Numerical Ranges in II$_1$ Factors

TL;DR

This paper extends the concept of $C$-numerical range from matrices to operators in tracial von Neumann algebras, providing explicit descriptions for various operators and classes.

Contribution

It introduces a generalized $C$-numerical range for operators in von Neumann algebras, expanding the scope beyond matrices and offering explicit characterizations.

Findings

$C$-numerical range is a compact, convex subset of $ ext{C}$ for operators.

Explicit descriptions of $C$-numerical ranges for several operators.

Generalization from matrices to operators in von Neumann algebras.

Abstract

In this paper, we generalize the notion of the $C$-numerical range of a matrix to operators in arbitrary tracial von Neumann algebras. For each self-adjoint operator $C$, the $C$-numerical range of such an operator is defined; it is a compact, convex subset of $\mathbb{C}$. We explicitly describe the $C$-numerical ranges of several operators and classes of operators.