Study of noncommutative analysis and free probability theory on operator algebras

我们研究了Hubert空间上的算子理论和算子代数理论及其在非交换概率论中的应用。近年来，由D.Voiculescu提出的自由概率论作为非对易概率论的一个分支得到了广泛的发展。研究了随机矩阵的大偏差理论和与自由概率论相关的自由相对熵。利用自由积方法研究了（内插）自由群因子上的自同构。进一步研究了算子平均的范数不等式，得到了算子算术-几何范数不等式的若干推广。更具体地说，（1）我们得到了一个新的关于算子单调函数和酉不变范数的矩阵的范数不等式。(2)We研究了自由相对熵将自由熵推广到两个变量，给出了它的双对数积分和矩阵近似两种定义，并利用有关随机矩阵经验特征值分布的大偏差证明了它们的一致性。(3)We用自由积的方法证明了在每个（内插）自由群因子上连续存在许多不同的非周期自同构，并阐明了它们的交叉积的结构.（4）利用二重积分变换理论和Schur乘子方法，得到了算子酉不变范数的范数不等式，加强了算术-几何平均不等式。(5)We给出了半正定矩阵的一个不等式，比较了酉不变范数的次可乘性与次可加性。(6)We利用随机矩阵逼近的方法，重新证明了Biane和Voiculescu测度的Talagrand运输费用不等式的自由概率模拟.

We studied operator theory and operator algebra theory on Hubert spaces with applications to noncommutative probability theory. In recent years, free probability theory initiated by D.Voiculescu has been extensively developed as a branch of noncommutative probability theory. We investigated large deviation theory for random matrices and free relative entropy related to free probability theory. We investigated automorphsims on (interpolated) free group factors by the method of free products. Moreover, we studied norm inequalities for means of operators and obtained a number of inequalities extending the arithmetic-geometric norm inequalities for operators. More concretely,(1)We obtained a new norm inequality for matrices involving operator monotone functions and unitarily invariant norms.(2)We investigated free relative entropy extending free entropy into two variables We gave its two definitions by double logarithmic integral and by matrix approximation, and proved their coincidence by making use of large deviation for the empirical eigenvalue distribution of relevant random matrices.(3)We showed that there are continuously many different aperiodic automorphisms on each (interpolated) free group factor by the method of free products and clarified the structure of their crossed-products.(4)Based on theory of double integral transformation and the method of Schur multipliers, we obtained norm inequalities for unitarily invariant nroms of operators, which strengthen the arithmetic-geometric mean inequality.(5)We gave an inequality for positive semidefinite matrices, comparing submultiplicativity with subadditivity for unitarily invariant norms.(6)We re-proved the free probabilistic analog of Talagrand transportation cost inequality for measures due to Biane and Voiculescu by means of random matrix approximation.