## Annals of Functional Analysis

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

#### Abstract

We prove that for any two elements $A$, $B$ in a factor ${\mathcal{M}}$, 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 ${\mathcal{M}}$, $f(\bigtriangleup_{\lambda}(T))$ is in the weak operator closure of the set $\{\sum_{i=1}^{n}z_{i}U_{i}f(T)U_{i}^{*}\mid n\in\mathbb{N},(U_{i})_{1\leq i\leq n}\in\mathscr{U}({\mathcal{M}}),\sum_{i=1}^{n}\vert z_{i}\vert \leq1\}$, where $f$ is a polynomial, $\bigtriangleup_{\lambda}(T)$ is the $\lambda$-Aluthge transform of $T$, and $0\leq\lambda\leq1$.

#### Article information

Source
Ann. Funct. Anal., Volume 9, Number 4 (2018), 463-473.

Dates
Accepted: 16 October 2017
First available in Project Euclid: 23 April 2018

https://projecteuclid.org/euclid.afa/1524470416

Digital Object Identifier
doi:10.1215/20088752-2017-0061

Mathematical Reviews number (MathSciNet)
MR3871907

Zentralblatt MATH identifier
07002084

Subjects
Primary: 47A12: Numerical range, numerical radius
Secondary: 46L10: General theory of von Neumann algebras

#### Citation

Zhou, Xiaoyan; Fang, Junsheng; Wen, Shilin. A note on the $C$ -numerical radius and the $\lambda$ -Aluthge transform in finite factors. Ann. Funct. Anal. 9 (2018), no. 4, 463--473. doi:10.1215/20088752-2017-0061. https://projecteuclid.org/euclid.afa/1524470416

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