距離空間内の勾配流とその応用

在这个学年里，研究人员继续把洛厄纳定理推广到几个变量上。第一个结果是一个新的算子凹单调函数的LMI表征。这将函数表征为辅助希尔伯特空间上线性矩阵不等式的唯一极值解。这一结果发表在RIMS的“算子代数的最新发展”研讨会上。后来利用这种LMI表示建立了一个精确解公式，这似乎是获得Loewner定理解析延拓部分的关键。这部分目前还在进行中。作为副产物，建立了矩阵凸集理论的无限维版本和非交换哈恩-巴拿赫定理作为关键工具。

In this academic year the researcher continued the work on extending Loewner’s theorem to several variables. The first result is an new LMI characterization of operator concave and monotone functions. This characterizes the functions as unique extremal solutions of linear matrix inequalities over some auxiliary Hilbert space. This result was presented at the ’Recent developments in operator algebras’ workshop in RIMS.Later using this LMI representation an exact solution formula was established which seems to be the key to obtain the analytic continuation part of Loewner’s theorem. This part is still being carried out at the moment. As a by-product an infinite dimensional version of the theory of matrix convex sets and non-commutative Hahn-Banach theorems were established as key tools.