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# Linear Maps on $C^*$-Algebras Preserving the Set of Operators that are Invertible in $\mathcal{A}/\mathcal{I}$

Published:2010-08-03
Printed: Mar 2011
• Sang Og Kim,
Department of Mathematics, Hallym University, Chuncheon 200-702, Republic of Korea
• Choonkil Park,
Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea
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## Abstract

For $C^*$-algebras $\mathcal{A}$ of real rank zero, we describe linear maps $\phi$ on $\mathcal{A}$ that are surjective up to ideals $\mathcal{I}$, and $\pi(A)$ is invertible in $\mathcal{A}/\mathcal{I}$ if and only if $\pi(\phi(A))$ is invertible in $\mathcal{A}/\mathcal{I}$, where $A\in\mathcal{A}$ and $\pi:\mathcal{A}\to\mathcal{A}/\mathcal{I}$ is the quotient map. We also consider similar linear maps preserving zero products on the Calkin algebra.
 Keywords: preservers, Jordan automorphisms, invertible operators, zero products
 MSC Classifications: 47B48 - Operators on Banach algebras 47A10 - Spectrum, resolvent 46H10 - Ideals and subalgebras