Study on von Neumann algebras, free probability and non-commutative function spaces

2015 Fiscal Year Final Research Report

• Project/Area Number: 24540214
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Global analysis
• Research Institution: Kyushu University
• Principal Investigator: UEDA Yoshimichi 九州大学, 数理(科)学研究科(研究院), 准教授 (00314724)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: フォンノイマン環 / 自由積 / 軌道自由エントロピー / 前共役

Outline of Final Research Achievements

I clarified the structure of free product von Neumann algebras from various viewpoints. Especially, I identified the resulting free product von Neumann algebras with a certain explicit one under some conditions. Also, we computed several invariants for free product von Neumann algebras in the general setup, and obtained a result on the positional rigidity for subalgebras as well as a uniqueness result for the free product decomposition. I studied the orbital free entropy. In particular, we made some nontrivial computation of it. We also introduced the concept of pressure related to the orbital free entropy motivated by statistical physics and investigated its relation to random matrices. Moreover, I generalized the previous definition of orbital free entropy and discussed its general properties. Finally, we studied von Neumann algebra preduals and investigated a strong property, which is rather opposite to uniform convexity, for them.

Free Research Field

作用素環論