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Commutativity via spectra of exponentials

Part of: Topological algebras, normed rings and algebras, Banach algebras Basic linear algebra Two-dimensional theory

Published online by Cambridge University Press:  02 November 2021

Rudi Brits ,
Francois Schulz  and
Cheick Touré
Rudi Brits
Affiliation:
Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg, South Africa e-mail: rbrits@uj.ac.za cheickkader89@hotmail.com
Francois Schulz*
Affiliation:
Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg, South Africa e-mail: rbrits@uj.ac.za cheickkader89@hotmail.com
Cheick Touré
Affiliation:
Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg, South Africa e-mail: rbrits@uj.ac.za cheickkader89@hotmail.com
*
e-mail: francoiss@uj.ac.za
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Abstract

Let A be a semisimple, unital, and complex Banach algebra. It is well known and easy to prove that A is commutative if and only $e^xe^y=e^{x+y}$ for all $x,y\in A$ . Elaborating on the spectral theory of commutativity developed by Aupetit, Zemánek, and Zemánek and Pták, we derive, in this paper, commutativity results via a spectral comparison of $e^xe^y$ and $e^{x+y}$ .

Keywords

Spectrumexponential functionBanach algebracommutativity

MSC classification

Primary: 46H05: General theory of topological algebras
Secondary: 31A05: Harmonic, subharmonic, superharmonic functions 15A16: Matrix exponential and similar functions of matrices 15A27: Commutativity
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 4 , December 2022 , pp. 815 - 824
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Copyright
© Canadian Mathematical Society, 2021

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