Stability of Real $C^*$-Algebras

http://dx.doi.org/10.4153/CMB-2011-019-0
Canad. Math. Bull. 54(2011), 593-606

  Published:2011-03-05
 Printed: Dec 2011

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.