无界算子的B值自由卷积
B-Valued Free Convolution for Unbounded Operators
摘要：
考虑$\mathcal{B}$值概率空间$(\mathcal{A}, E, \mathcal{B})$，其中$\mathcal{A}$是迹vonNeumann代数.本文将算子值自由概率理论推广到关联算子代数$\tilde{\mathcal{A}}$上。对于随机变量$X \in \tilde{\mathcal{A}}^{sa}$，我们研究了Cauchy变换$G_{X}$，并证明了算子代数$(\mathcal{B} \cup \{X\})"$可以从该函数恢复.在$\mathcal{B}$是有限维的情况下，我们表明，当$X, Y \in \tilde{\mathcal{A}}^{sa}$被假定为$\mathcal{B}$ -免费的，$\mathcal{R}$ -变换定义在通用子集的预解式和满足$$ \mathcal{R}_{X} + \mathcal{R}_{Y} = \mathcal{R}_{X + Y}. $$的例子表明失败的理论为无限维$\mathcal{B}$提供。最后，我们证明了作为附属算子的柯西变换而出现的函数类是有界算子的柯西变换集的闭包。
英文摘要：
Consider the $\mathcal{B}$-valued probability space $(\mathcal{A}, E, \mathcal{B})$, where $\mathcal{A}$ is a tracial von Neumann algebra. We extend the theory of operator valued free probability to the algebra of affiliated operators $\tilde{\mathcal{A}}$. For a random variable $X \in \tilde{\mathcal{A}}^{sa}$ we study the Cauchy transform $G_{X}$ and show that the operator algebra $(\mathcal{B} \cup \{X\})"$ can be recovered from this function. In the case where $\mathcal{B}$ is finite dimensional, we show that, when $X, Y \in \tilde{\mathcal{A}}^{sa}$ are assumed to be $\mathcal{B}$-free, the $\mathcal{R}$-transforms are defined on universal subsets of the resolvent and satisfy $$ \mathcal{R}_{X} + \mathcal{R}_{Y} = \mathcal{R}_{X + Y}. $$ Examples indicating a failure of the theory for infinite dimensional $\mathcal{B}$ are provided. Lastly, we show that the class of functions that arise as the Cauchy transform of affiliated operators is, in a natural way, the closure of the set of Cauchy transforms of bounded operators.
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