Annals of Functional Analysis

A note on the C-numerical radius and the λ-Aluthge transform in finite factors

Xiaoyan Zhou, Junsheng Fang, and Shilin Wen

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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 CI. Then we obtain an equivalent condition for the situation that the C-numerical radius ωC(⋅) 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(△λ(T)) is in the weak operator closure of the set {∑i=1nziUif(T)Ui∗∣n∈N,(Ui)1≤i≤n∈U(M),∑i=1n|zi|≤1}, where f is a polynomial, △λ(T) is the λ-Aluthge transform of T, and 0≤λ≤1.

Article information

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

Dates
Received: 5 August 2017
Accepted: 16 October 2017
First available in Project Euclid: 23 April 2018

Permanent link to this document
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

Keywords
C-numerical radius finite factors weakly unitarily invariant norm λ-Aluthge transform

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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References

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