Research Article Open Access

On Numerical Ranges of Nilpotent Elements of C*-Algebra

A. Abdollahi, M. T. Heydari and M. Moosavi

Abstract

Problem statement: Let A be a C*-algebra with unit 1. For each a∈A, let V(a), v(a) v0(a) and denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. Approach: If a is a nilpotent element of A with the power of nilpotency n, i.e., an = 0 and v(a) = (n-1) v0(a). Results: We proved that V(a) = bW(An), where b is a scalar and An is the strictly upper triangular n-by-n matrix with all entries above the main diagonal equal to one. Conclusion/Recommendations: We also completely determined the numerical range of such elements, by determining the numerical range of W(An) and showed that the boundary of it does not contain any arc of circle.

Journal of Mathematics and Statistics
Volume 5 No. 4, 2009, 348-351

DOI: https://doi.org/10.3844/jmssp.2009.348.351

Submitted On: 5 September 2009 Published On: 31 December 2009

How to Cite: Abdollahi, A., Heydari, M. T. & Moosavi, M. (2009). On Numerical Ranges of Nilpotent Elements of C*-Algebra. Journal of Mathematics and Statistics, 5(4), 348-351. https://doi.org/10.3844/jmssp.2009.348.351

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Keywords

  • Numerical range
  • numerical radius
  • C*-algebra, nilpotent