#### Date of Award

Fall 2018

#### Project Type

Dissertation

#### Program or Major

Mathematics

#### Degree Name

Doctor of Philosophy

#### First Advisor

Donald Hadwin

#### Second Advisor

Eric Nordgren

#### Third Advisor

Rita Hibschweiler

#### Abstract

John von Neumann’s 1937 characterization of unitarily invariant norms on the n × n matrices in terms of symmetric gauge norms on Cn had a huge impact on linear algebra. In 2008 his results were extended to Ifactor von Neumann algebras by J. Fang, D. Hadwin, E. Nordgren and J. Shen. There already have been many important applications. The factor von Neumann algebras are the atomic building blocks from which every von Neumann algebra can be built. My work, which includes a new proof of the Ifactor case, extends von Neumann’s results to an arbitrary finite von Neumann algebra on a separable Hilberts space. A major tool is the theory of direct integrals.

The main idea is to associate to a von Neumann algebra R a measure space (Λ, λ) and a group G (R) of invertible measure-preserving transformations on L∞ (Λ, λ). Then we show that there is a one-to-one correspondence between the unitarily invariant norms on R and the normalized G (R)-symmetric gauge norms on (Λ, λ).

#### Recommended Citation

Fan, Haihui, "UNITARILY INVARIANT NORMS ON FINITE VON NEUMANN ALGEBRAS" (2018). *Doctoral Dissertations*. 2418.

https://scholars.unh.edu/dissertation/2418