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Journal of Operator Theory

Volume 83, Issue 2, Spring 2020  pp. 299-319.

Jordan maps and pseudospectrum in $C^*$-algebras

Authors:  A. Bourhim (1), J. Mashreghi (2)
Author institution:(1) Syracuse University, Department of Mathematics,\break 215 Carnegie Building, Syracuse, NY 13244, U.S.A.
(2) D\'epartement de math\'ematiques et de statistique, Universit\'e Laval, Qu\'ebec, QC, Canada, G1K 0A6

Summary: We show that a surjective map $\varphi$ between two unital $C^*$-algebras $\mathcal{A}$ and $\mathcal{B}$, with $\varphi(0)=0$, satisfies \[ \Lambda_\varepsilon(\varphi(x_1)-\varphi(x_2))=\Lambda_\varepsilon(x_1-x_2),\quad(x_1,~x_2\in\mathcal{A}), \] where $\Lambda_\varepsilon$ denotes the $\varepsilon$-pseudospectrum, if and only if $\varphi$ is a Jordan $*$-isomor\-phism. We also characterize maps $\varphi_1$ and $\varphi_2$ from $\mathcal{A}$ onto $\mathcal{B}$ that satisfy \[ \Lambda_\varepsilon (\varphi_1(x_1)\varphi_2(x_2) )=\Lambda_\varepsilon (x_1x_2 ), \quad(x_1,~x_2\in\mathcal{A}), \] or some other binary operations, in terms of Jordan $*$-isomorphisms. The main results imply several other characterizations of Jordan $*$-isomorphisms which are interesting in their own right.

Keywords: pseudospectrum, spectrum, nonlinear preservers, Jordan homomorphism

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