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# Stability of Real $C^*$-Algebras

Published:2011-03-05
Printed: Dec 2011
• Jeffrey L. Boersema,
Department of Mathematics, Seattle University, Seattle, WA 98122, U.S.A.
• Efren Ruiz,
Department of Mathematics, University of Hawaii Hilo, Hilo, Hawaii 96720, U.S.A.
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## Abstract

We will give a characterization of stable real $C^*$-algebras analogous to the one given for complex $C^*$-algebras by Hjelmborg and Rørdam. Using this result, we will prove that any real $C^*$-algebra satisfying the corona factorization property is stable if and only if its complexification is stable. Real $C^*$-algebras satisfying the corona factorization property include AF-algebras and purely infinite $C^*$-algebras. We will also provide an example of a simple unstable $C^*$-algebra, the complexification of which is stable.
 Keywords: stability, real C*-algebras
 MSC Classifications: 46L05 - General theory of $C^*$-algebras