# A note on the $C$-numerical radius and the $\lambda$-Aluthge transform in finite factors

## TL;DR

This paper investigates properties of the $C$-numerical radius and the $ ext{Aluthge}$ transform in finite factors, establishing conditions for invariance, inequalities, and operator approximations within the framework of operator algebras.

## Contribution

It provides new characterizations of the $C$-numerical radius, inequalities related to it, and an approximation result for the $ ext{Aluthge}$ transform in finite factors.

## Key findings

- Characterization of elements commuting with all unitary conjugates
- Equivalent condition for $C$-numerical radius invariance
- Approximation of $ ext{Aluthge}$ transform via convex combinations of unitarily conjugated operators

## Abstract

We prove that for any two elements $A$, $B$ in a factor $M$, if $B$ commutes with all the unitary conjugates of $A$, then either $A$ or $B$ is in $\mathbb{C}I$. Then we obtain an equivalent condition for the situation that the $C$-numerical radius $\omega_{C}(\cdot)$ is a weakly unitarily invariant norm on finite factors and we also prove some inequalities on the $C$-numerical radius on finite factors. As an application, we show that for an invertible operator $T$ in a finite factor $M$, $f(\bigtriangleup_{\lambda}(T))$ is in the weak operator closure of the set $\{\sum_{i=1}^{n}z_{i}U_{i}f(T)U_{i}^{*}|n\in\mathbb{N},(U_{i})_{1\leq i\leq n}\in \mathscr{U}(M),\sum_{i=1}^{n}|z_{i}|\leq 1\}$, where $f$ is a polynomial, $\bigtriangleup_{\lambda}(T)$ is the $\lambda$-Aluthge transform of $T$ and $0\leq\lambda \leq 1$.