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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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)UinN,(Ui)1inU(M),i=1n|zi|1}, where f is a polynomial, λ(T) is the λ-Aluthge transform of T, and 0λ1.

Article information

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • [1] A. Aluthge, On $p$-hyponormal operators for $0<p<1$, Integral Equations Operator Theory 13 (1990), no. 3, 307–315.
  • [2] J. Antezana, E. R. Pujals, and D. Stojanoff, The iterated Aluthge transforms of a matrix converge, Adv. Math. 226 (2011), no. 2, 1591–1620.
  • [3] R. Bhatia, Matrix Analysis, Grad. Texts in Math. 169, Springer, New York, 1997.
  • [4] K. Dykema and H. Schultz, Brown measure and iterates of the Aluthge transform for some operators arising from measurable actions, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6583–6593.
  • [5] K. Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951), no. 11, 760–766.
  • [6] J. Fang, D. Hadwin, E. Nordgren, and J. Shen, Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property, J. Funct. Anal. 225 (2008), no. 1, 142–183.
  • [7] I. C. Gohberg and M. G. Kreĭn, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr. 18, Amer. Math. Soc., Providence, 1969.
  • [8] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, II: Structure and Analysis for Compact Groups, Grundlehren Math. Wiss. 152, Springer, New York, 1970.
  • [9] T. Huruya, A note on $p$-hyponormal operators, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3617–3624.
  • [10] I. B. Jung, E. Ko, and C. Pearcy, Aluthge transforms of operators, Integral Equations Operator Theory 37 (2000), no. 4, 437–448.
  • [11] K. Okubo, On weakly unitarily invariant norm and the Aluthge transformation, Linear Algebra Appl. 371 (2003), 369–375.
  • [12] K. Okubo, On weakly unitarily invariant norm and the $\lambda$-Aluthge transformation for invertible operator, Linear Algebra Appl. 419 (2006), no. 1, 48–52.
  • [13] A. M. Sinclair and R. R. Smith, Finite von Neumann Algebras and Masas, London Math. Soc. Lecture Note Ser. 351, Cambridge Univ. Press, Cambridge, 2008.
  • [14] J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937), 286–300.