, we determi