关于代数值 R-对角元素（附勘误）
On algebra-valued R-diagonal elements (with erratum)
AI总结：
本文研究了代数值 R-对角元素的等价条件、矩与累积量的形式幂级数关系及其酉元与自伴元的乘积分解，并应用其描述了拟幂零 DT-算子的极分解。
摘要：
对于代数值 *-非交换概率空间中的元素，本文证明了代数值 R-对角性（由 Sniady 和 Speicher 引入的概念）的等价条件。证明了涉及此类 R-对角元素的矩和累积量的形式幂级数关系。研究了代数值 R-对角元素分解为酉元乘自伴元形式的乘积；以累积量表示，证明了酉部分和自伴部分 *-自由性的充分条件，并给出了 *-自由性失效的迹示例。考虑了代数值圆元素的特例；作为应用，描述了拟幂零 DT-算子的极分解。附有勘误。（上一段最后一句话由此作废。）
英文摘要：
For an element in an algebra-valued *-noncommutative probability space, equivalent conditions for algebra-valued R-diagonality (a notion introduced by Sniady and Speicher) are proved. Formal power series relations involving the moments and cumulants of such R-diagonal elements are proved. Decompositions of algebra-valued R-diagonal elements into products of the form unitary times self-adjoint are investigated; sufficient conditions, in terms of cumulants, for *-freeness of the unitary and the self-adjoint part are proved, and a tracial example is given where *-freeness fails. The particular case of algebra-valued circular elements is considered; as an application, the polar decompostion of the quasinilpotent DT-operator is described. An erratum is appended. (The last sentence in the above paragraph is thereby nullified.)
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