K-Theory for Real C*-algebras via Unitary Elements with Symmetries

We prove that all eight K O groups for a real C*-algebra can be constructed from homotopy classes of unitary matrices that respect a variety of symmetries. In this manifestation of the K O groups, all eight boudary maps in the 24-term exact sequences associated to an ideal in a real C*-algebra can be computed as exponential or index maps with formulas that are nearly identical to the complex case.