Twisted K-theory, Real A-bundles and Grothendieck-Witt groups

We introduce a general framework to unify several variants of twisted topological K-theory. We focus on the role of finite dimensional real simple algebras with a product-preserving involution, showing that Grothendieck-Witt groups provide interesting examples of twisted K-theory. These groups are linked with the classification of algebraic vector bundles on real algebraic varieties.