A Bott periodicity proof for real graded C^ast-algebras

We give a proof of Bott periodicity for real graded $C^\ast$-algebras in terms of K- theory and E-theory. Guentner and Higson proved a similar result in the complex graded case but we extend this to cover all graded $C^\ast$-algebras. We obtain the 8-fold periodicity in E-theory by constructing two maps that are inverse to each other.