Algebra of operators affiliated with a finite type I von Neumann algebra

TL;DR

This paper proves that the algebra of operators affiliated with a finite type I von Neumann algebra has a unique, normal center-valued trace, a property not shared by other von Neumann algebras.

Contribution

It establishes the existence and uniqueness of a normal center-valued trace for operators affiliated with finite type I von Neumann algebras, highlighting a special property of this class.

Findings

Unique center-valued trace exists for finite type I von Neumann algebras

The trace is normal in a specific sense

Such a trace cannot be constructed for other types of von Neumann algebras

Abstract

It is shown that the *-algebra of all (closed densely defined linear) operators affiliated with a finite type I von Neumann algebra admits a unique center-valued trace, which turns out to be, in a sense, normal. It is also demonstrated that for no other von Neumann algebras similar constructions can be performed.