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Commutativity via spectra of exponentials

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

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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References

Aupetit, B., Caractérisation spectrale des algèbres de Banach commutatives. Pacific J. Math. 63(1976), 23–35. http://doi.org/10.2140/pjm.1976.63.23 CrossRefGoogle Scholar
Aupetit, B., A primer on spectral theory . In: Universitext (1979), Springer-Verlag, Berlin, 1991. http://doi.org/10.1007/978-1-4612-3048-9 Google Scholar
Blackadar, B., Operator algebras: theory of ${C}^{\star }$ -algebras and von Neumann algebras. In: Encyclopaedia of Mathematical Sciences, Springer, Berlin, 2006. http://doi.org/10.1007/3-540-28517-2 Google Scholar
Brešar, M. and Šemrl, P., Derivations mapping into the socle. Math. Proc. Camb. Phil. Soc. 120(1996), 339–346. http://doi.org/10.1017/s0305004100074892 CrossRefGoogle Scholar
Duncan, J. and Tullo, A., Finite dimensionality, nilpotents and quasinilpotents in Banach algebras. Proc. Edinburgh Math. Soc. 19(1974), no. 2, 45–49. http://doi.org/10.1017/s0013091500015352 CrossRefGoogle Scholar
Fréchet, M., Les Solutions non commutables de L’équation matricielle ${e}^{x+y}={e}^x{e}^y$ . Rend. Circ. Math. Palermo 2(1952), 11–27. http://doi.org/10.1007/bf02843715 CrossRefGoogle Scholar
Hirschfeld, R. and Johnson, B., Spectral characterization of finite dimensional algebras. Nedrl. Akad. Wet. Proc. Ser A 75(1972), 19–23. http://doi.org/10.1016/1385-7258(72)90023-6 Google Scholar
Huff, C., On Pairs of matrices (of order two) $A,B$ satisfying the condition ${e}^{A+B}={e}^A{e}^B\ne {e}^B{e}^A$ . Rend. Circ. Math. Palermo 2(1953), 326–330. http://doi.org/10.1007/bf02843709 CrossRefGoogle Scholar
Kakar, A., Non-commuting solutions of the matrix equation $\mathit{\exp}(X+Y)=\mathit{\exp}(X)\mathit{\exp}(Y)$ . Rend. Circ. Math. Palermo 2(1953), 331–345. http://doi.org/10.1007/bf02843710 CrossRefGoogle Scholar
Morinaga, K. and Nôno, T., On the non-commutative solutions of the exponential equation ${e}^{x+y}={e}^x{e}^y$ . J. Sci. Hiroshima Univ. 17(1954), 345–358. http://doi.org/10.32917/hmj/1557281146 Google Scholar
Morinaga, K. and Nôno, T., On the non-commutative solutions of the exponential equation ${e}^{x+y}={e}^x{e}^y$ II. J. Sci. Hiroshima Univ. 18(1954), 137–178. http://doi.org/10.32917/hmj/1556935270 Google Scholar
Pták, V. and Zemánek, J., On uniform continuity of the spectral radius in Banach algebras. Manuscripta Math. 20(1977), no. 2, 177–189. http://doi.org/10.1007/bf01170724 CrossRefGoogle Scholar
Rudin, W., Functional analysis . In: International series in pure and applied mathematics, McGraw-Hill, New York, 1991.Google Scholar
de Seguins Pazzis, C., On commuting matrices and exponentials. Proc. Amer. Math. Soc. 141(2013), no. 3, 763–774. http://doi.org/10.1090/s0002-9939-2012-11396-6 CrossRefGoogle Scholar
Vesentini, E., On the subharmonicity of the spectral radius. Boll. Un. Mat. Ital. 4(1968), no. 1, 427–429.Google Scholar
Zemánek, J., Spectral radius characterizations of commutativity in Banach algebras. Studia Math. 61(1977), no. 3, 257–268. http://doi.org/10.4064/sm-61-3-257-268 CrossRefGoogle Scholar